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8/20/2019 Libro Polochic Matemática 1er. Sem
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1.³ M`hjbnngr`tf - Irupf [fnfhjbh[rbegr sgegstrg - BIG]0;
E`tgeàtbh`
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1.³ M`hjbnngr`tf ‘ Irupf [fnfhjbh[rbegr sgegstrg ‘ BIG]
E`tgeàtbh` 0;
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E`tgeàtbh` 0;
[rbegr sgegstrg
]gsgrv`afs tfafs nfs agrghjfs. Puga` rbiurfs`egltg prfjbmba` n` rgprfauhhbøl tft`n f p`rhb`nag gstg e`tgrb`n gauh`tbvf, pfr hu`nqubgr egabf f prfhgabebgltf, sbl n` `utfrbz`hbøl agn BlstbtutfIu`tge`ntghf ag Gauh`hbøl ]`abfdølbh`, BIG]. Xgiól `rtåhunf 1= ag n` Hflstbtuhbøl [fnåtbh` agIu`tge`n` qug sg rgdgrg n` utfrå`.
© Blstbtutf Iu`tge`ntghf ag Gauh`hbøl ]`abfdølbh`, BIG].Gs ul` fmr` prfauhba` pfr gn Agp`rt`egltf ag ]ga`hhbøl y Absgðf, p`r` gn
Blstbtutf Iu`tge`ntghf ag Gauh`hbøl ]`abfdølbh`, BIG].
00 vglba` 0:-17, Hbua`a Lugv`, zfl` =Hbua`a ag Iu`tge`n`.
[MZ3 =10= 4444D`x3 =10= 49;1
Hfrrgf gnghtrølbhf3 bigrKbigr.gau.it[àibl` wgm3 www.bigr.gau.it
Gabhbøl =;04Beprgsf gl BIG] t`nngrgs iràdhfs
Høabif3 0000;;1=;0
BXML 29:22=240170=
8/20/2019 Libro Polochic Matemática 1er. Sem
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Ålabhg
Ålabhg ............................................................................................................................................................... B
¡Mbglvglba` y mbglvglbaf! .................................................................................................................... 0
Xge`l` 0Nøibh` B .................... ...................... ..................... ..................... ..................... ..................... ..................... ................. 08
¡[`r` hfeglz`r! Gn huepng`ðfs ag @ngc`lar` ..................... ..................... ..................... ..................... ............... 01
Gn eulaf ag n` e`tgeàtbh`
0. Nøibh` prfpfsbhbfl`n .................... ..................... ..................... ..................... ..................... ..................... .......... 07
=. [rfpfsbhbflgs sbepngs ..................... ..................... ..................... ..................... ..................... ..................... ...... 07
=.0 Lgi`hbøl nøibh` (~) ................... ..................... ..................... ..................... ..................... ..................... .. 09
8. Nøibh` ag prgabh`afs.................... ..................... ..................... ..................... ..................... ..................... ......... 0: 8.0 Hu`ltbh`afr ulbvgrs`n (∋) .................... ..................... ..................... ..................... ..................... ........ 0:
8.= Hu`ltbh`afr gxbstglhb`n (∈) ..................... ..................... ..................... ..................... ..................... ..... 0:
]gsuegl ................... ..................... ..................... ..................... ..................... ..................... ..................... ...................... .... 02
@utfhfltrfn .................... ..................... ..................... ..................... ..................... ..................... ..................... .................. =;
@ibnba`a ag hànhunf eglt`n ................... ..................... ..................... ..................... ..................... ..................... ......... ==
]`zfl`ebgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................. =8
Ags`rrfnng lugv`s j`mbnba`ags ................... ..................... ..................... ..................... ..................... ..................... .. =1
Xge`l` =Nøibh` BB ..................... ....................... ..................... ..................... ..................... ..................... ..................... ............. =7
¡[`r` hfeglz`r! Nfs hflghtbvfs ................... ..................... ..................... ..................... ..................... ..................... .. =4
Gn eulaf ag n` e`tgeàtbh`
0. [rfpfsbhbflgs hfepugst`s .................... ..................... ..................... ..................... ..................... ................... =9
=. Hflghtbvfs nøibhfs ..................... ..................... ..................... ..................... ..................... ..................... ............. =9
=.0 Hflculhbøl (∩) ..................... ..................... ..................... ..................... ..................... ..................... .......... =9
=.= Absyulhbøl (∪) ................... ..................... ..................... ..................... ..................... ..................... ............. =:
=.8 Hflabhbfl`n (↝) .................. ..................... ..................... ..................... ..................... ..................... .......... =:
=.1 Mbhflabhbfl`n (↑) ................... ..................... ..................... ..................... ..................... ..................... ...... =:8. ]gin`s ag nfs hflghtbvfs nøibhfs .................... ..................... ..................... ..................... ..................... ........ =2
8.0 ]gin` ag n` hflculhbøl .................... ..................... ..................... ..................... ..................... ................ =2
8.= ]gin` ag n` absyulhbøl ..................... ..................... ..................... ..................... ..................... ................ 8;
8.8 ]gin` agn hflabhbfl`n .................... ..................... ..................... ..................... ..................... ................... 80
8.1 ]gin` agn mbhflabhbfl`n ................... ..................... ..................... ..................... ..................... ................ 8=
]gsuegl ................... ..................... ..................... ...................... ..................... ..................... ..................... ..................... .... 87
Blvgstbiug gl n` rga ................... ..................... ..................... ..................... ..................... ..................... ..................... ... 87
BE`tgeàtbh` ∝ Ålabhg
8/20/2019 Libro Polochic Matemática 1er. Sem
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@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 84
@ibnba`a ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ 8:
]`zfl`ebgltf nøibhf ................... ..................... ..................... ..................... ..................... ..................... ................... 82
Ags`rrfnng lugv`s j`mbnba`ags .................... ..................... ..................... ..................... ..................... ..................... . 1;
Xge`l` 8Lóegrfs rg`ngs .................. ..................... ..................... ..................... ..................... ..................... ........ 10
¡[`r` hfeglz`r! Nfs pbt`iørbhfs y nfs lóegrfs brr`hbfl`ngs .................. ..................... ..................... ............ 1=
Gn eulaf ag n` e`tgeàtbh`
0. Lóegrfs rg`ngs (]) ..................... ..................... ..................... ..................... ..................... ..................... ............ 18
0.0 H`r`htgråstbh s ag nfs lóegrfs rg`ngs ..................... ..................... ..................... ..................... ...... 18
0.= T`nfr `msfnutf ag ul lóegrf rg`n ................... ..................... ..................... ..................... ................ 11
=. Fpgr`hbflgs hfembl`a`s .................... ..................... ..................... ..................... ..................... ..................... 17
8. [rfpbga`ags ag nfs lóegrfs rg`ngs .................. ..................... ..................... ..................... ..................... ... 14
1. [ftglhb hbøl ag lóegrfs rg`ngs ..................... ..................... ..................... ..................... ..................... ....... 1:
1.0 [rfpbga`ags ag nfs gxpflgltgs p`r` lóegrfs rg`ngs ..................... ..................... .................. 1:
7. ]`abh`hbøl ag lóegrfs rg`ngs .................. ..................... ..................... ..................... ..................... .............. 7;
7.0 [rfpbga`ags ag nfs r`abh`ngs p`r` lóegrfs rg`ngs .................... ..................... ..................... .... 7;
]gsuegl ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 70
@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 7=
@ibnba`a ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ 71
]`zfl`ebgltf nøibhf ................... ..................... ..................... ..................... ..................... ..................... ................... 77
Ags`rrfnng lugv`s j`mbnba`ags .................... ...................... ..................... ..................... ..................... ..................... 74
Xge`l` 1 Lóegrfs hfepngcfs ..................... ..................... ..................... ..................... ..................... .......... 79
¡[`r` hfeglz`r! ]`åhgs p`rgs g bep`rgs .................. ..................... ..................... ..................... ..................... ....... 7:
Gn eulaf ag n` e`tgeàtbh`
0. Lóegrfs hfepngcfs (H) .................... ..................... ..................... ..................... ..................... ..................... .... 72
0.0 [ftglhb`hbøl ag b .................. ..................... ..................... ..................... ..................... ..................... ........ 40
0.= Dfre` mblfeb`n ag ul lóegrf hfepngcf ................... ..................... ..................... ..................... .. 40
=. Xue` y rgst` ag lóegrfs hfepngcfs ................... ..................... ..................... ..................... ..................... 4=
8. Euntbpnbh`hbøl ag lóegrfs hfepngcfs ..................... ..................... ..................... ..................... ................. 48
1. Abvbsbøl ag lóegrfs hfepngcfs .................. ..................... ..................... ..................... ..................... ........... 47
1.0 ]`hbfl`nbz`hbøl agn aglfebl`afr egab`ltg gn hflcui`af .................. ..................... ........... 47
]gsuegl ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 49
Blvgstbiug gl n` rga .................... ..................... ..................... ..................... ..................... ..................... ..................... .. 49
@utfhfltrfn ..................... ..................... ..................... ..................... ..................... ..................... ..................... ................. 4:
@ibnba`a ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ 9;
]`zfl`ebgltf nøibhf ................... ..................... ..................... ..................... ..................... ..................... ................... 90
Ags`rrfnng lugv`s j`mbnba`ags .................... ..................... ..................... ..................... ..................... ..................... . 9=
BBBIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
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Xge`l` 7Xue` y rgst` ag e`trbhgs .................. ..................... ..................... ..................... ........... 98
¡[`r` hfeglz`r! Xuafou3 ul cugif ag bliglbf .................... ..................... ..................... ..................... ............... 91
Gn eulaf ag n` e`tgeàtbh`
0. E`trbhgs .................. ..................... ..................... ..................... ..................... ..................... ...................... .............. 97
=. Qbpfs ag e`trbhgs ................... ..................... ..................... ..................... ..................... ..................... ................. 94
=.0 E trbz tr`lspugst` ..................... ..................... ..................... ..................... ..................... ..................... .. 94
=.= E`trbz sbeâtrbh` ................... ..................... ..................... ..................... ..................... ..................... .......... 94
=.8 E`trbz bagltba`a ................... ..................... ..................... ..................... ..................... ..................... ......... 94
=.1 E`trbz blvgrtbmng f lf sbliun`r .................... ..................... ..................... ..................... ..................... . 94
=.7 E trbz sbliun`r .................. ..................... ..................... ..................... ..................... ..................... ............. 94
8. Fpgr`hbflgs gltrg e`trbhgs .................. ..................... ..................... ..................... ..................... ................... 99
8.0 Xue` y rgst` ................... ..................... ..................... ..................... ..................... ..................... ................ 99
8.= [rfpbga`ags ag n` sue` ag e`trbhgs ................... ..................... ..................... ..................... ........... 92
]gsuegl .................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... .... :0
Blvgstbiug gl n` rga ................... ..................... ..................... ..................... ..................... ..................... ..................... ... :0
@utfhfltrfn .................... ..................... ..................... ..................... ..................... ..................... ..................... .................. :=
@ibnba`a ag hànhunf eglt`n ................... ..................... ..................... ..................... ..................... ..................... ......... :1
]`zfl`ebgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................. :7
Ags`rrfnng lugv`s j`mbnba`ags ................... ..................... ..................... ..................... ..................... ..................... .. :4
Xge`l` 4Euntbpnbh`hbøl ag e`trbhgs .................... ..................... ..................... ..................... ....... :9
¡[`r` hfeglz`r! Gn `cgargz y n` ngygla` ag Xbs` ..................... ..................... ..................... ..................... ........... ::
Gn eulaf ag n` e`tgeàtbh`
0. Euntbpnbh`hbøl ag e`trbhgs .................... ..................... ..................... ..................... ..................... ................... :2
0.0 [rfauhtf ag ul lóegrf rg`n pfr ul` e`trbz ................... ..................... ..................... ................. :2
0.= [rfauhtf ag e`trbhgs .................... ..................... ..................... ..................... ..................... ................... 20
]gsuegl ................... ...................... ..................... ..................... ..................... ..................... ..................... ..................... .... 27
Blvgstbiug gl n` rga ................... ..................... ..................... ..................... ..................... ..................... ..................... ... 27
@utfhfltrfn .................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 24
@ibnba`a ag hànhunf eglt`n ................... ..................... ..................... ..................... ..................... ..................... ......... 2:
]`zfl`ebgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................. 22Ags`rrfnng lugv`s j`mbnba`ags ................... ..................... ..................... ..................... ..................... ..................... .. 0;;
BBBE`tgeàtbh` ∝ Ålabhg
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Xge`l` 9Xuhgsbflgs e`tgeàtbh`s ................... ..................... ..................... ..................... ............. 0;0
¡[`r` hfeglz`r! N` suhgsbøl ag Dbmfl`hhb ................... ...................... ..................... ..................... ..................... .. 0;=
Gn eulaf ag n` e`tgeàtbh`
0. Xuhgsbflgs e`tgeàtbh s ..................... ..................... ..................... ..................... ..................... ..................... . 0;8 0.0 Xuhgsbflgs `rbteâtbh`s .................. ..................... ..................... ..................... ..................... .................. 0;8
0.= Xuhgsbflgs igfeâtrbh`s.................................. ..................... ..................... ..................... ..................... 0;9
]gsuegl ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 000
Blvgstbiug gl n` rga .................... ..................... ..................... ..................... ..................... ..................... ..................... .. 000
@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 00=
@ibnba`a ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ 001
]`zfl`ebgltf nøibhf ................... ..................... ..................... ..................... ..................... ..................... ................... 007
Ags`rrfnng lugv`s j`mbnba`ags .................... ..................... ..................... ..................... ..................... ..................... . 004
Xge`l` :]gp`sf3 sge`l`s 0 ` 9 .................... ..................... ..................... ...................... ................ 009
Gn eulaf ag n` e`tgeàtbh`
Nøibh` B .................. ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 002
Nøibh` BB ..................... ..................... ..................... ..................... ..................... ..................... ..................... .................... 0=0
Lóegrfs rg`ngs ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 0=8
Lóegrfs hfepngcfs ................... ..................... ..................... ..................... ..................... ..................... ................... 0=7
Xue` y rgst` ag e`trbhgs .................... ..................... ..................... ..................... ..................... ..................... ........ 0=9
Euntbpnbh`hbøl ag e`trbhgs .................. ..................... ..................... ..................... ..................... ..................... ........ 0=2
Xuhgsbflgs e`tgeàtbh s .................. ..................... ..................... ..................... ..................... ..................... ............ 080
@ibnba`a ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ 088
Frbglt`hbflgs sfmrg n` prugm` p`rhb`n .............................................................................................................. 081
Xge`l` 2[ftglhb`hbøl ag gxprgsbflgs `nigmr`bh`s .................................. 087
¡[`r` hfeglz`r! ]gp`sf ag n` pftglhb`hbøl ag lóegrfs rg`ngs ................... ..................... ..................... .... 084
Gn eulaf ag n` e`tgeàtbh`
0. ]gin`s ag n` pftglhb`hbøl ................... ..................... ..................... ..................... ..................... ..................... . 089 0.0 ]gin` agn gxpflgltg hgrf........................... ..................... ..................... ..................... ..................... .... 089
0.= ]gin` agn prfauhtf ag pftglhb`s hfl biu`n m`sg .................... ..................... ..................... ........ 089
0.8 ]gin` agn hfhbgltg ag pftglhb`s ag biu`n m`sg .................... ..................... ..................... ............ 08:
0.1 ]gin` ag ul` pftglhb` gngv`a` ftr` pftglhb` ..................... ..................... ..................... ........ 08:
0.7 ]gin` ag n` pftglhb` ag ul` dr`hhbøl `nigmr`bh` ................... ..................... ..................... ........ 08:
0.4 ]gin` agn gxpflgltg lgi`tbvf ................... ..................... ..................... ..................... ..................... ... 01;
=. Xbepnbh`hbøl ag gxprgsbflgs nigmr`bh`s ..................... ..................... ..................... ..................... ......... 010
BTBIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
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]gsuegl ................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..... 018
Blvgstbiug gl n` rga... ................... ..................... ..................... ..................... ..................... ..................... ..................... 018
@utfhfltrfn .................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 011
@ibnba`a ag hànhunf eglt`n ................... ..................... ..................... ..................... ..................... ..................... ......... 01:
]`zfl`ebgltf nøibhf ................... ..................... ..................... ..................... ..................... ..................... ..................... 012
Ags`rrfnng lugv`s j`mbnba`ags ................... ..................... ..................... ..................... ..................... ..................... .. 07;
Xge`l` 0;]`abh`hbøl ag gxprgsbflgs `nigmr`bh`s ......................................... 070
¡[`r` hfeglz`r! Biu`n lóegrf ag dbn`s y hfnuel`s .................. ..................... ..................... ..................... ....... 07=
Gn eulaf ag n` e`tgeàtbh`
0. ]gin`s ag nfs r`abh`ngs .................... ..................... ..................... ..................... ..................... ..................... ...... 078
=. Hflvgrsbøl ag ul` r`åz ` ul` pftglhb` dr`hhbfl`rb` ..................... ..................... ..................... .......... 071
8. ]`abh`ngs sgegc`ltgs ................... ..................... ..................... ..................... ..................... ..................... .......... 077
8.0 Xue` y rgst` ag r`abh`ngs sgegc`ltgs ..................... ..................... ..................... ..................... ....... 077 8.= Euntbpnbh`hbøl ag r`abh`ngs .................... ..................... ..................... ..................... ..................... ........ 074
8.8 Abvbsbøl ag r`abh`ngs .................. ..................... ..................... ..................... ..................... ..................... .. 07:
8.1 Gnbebl`hbøl ag d`htfrgs gl nfs r`abh`ngs .................... ..................... ..................... ..................... .... 072
1. ]`hbfl`nbz`hbøl ag aglfebl`afrgs .................... ..................... ..................... ..................... ..................... . 04;
]gsuegl ................... ..................... ..................... ..................... ..................... ..................... ..................... ...................... .... 04=
Blvgstbiug gl n` rga... ................... ..................... ..................... ..................... ..................... ..................... ..................... 04=
@utfhfltrfn .................... ..................... ..................... ..................... ..................... ..................... ..................... .................. 048
@ibnba`a ag hànhunf eglt`n ................... ..................... ..................... ..................... ..................... ..................... ......... 044
]`zfl`ebgltf nøibhf ................... ..................... ..................... ..................... ..................... ..................... ..................... 049Ags`rrfnng lugv`s j`mbnba`ags ................... ..................... ..................... ..................... ..................... ..................... .. 04:
Xge`l` 00Fpgr`hbflgs pfnblfeb`ngs .................. ..................... ..................... ..................... ........... 042
¡[`r` hfeglz`r! N` bepfrt`lhb` agn ànigmr` .................... ..................... ..................... ..................... .................. 09;
Gn eulaf ag n` e`tgeàtbh`
0. [fnblfebfs ..................... ..................... ..................... ..................... ..................... ..................... ..................... ....... 090
0.0 Hn`sbh`hbøl ag nfs pfnblfebfs ................... ..................... ..................... ..................... ..................... . 090
=. Xue` y rgst` ag pfnblfebfs .................... ..................... ..................... ..................... ..................... ................ 09=
8. Euntbpnbh`hbøl ag pfnblfebfs .................. ..................... ..................... ..................... ..................... ................ 091
8.0 Euntbpnbh`hbøl ag ul lóegrf pfr ul pfnblfebf ................... ..................... ..................... .......... 091
8.= Euntbpnbh`hbøl ag ul eflfebf pfr ul pfnblfebf ................... ..................... ..................... ...... 091
8.8 Euntbpnbh`hbøl gltrg pfnblfebfs .................... ...................... ..................... ..................... .................. 097
1. Abvbsbøl ag ul pfnblfebf gltrg ul eflfebf ................... ..................... ..................... ..................... .... 094
TE`tgeàtbh` ∝ Ålabhg
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]gsuegl ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 099
Blvgstbiug gl n` rga... ..................... ..................... ..................... ..................... ..................... ..................... ................... 099
@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 09:
@ibnba`a ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ 0:;
]`zfl`ebgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................... 0:0
Ags`rrfnng lugv`s j`mbnba`ags .................... ..................... ..................... ..................... ..................... ..................... . 0:=
Xge`l` 0=[rfauhtfs lft`mngs .................. ..................... ..................... ..................... ..................... .............. 0:8
¡[`r` hfeglz`r! @n-Cw`rbzeb, gn hrg`afr agn ànigmr` .................... ..................... ..................... ..................... .. 0:1
Gn eulaf ag n` e`tgeàtbh`
0. [rfauhtfs lft`mngs ..................... ..................... ...................... ..................... ..................... ..................... ........... 0:7
0.0 Hu`ar`af ag n` sue` ag ul mblfebf (` + m)= .................... ..................... ..................... ............... 0:7
0.= Hu`ar`af ag n` abdgrglhb` ag ul mblfebf (` ‘ m)= .................. ..................... ..................... ........ 0:4
0.8 [rfauhtf ag n` sue` pfr n` abdgrglhb` (` + m)(` ‘ m) ..................... ..................... ..................... . 0:4 0.1 [rfauhtf ag afs mblfebfs hfl ul târeblf hfeól y sbilfs biu`ngs .................. ............. 0:9
0.7 Humf ag n` sue` ag ul mblfebf (` + m)8 ..................... ..................... ..................... ..................... .. 0:2
0.4 Humf ag n` abdgrglhb` ag ul mblfebf (` ‘ m)8 .................... ..................... ..................... ............... 02;
]gsuegl ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... 020
Blvgstbiug gl n` rga... ..................... ..................... ..................... ..................... ..................... ..................... ................... 020
@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... 02=
@ibnba`a ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ 021
]`zfl`ebgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................... 027
Ags`rrfnng lugv`s j`mbnba`ags .................... ..................... ..................... ..................... ..................... ...................... 024
Xge`l` 08D`htfrbz`hbøl B .................. ..................... ......................... ..................... ..................... ..................... ........ 029
¡[`r` hfeglz`r! D`htfrgs prbefs ..................... ..................... ..................... ..................... ..................... .................. 02:
Gn eulaf ag n` e`tgeàtbh`
0. D`htfrbz hbøl ag pfnblfebfs .................. ..................... ..................... ..................... ..................... .................. 022
0.0 D`htfr hfeól eflfebf ................... ..................... ..................... ..................... ..................... .............. 022
0.= D`htfr hfeól pfr `irup`hbøl ag târeblfs ................... ..................... ..................... ................... =;0
0.8 Qrblfebf hu`ar`af pgrdghtf.............................. ..................... ..................... ..................... ................. =;8
0.1 Abdgrglhb` ag hu`ar`afs ................... ..................... ..................... ..................... ..................... .............. =;7
]gsuegl ..................... ..................... ..................... ..................... ..................... ..................... ..................... ..................... ... =;9
Blvgstbiug gl n` rga... ..................... ..................... ..................... ..................... ..................... ..................... ................... =;9
@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... =;:
@ibnba`a ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ =0;
]`zfl`ebgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................... =00
Ags`rrfnng lugv`s j`mbnba`ags .................... ..................... ..................... ..................... ..................... ...................... =0=
TBBIG] ∝ [fnfhjbh
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Blvgstbiug gl n` rga... ..................... ..................... ..................... ..................... ..................... ..................... ................... =7;
@utfhfltrfn .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... =70
@ibnba`a ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ =71
]`zfl`ebgltf nøibhf ..................... ..................... ..................... ..................... ..................... ..................... ................... =77
Ags`rrfnng lugv`s j`mbnba`ags .................... ..................... ..................... ..................... ..................... ..................... . =74
Xge`l` 09]gp`sf3 sge`l`s 2 ` 04 ................... ..................... ..................... ..................... ............. =79
Gn eulaf ag n` e`tgeàtbh`
[ftglhb hbøl ag gxprgsbflgs nigmr`bh`s ................... ..................... ..................... ..................... ..................... =72
]`abh`hbøl ag gxprgsbflgs `nigmr`bh`s ................... ..................... ..................... ..................... ..................... ... =40
Fpgr`hbflgs pfnblfeb`ngs .................. ..................... ..................... ..................... ..................... ..................... ........ =48
[rfauhtfs lft`mngs ..................... ..................... ..................... ..................... ..................... ..................... ................... =47
D`htfrbz hbøl B .................... ..................... ..................... ..................... ..................... ..................... ..................... ......... =49
D`htfrbz hbøl BB ................... ..................... ..................... ..................... ..................... ..................... ..................... ......... =42Dr`hhbflgs `nigmr`bh`s B ................... ..................... ..................... ..................... ..................... ..................... ............ =90
Dr`hhbflgs `nigmr`bh`s BB ................... ..................... ..................... ..................... ..................... ..................... ........... =98
@ibnba`a ag hànhunf eglt`n .................... ..................... ..................... ..................... ..................... ..................... ........ =97
Frbglt`hbflgs sfmrg n` prugm` dbl`n ................................................................................................................... =94
Hn`vgs ............................................................................................................................................................... =99
Q`nngr e`tgeàtbhf[rugm` @ .................. ..................... ..................... ..................... ..................... ..................... ..................... .................... =22
[rugm` M ................... ..................... ..................... ..................... ..................... ..................... ..................... .................... 8;9
Mbmnbfir`då` ................................................................................................................................................... 808
TBBBBIG] ∝ [fnfhjbh
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‘0 5 b ‘0 5 b
0 = 8 1
8
1
7
=
0
;
x
y
0E`tgeàtbh` ∝ ¡Mbglvglba` y mbglvglbaf!
¡Mbglvglba` y mbglvglbaf!
Blbhb`efs hfl gltusb`sef gn hursf ag e`tgeàtbh` ag 1.» m`hjbnngr`tf. Gstg nbmrf hfrrgspflag`n prbegr sgegstrg agn irupf [fnfhjbh y hflst` ag abghbsbgtg sge`l`s, gl n`s qug sg ags`-rrfnn`ràl hblhf hfepgtglhb`s e`rh`a`s pfr gn Hurråhunf L`hbfl`n M`sg (HLM).
Gn hursf gstà gstruhtur`af ag t`n dfre` qug nfs hfltglbafs sfl ir`au`ngs y afsbh`afs p`r`dfrt`nghgr y glrbqughgr n`s r`e`s sbiubgltgs3
• @rbteâtbh`3 pgrebtg rg`nbz`r fpgr`hbflgs `rbteâtbh`s hfl nfs lóegrfs rg`ngs.
• Ànigmr`3 `yua` ` nggr, fpgr`r y rgprgsglt`r bldfre`hbøl gl gn ngliu`cg sbemønbhf ag n`e`tgeàtbh`.
• Igfegtrå`3 rgdbgrg n`s prfpbga`ags ag n`s dbiur`s igfeâtrbh`s hfef àrg`s y vfnóeglgsag pfnåiflfs. Q`embâl rgn`hbfl` n`s prfpbga`ags gltrg rght`s.
• Qrbiflfegtrå`3 gst` r`e` ag n` e`tgeàtbh` sg glh`ri` agn gstuabf ag nfs n`afs y àliu-nfs ag trbàliunfs rghtàliunfs, ` tr`vâs ag n`s r`zflgs trbiflfeâtrbh`s.
• Hànhunf abdgrglhb`n g bltgir`n3 pgrebtg rgn`hbfl`r n`s dulhbflgs y sus abdgrgltgs `pnbh`-hbflgs hfef r`zøl ag h`embf.
@ltgs ag gltr`r `n agt`nng ag n`s hfepgtglhb`s p`r` gstg hursf, fmsgrvgefs n` pfrt`a`.
Gn eulaf ag n` pfrt`a` lfs rghugra` qug sfefs p`rtg agn pn`lgt` Qbgrr` y qug gl ân, n`e`tgeàtbh` gs ul` jgrr`ebglt` qug lfs `yua` ` agshumrbr, `prglagr y absdrut`r n`s `rtgs yn` huntur`.
Nfs lóegrfs e`y`s lfs rghugra`l qug pgrtglghgefs ` ul pugmnf euntbâtlbhf g bltgr-huntur`n, hfl ul hflfhbebgltf e`tgeàtbhf qug
agmgefs v`nfr`r y hflsgrv`r.
N` e`lf a`laf ul hnbh lfs blvbt` ` gxpnfr`r,agshumrbr y hfep`rtbr hflfhbebgltfs hfl gnrgstf agn eulaf ` tr`vâs ag bltgrlgt.
N` be`igl ag ul irupf ag e`lfs `nrgagafrag ul hfr`zøl lfs j`mn` ag gst`mnghgr rgn`-hbflgs hfl n`s pgrsfl`s qug lfs rfag`l.
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¶Høef `nh`lz`rà n`s hfepgtglhb`s e`tgeàtbh`s6
Lfs gldfh`rgefs gl hblhf hfepgtglhb`s gspghåh`s. [`r` s`mgr sb n`s j` `nh`lz`af, gn HurråhunfL`hbfl`n M`sg prfpflg blabh`afrgs ag nfirf, gstfs hrbtgrbfs sfl hfef ul tgreøegtrf qug ebagsu agsgepgðf gl h`a` hfepgtglhb`. Brgefs `v`lz`laf p`sf ` p`sf.
T`y` càlafsg quâ sghhbflgs agn nbmrf j`hgl pfsbmng qug ustga ags`rrfnng h`a` hfepgtglhb` qugprgsglt`efs ` hfltblu`hbøl.
Hfepgtglhb 0 Xghhbøl agn nbmrfXge`l`
0 = 8 1 7 4 9 : 2 0; 00 0= 08 01 07 04 09
[rfauhg p`trflgs`rbteâtbhfs,`nigmr`bhfs yigfeâtrbhfs
`pnbh`laf prfpbg-a`ags, rgn`hbflgs,iur`s igfeâ-trbh`s, såemfnfsy sgð`ngs agdgløeglfs l`tu-r`ngs, sfhb`ngs yhuntur`ngs.
¡[`r`hfeglz`r! Z Z Z Z Z
Gn eulaf agn` e`tgeàtbh` Z Z Z Z Z Z Z Z Z
]gsuegl
Blvgstbiug gln` rga Z Z Z Z Z Z
@utfhfltrfn Z Z Z Z Z Z Z Z
@ibnba`a aghànhunf eglt`n
Z Z Z Z Z Z Z
]`zfl`ebgltfnøibhf Z Z Z Z Z Z Z Z Z Z Z Z
Ags`rrfnng lugv`sj`mbnba`ags Z Z Z Z Z Z Z Z Z Z Z
Hfepgtglhb = Xghhbøl agn nbmrf Xge`l`0 = 8 1 7 4 9 : 2 0; 00 0= 08 01 07 04 09
]gsugnvgsbtu`hbflgs,prfmnge`s agh`ràhtgr dfre`nqug age`la`l gnafeblbf agn pgl-s`ebgltf nøibhfe`tgeàtbhf yn`s fpgr`hbflgs
e`tgeàtbh`sag `rbteâtbh` yànigmr` gl nfshflcultfs lu-eârbhfs rg`ngs yhfepngcfs.
¡[`r`hfeglz`r! Z Z Z Z Z Z Z Z Z Z
Gn eulaf agn` e`tgeàtbh` Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z
]gsuegl
Blvgstbiug gln` rga Z Z Z Z Z Z Z Z Z Z Z Z Z
@utfhfltrfn Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z
@ibnba`a aghànhunf eglt`n
Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z
]`zfl`ebgltfnøibhf Z Z Z Z Z Z Z Z Z Z Z Z Z Z
Ags`rrfnng lugv`sj`mbnba`ags Z Z Z Z Z Z Z Z Z Z Z Z Z
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Hfepgtglhb` 8 Xghhbøl agn nbmrfXge`l`
0 = 8 1 7 4 9 : 2 0; 00 0= 08 01 07 04 09@pnbh` hflf-hbebgltfs sfmrgdulhbflgs, e`-trbhgs, igfegtrå`y vghtfrgs, glsbtu`hbflgs qugprfeugvgl gnegcfr`ebgltf ytr`lsdfre`hbølagn egabf l`tur`n,
sfhb`n y huntur`nag su hfltgxtf.
¡[`r`hfeglz`r!
Gn eulaf agn` e`tgeàtbh` Z Z Z
]gsuegl
Blvgstbiug gln` rga Z
@utfhfltrfn Z
@ibnba`a aghànhunf eglt`n
]`zfl`ebgltfnøibhf
Ags`rrfnng lugv`sj`mbnba`ags
Hfepgtglhb` 1 Xghhbøl agn nbmrfXge`l`
0 = 8 1 7 4 9 : 2 0; 00 0= 08 01 07 04 09
Wtbnbz` tâhlbh`sag suhgsbflgs
y sgrbgs p`r`bltgrprgt`rjghjfs sfhb`ngs,ghfløebhfs yigfiràhfs.
¡[`r`hfeglz`r! Z
Gn eulaf ag
n` e`tgeàtbh` Z Z Z
]gsuegl
Blvgstbiug gln` rga Z
@utfhfltrfn Z
@ibnba`a aghànhunf eglt`n
]`zfl`ebgltfnøibhf Z
Ags`rrfnng lugv`sj`mbnba`ags Z
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Hfl su gsdugrzf, pfarà hflqubst`r gst`s hfepgtglhb`s ` tr`vâs ag trgs tbpfs ag hfltglbafs3
Aghn`r`tbvfs3 Nfs hfltglbafs aghn`r`tbvfs ng `pfrt`ràl gn hflfhbebgltf ag nfsabstbltfs `spghtfs tgørbhfs, hflhgptu`ngs y hbgltådbhfs agn àrg`.
[rfhgabeglt`ngs3 Hfef blabh` su lfemrg, nfs hfltglbafs prfhgabeglt`ngs sgags`rrfnn`ràl ` m`sg ag gcgrhbhbfs, prfhgabebgltfs f `lànbsbs ag prfmnge`s gl nfshu`ngs puga` agefstr`r gn afeblbf y n` pugst` gl pràhtbh` ag nfs hflfhbebgltfsaghn`r`tbvfs.
@htbtuabl`ngs3 Nfs hfltglbafs `htbtuabl`ngs sfl tfaf `qugnnf qug tbglg qug vgrhfl su `htbtua `ltg gn gstuabf y `ltg n` vba`, gl iglgr`n. Gl gn nbmrf glhfltr`rà `htb-vba`ags y rgdngxbflgs sfmrg sbtu`hbflgs hftbab`l`s aflag bltgrvbglg n` e`tgeàtbh`.
Hfepgtglhb` 7 Xghhbøl agn nbmrfXge`l`
0 = 8 1 7 4 9 : 2 0; 00 0= 08 01 07 04 09Gepng` n`s tgfrå`sag igfegtrå` ytrbiflfegtrå`p`r` bltgrprgt`rbldfre`hbflgs ygn`mfr`r bldfregssfmrg sbtu`hbflgsrg`ngs.
¡[`r`hfeglz`r! Z
Gn eulaf agn` e`tgeàtbh`
]gsuegl
Blvgstbiug gln` rga
@utfhfltrfn
@ibnba`a ag
hànhunf eglt`n ]`zfl`ebgltf
nøibhf Z Z Z Z Z
Ags`rrfnng lugv`sj`mbnba`ags Z Z Z Z
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Nbmrf, hn`sg r`ab`n y hårhunf ag gstuabf
¡Xu gqubpf ag tr`m`cf!
Gn nbmrf, hfl sgr ul` mugl` jgrr`ebglt`, lf nf gs tfaf. [`r` qug ustga `nh`lhg gn lbvgn aghfepgtglhb` agsg`af, lugstrf sbstge` pflg ` su abspfsbhbøl3 gn nbmrf, n` hn`sg r`ab`n y n`
blvbt`hbøl ` p`rtbhbp`r gl ul hårhunf ag gstuabf.
• N` hn`sg r`ab`n tbglg hfef dulhbøl prblhbp`ngxpnbh`r y d`hbnbt`r n` hfeprglsbøl ag nfs tge`s
tr`t`afs gl gn nbmrf.
[ugag gshuhj`r n` hn`sg r`ab`n gl ul` gebsfr`ag su nfh`nba`a, agsh`ri`rn`s gl lugstr` pà-ibl` www.bigr.gau.it f `aqubrbrn`s gl HA gl n`hffrabl`hbøl rgibfl`n.
• Gn hårhunf ag gstuabf gs gn nui`rp`r` hfep`rtbr y `prglagr cultfs.
@prfvghjg gstfs rghursfs y pøygsg gl pgrsfl`s ag su hfeulba`a p`r` rgsfnvgr sus aua`s.
• Gn nbmrf huepng hu`trf dulhbflgs3
`. Qgxtf, gl gn qug glhugltr` n` bldfre`hbøl y gnags`rrfnnf ag nfs hfltglbafs ` gstuab`r.
m. [bz`rrøl, p`r` qug aur`ltg n` hn`sg r`ab`nsumr`yg bag`s bepfrt`ltgs f rg`nbhg abstblt`s`htbvba`ags.
h. Hu`agrlf ag tr`m`cf, hfl gcgrhbhbfs p`r` pr`h-tbh`r nf `prglabaf.
a. Jgrr`ebglt` ag `utfgv`nu`hbøl, hu`laf rg-sugnvg su `utfhfltrfn h`a` sge`l`.
D`nt` h`emb`rpfrt`abt`
8/20/2019 Libro Polochic Matemática 1er. Sem
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Lugstr` egtfafnfiå` p`sf ` p`sf
[`r` d`hbnbt`r su `prglabz`cg y `prfvghj`r eàs y egcfr gn gstuabf h`a` sge`l`, sbi` gstfsp`sfs. ¡Lf sg s`ntg lbliulf!
Ng` gn hfltglbaf ag n` sge`l`
Nggr gn hfltglbaf lfs pgrebtg tglgr ul` bag` iglgr`n agn tge`3 quâ s`-
mgefs, hfl quâ nf rgn`hbfl`efs, gth. Gstg prbegr hflt`htf t`embâl lfs
j`rà h`gr gl n` huglt` agn gsdugrzf rg`nbz`r p`r` prglagr nf lugvf y lfs
pflarà "gl fla`" p`r` n` hn`sg r`ab`n.
0
Gshuhjg n` hn`sg r`ab`nHfl nfs 7 sgltbafs
N` hn`sg r`ab`n gs lugstr` e`gstr`. Ag `jå qug gn prfir`e` sg nn`eg "GnE`gstrf gl H`s`". N`s e`gstr`s y e`gstrfs nfhutfrgs gxpnbh`l gn hfltglb-
af, prfpflgl gcgrhbhbfs y ftrfs gcgepnfs p`r` `epnb`r gn tge`.
=
Agspuâs ag n` hn`sg r`ab`n, su tr`m`cf pgrsfl`n Gstuabf y `utfhfltrfn
Dbl`nbz`a` n` hn`sg r`ab`n gs gn efegltf ag su tr`m`cf pgrsfl`n. Abstrbmuy`
su tbgepf3 gs egcfr ul pfhf h`a` aå`, qug tfaf n` våspgr`.
8
Hflsuntg sus aua`sWl gstuab`ltg bltgnbigltg s`mg huàlaf pgabr `yua`
Hflsuntg nfs tge`s qug lf ng j`l quga`af hn`rfs gl ftrfs nbmrfs, gl bltgr-
lgt, hfl d`ebnb`rgs f `ebifs. Xgiurf qug glhfltr`rà pgrsfl`s abspugst`s
` `yua`rng.
1
[`rtbhbpg gl ul hårhunf ag gstuabf
@prglagr cultfs[øli`sg ag hugraf hfl ftrfs gstuab`ltgs y fri`lbhgl ul hårhunf ag gstu-
abf. Xfnbhbtgl n` `yua` ag `niul` pgrsfl` vfnult`rb` ag n` hfeulba`a. Gsf
ngs `yua`rà ` rgsfnvgr aua`s y rgdfrz`r nf `prglabaf. @ageàs, tglaràl n`
fpfrtulba`a ag bltgrh`emb`r `prglabz`cgs, bag`s y sgltbebgltfs.
]ghugrag qug sbgeprg pugag `huabr ` su tutfr `sbil`af.
7
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¶Høef `prfvghj`r egcfr su gstuabf6
• Musqug ul nui`r høefaf y hfl mugl` bnuebl`hbøl. Gs bepfrt`ltg qug sg`ngcg agn rubaf y ag n`s abstr`hhbflgs.
• Gnbc` ul jfr`rbf p`r` tr`m`c`r y gstuab`r. N` hflst`lhb` y n` abshbpnbl` sflsus egcfrgs hfep`ðgr`s ag gstuabf.
• Ng` hfl `tglhbøl n`s blstruhhbflgs ag nfs gcgrhbhbfs `ltgs ag rgsfnvgrnfs.
• Hflsuntg sus aua`s hfl ftr`s pgrsfl`s ag su hfeulba`a qug puga`l`yua`rng.
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00
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
Nøibh` B
¶Puâ glhfltr`rà gst` sge`l`6
Gn huepng`ðfs ag @ngc`lar`
Nøibh` prfpfsbhbfl`n
@ibnba`a ag hànhunf hfl euntbpnbh`hbflgs y abvbsbflgs
[rfmnge`s ag nøibh`
Gst` sge`l` nfir`rà3
Hflfhgr y hfeprglagr gn h`epf ag gstuabf ag n` nøibh`.
Bagltbdbh`r y hflstrubr prfpfsbhbflgs sbepngs.
Agtgrebl`r gn v`nfr ag vgra`a ag prfpfsbhbflgs sbepngs. Gcgrhbt`r n` lgi`hbøl ag prfpfsbhbflgs sbepngs.
Hfeprglagr y pr`htbh`r gn usf ag hu`ltbdbh`afrgs.
]gsfnvgr fpgr`hbflgs `rbteâtbh`s màsbh`s hfl `ibnba`a.
08E`tgeàtbh` ∝ Xge`l` 0
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¡[`r` hfeglz`r!Gn huepng`ðfs ag @ngc`lar`
Tg`efs y sfnuhbflgefs ul pgqugðf prfmnge` gepng`laf lugstr` nøibh`.
Gn sbiubgltg gcgrhbhbf dug pn`ltg`af gl n`s fnbepb`a`s ag e`tgeàtbh`s ag @sb` yXbli`pur, gl `mrbn ag =;07. Nâ`nf, pfli` ` tr`m`c`r sus lgurfl`s y rgsuânv`nf3
@nmgrtf y Mgrl`raf sg `h`m`l ag j`hgr `ebifs ag @ngc`lar` y qubgrgl s`mgr
huàlaf gs su huepng`ðfs.
@ngc`lar` ngs a` ul` nbst` hfl 0; pfsbmngs dghj`s3
¡@ tr`m`c`r!
Hfl n` bldfre`hbøl a`a`, rgspfla` n`s prgiult`s sbiubgltgs.
0) ¶Huàlaf gs gn huepng`ðfs ag @ngc`lar`6
=) Gxpnbqug høef j`nnø n` rgspugst`, ¶huàn dug su r`zfl`ebgltf6
Nugif, @ngc`lar` ngs abhg n` dghj` ag su huepng`ðfs, pgrf pfr sgp`r`af3 `
@nmgrtf ng abhg gn egs y ` Mgrl`raf, gn aå`.
@emfs cøvglgs sg pn`ltg`l n` sbiubgltg bldfre`hbøl3
• @nmgrtf3 "Lf sâ huàlaf gs gn huepng`ðfs ag @ngc`lar`, pgrf sâ qug
Mgrl`raf t`epfhf nf s`mg".
• Mgrl`raf3 "@n prblhbpbf lf s`må` huàlaf gr` gn huepng`ðfs ag @ngc`lar`,
pgrf `jfr` y` nf sâ".
• @nmgrtf3 "Gltflhgs, yf t`embâl sâ huàlaf gs su huepng`ðfs".
@a`pt`af ag iff.in/=D2Icx
e`yf 07 e`yf 04 e`yf 02
culbf 09 culbf 0:
cunbf 01 cunbf 04
`ifstf 01 `ifstf 07 `ifstf 09
01BIG] ∝ [fnfhjbh
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0. Nøibh` prfpfsbhbfl`n@ltgs ag blbhb`r, rghfragefs qug hfl n` `yua` ag n` nøibh` gstuab`efs nfs r`zf-
l`ebgltfs y agauhbefs hflhnusbflgs ` p`rtbr ag supugstfs pn`ltg`afs.Wl` r`e` ag n` hbglhb` ltgrbfr gs n` nøibh` prfpfsbhbfl`n, qug gstuab` glulhb`-afs y gv`nó` su lbvgn ag vgra`a, gs aghbr, agtgrebl` sb nf qug sg `re` gs d`nsff vgra`agrf.
=. [rfpfsbhbflgs sbepngsNfs glulhb`afs qug gstuab` n` nøibh` prfpfsbhbfl`n sg aglfebl`l prfpfsbhbf-lgs, gst`s pugagl tfe`r afs v`nfrgs nøibhfs f ag vgra`a3 vgra`agrf f d`nsf. Xblf pfagefs hflhnubr sb ul glulhb`af gs d`nsf f vgra`agrf, gltflhgs lf gs ul`prfpfsbhbøl.
N`s prfpfsbhbflgs sbepngs sfl `qugnn`s qug lf sg pugagl abvbabr sbl `dght`rsu sgltbaf. Xg rgprgsglt`l utbnbz`laf ngtr`s eblóshun`s. Tg`efs ulfs gcgepnfs3
q3 Rf l`hå gl Jugjugtgl`lif.
r 3 Iu`tge`n` gs p`rtg ag Hgltrf`eârbh`.
w 3 Nfs e`eådgrfs sfl vbvåp`rfs.
Hfef vgefs, q, r y w sfl prfpfsbhbflgs sbepngs pfrqug lf sg pugagl sgp`r`r.
Gl tfa`s sg pugag gv`nu`r su v`nfr ag vgra`a, gl n` prbegr`, pugag sgr d`nsf fvgra`agrf, agpglabglaf sb n` pgrsfl` l`hbø gl Jugjugtgl`lif f gl niól ftrfnui`r. Ebgltr`s t`ltf, n` sgiula` y n` tgrhgr` prfpfsbhbøl tfe`l gn v`nfr nøibhfag vgra`agrf.
Lf tfafs nfs glulhb`afs sfl prfpfsbhbflgs. Hfef eglhbfl`efs `ltgrbfr-egltg, sb lf pfagefs agtgrebl`r gn v`nfr nøibhf ag ul` fr`hbøl, lf gs ul`prfpfsbhbøl.
Gstf suhgag hfeólegltg gl glulhb`afs bepgr`tbvfs, ag prgiult` f `qugnnfsqug prgsglt`l ul cubhbf ag v`nfr.
Tg`efs `niul`s dr`sgs qug lf sfl prfpfsbhbflgs3
¡Lf tbrg m`sur` gl n` h`nng!
¶Puâ gstfy j`hbglaf6
Gs bepfrt`ltg gst`r gl n` rgulbøl.
Hfef vgefs, ` nfs glulhb`afs `ltgrbfrgs lf sg ngs pugag agtgrebl`r ul v`nfrnøibhf.
Cubhbf ag v`nfr3glulhb`af qug
gxprgs` ul` fpblbøl sumcgtbv`.
07E`tgeàtbh` ∝ Xge`l` 0
Gn eulaf ag n` e`tgeàtbh`
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Gcgrhbhbf 0@. Ng` n`s fr`hbflgs sbiubgltgs. Nugif, e`rqug hfl ul` "Z" gn hu`arf, sgiól sg` ul` prfpfsbhbøl
f lf. Ng a`efs ul gcgepnf.
Xå Lf
;) Iu`tge`n` gs p`rtg ag @eârbh`.
0) ¶Puâ jfr` gs6
=) [fr d`vfr, `vås`ng ` ân.
8) = gs ul abvbsfr ag 01.
1) Ebiugn Àlign @sturb`s dug ul gshrbtfr.
7) ¶Puâ eósbh` gshuhj`6
4) Gn tâ gs egcfr qug gn h`dâ.
9) Gn `zun gs ul hfnfr sghula`rbf.
:) Gn `e`rbnnf gs ul hfnfr prbe`rbf.
2) N` Qbgrr` gs pn`l`.
0;) Nggr gs euy bepfrt`ltg.
M. @jfr` gs su turlf. [bglsg y gshrbm` gl n`s nålg`s sbiubgltgs fhjf prfpfsbhbflgs. Nugif, agtgreblgsu v`nfr nøibhf hfnfh`laf ul` T (vgra`agrf) f ul` D (d`nsf) gl gn gsp`hbf ` n` agrghj`. Iuågsg pfr
gn gcgepnf.
;)
0)
=)
8)
1)
7)
4)
9)
:)
N`s e`rbpfs`s pugagl vfn`r ir`lags abst`lhb`s. T
x
04BIG] ∝ [fnfhjbh
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8. Nøibh` ag prgabh`afsN` nøibh` ag prgabh`afs pgrebtg j`hgr `re`hbflgs hfl rgspghtf ` rgn`hbflgsgltrg fmcgtfs qug s`tbsd`hgl gngegltfs ag ul hflcultf ulbvgrsf gst`mnghbaf.
Gl gst` nøibh` sg gepng`l nfs hu`ltbh`afrgs, nfs hu`ngs sfl såemfnfs utbnbz`-
afs p`r` gst`mnghgr huàltfs gngegltfs ag ul hflcultf agtgrebl`af huepnglhfl hbgrt` prfpbga`a.
Nfs hu`ltbh`afrgs pgrebtgl n` hflstruhhbøl ag prfpfsbhbflgs ` p`rtbr ag dul-hbflgs prfpfsbhbfl`ngs, gs aghbr, ag cubhbfs qug gst`mnghgl p`ràegtrfs hfl nfshu`ngs agmgl huepnbr nfs gngegltfs ag ul hflcultf ulbvgrsf.
Nfs hu`ltbh`afrgs eàs utbnbz`afs sfl gn ulbvgrs`n y gn gxbstglhb`n.
8.0 Hu`ltbdbh`afr ulbvgrs`n (∋)
Xg gepng` p`r` `re`r qug tfafs nfs gngegltfs ag ul hflcultf huepngl hflul` hflabhbøl f p`ràegtrf agtgrebl`af. Xg rgprgsglt` hfl gn såemfnf ∋ y sgngg "p`r` tfaf…". Tg`efs ul gcgepnf p`r` gltglagr egcfr3
∋ x ∊ ] , = x ∊ ]
N` gxprgsbøl `ltgrbfr sg ngg3 "p`r` tfa` x qug pgrtglghg ` ] (hflcultf ag nfslóegrfs rg`ngs), sg huepng qug = x pgrtglghg ` ] "
8.= Hu`ltbdbh`afr gxbstglhb`n (∈)
Xg utbnbz` p`r` blabh`r qug j`y n eglfs ul gngegltf gl ul hflcultf qug huepnghfl ul` hflabhbøl agtgrebl`a`. Xg rgprgsglt` hfl gn såemfnf ∈ y sg ngg "gxbstg`n eglfs ul…". Tg`efs ul gcgepnf.
Qfe`laf hfef rgdgrglhb` nfs hflcultfs sbiubgltgs3 @ 5 {0, =, 8, 1, 7}, M 5 {1, 7,4, 9, :}
∈ x ∊ M ∩ x ∊ @
N` gxprgsbøl `ltgrbfr sg ngg3 "gxbstg n eglfs ul gngegltf x qug pgrtglghg ` My pgrtglghg t`embâl ` @".
Gcgrhbhbf 8@. Ng` n`s gxprgsbflgs sbemønbh`s sbiubgltgs. Nugif, rggshråm`n`s gl ngliu`cg hfeól. Iuågsg pfr gn
gcgepnf.
;) ∈ x ∊ M ∩ x ∊ P
]ghugrag qug x gsul` v`rb`mng y pugagrgprgsglt`r ` hu`nqubgrlóegrf.
Gn såemfnf ∩ sg ngg "y".
Gxbstg `n eglfs ul` x qug pgrtglghg ` M y t`embâl pgrtglghg ` P.
0:BIG] ∝ [fnfhjbh
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0) ∋ x ∊ W, 7 x ∊ W
=) ∈ x ∊ D ∩ x ∊ E
M. Ng` n`s gxprgsbflgs sbiubgltgs. Nugif, tr`lsdøregn`s ngliu`cg sbemønbhf y rggshråm`n`s. Ng a`efsul gcgepnf.
;) [`r` tfa` x qug pgrtglghg ` nfs lóegrfs l`tur`ngs, sg huepng qug : x pgrtglghg t`embâl ` nfslóegrfs l`tur`ngs.
0) Gxbstg `n eglfs ul gngegltf x qug pgrtglghg ` @ y pgrtglghg t`embâl ` M.
=) [`r` tfa` x qug pgrtglghg ` nfs lóegrfs rg`ngs, sg huepng qug 4 x gs ul lóegrf rg`n.
8) Gxbstg `n eglfs ul` x qug pgrtglghg ` F y pgrtglghg t`embâl ` [.
∋ x ∊ L, : x ∊ L
0. N` nøibh` prfpfsbhbfl`n gstuab` glulhb`afs y gv`nó` su v`nfr ag vgra`a, gs aghbr, agtgrebl` sbsfl vgra`agrfs f d`nsfs.
=. N`s prfpfsbhbflgs sfl glulhb`afs qug pugagl sgr `l`nbz`afs y `så agtgrebl`r su v`nfr nøibhf.
Wl glulhb`af lf gs ul` prfpfsbhbøl sb lf sg pugag agtgrebl`r su v`nfr ag vgra`a. Gstf suhgaghfeólegltg hfl glulhb`afs bepgr`tbvfs, ag prgiult` f `qugnnfs qug prgsglt`l ul cubhbf agv`nfr.
N`s prfpfsbhbflgs sbepngs sfl qugnn`s qug lf pugagl sgr abvbaba`s sbl dght`r su sgltbaf. Gst`ssfl rgprgsglt`a`s hfl ul` ngtr` eblóshun`.
=.0 N` lgi`hbøl gs ul` fpgr`hbøl qug hflsbstg gl blvgrtbr gn v`nfr nøibhf ag ul` prfpfsbhbøl? sgblabh` `ltgpflbglaf gn sbilf "~" ` n` prfpfsbhbøl.
8. Nfs hu`ltbdbh`afrgs sfl såemfnfs qug sg utbnbz`l p`r` gst`mnghgr huàltfs gngegltfs ag ul hfl- cultf agtgrebl`af huepngl hfl hbgrt` prfpbga`a.
8.0 Gn hu`ltbdbh`afr ulbvgrs`n (∋) sg gepng` p`r` dbre`r qug tfafs nfs gngegltfs ag ul hflcultfhuepngl hfl ul` hflabhbøl a`a`.
8.= Gn hu`ltbdbh`afr gxbstglhb`n (∈) sg utbnbz` p`r` blabh`r qug `n eglfs ul gngegltf ag ul hfl- cultf huepng hfl ul` hflabhbøl a`a`.
]gsuegl
02E`tgeàtbh` ∝ Xge`l` 0
8/20/2019 Libro Polochic Matemática 1er. Sem
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@htbvba`a 0. Ageugstrg nf prglabaf
@. Ng` nfs glulhb`afs sbiubgltgs. Nugif, gshrbm` ` n` agrghj` sb gs ul` prfpfsbhbøl f lf. Ng a`efs ulgcgepnf.
;) Gn tbgepf hfrrg ràpbaf.
0) 8 gs ul d`htfr ag 2.
=) Gn abhhbfl`rbf sbrvg p`r` nggr hugltfs ag lbðfs.
8) ¶[fr quâ lf gstàs gl h`s`6
1) Gn fhâ`lf [`hådbhf hfnbla` hfl n`s hfst`s n sur ag Iu`tge`n`.
7) Gn dåsbhf eàs bepfrt`ltg agn sbinf ZZ dug @nmgrt Gblstgbl.
4) ¡[fr d`vfr, j`z tu t`rg`!
9) Gn `zóh`r gs aunhg.
:) Iu`tge`n` gs gn egcfr p`ås.
M. Ng` nfs glulhb`afs sbiubgltgs. Nugif, tr`lsdøregnfs ` ngliu`cg sbemønbhf. ]ghugrag nf qug `prglabøsfmrg nfs hu`ltbdbh`afrgs.
0) [`r` tfa` x qug pgrtglghg ` nfs lóegrfs rg`ngs, sg huepng qug 0; x gs ul lóegrf rg`n.
=) Gxbstg `n eglfs ul gngegltf z qug pgrtglghg ` H y pgrtglghg t`embâl ` Y.
8) [`r` tfa` x qug pgrtglghg ` @, gxbstg `n eglfs ul` y qug pgrtglghg ` M.
1) [`r` tfa` a qug pgrtglghg ` Z, gxbstg `n eglfs ul` m qug pgrtglghg ` S.
7) Gxbstg `n eglfs ul gngegltf x qug pgrtglghg ` ] y pgrtglghg t`embâl ` L.
@utfhfltrfn
Lf gs prfpfsbhbøl.
=;BIG] ∝ [fnfhjbh
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@htbvba`a =. [r`htbqug nf prglabaf
@. [bglsg y gshrbm` fhjf prfpfsbhbflgs sbepngs. Nugif, gl n` nålg` hfrrgspflabgltg, hfnfqug ul` T, sbgs vgra`agr` f ul` D, sb gs d`ns`. Iuågsg pfr gn gcgepnf.
;)
0)
=)
8)
1)
7)
4)
9)
:)
M. Ng` n`s prfpfsbhbflgs ag n` t`mn` sbiubgltg, gv`nógn`s y agtgreblg su v`nfr nøibhf. Nugif, gshrbm`n` lgi`hbøl ag h`a` ul` gl gn gsp`hbf hfrrgspflabgltg. ]ghugrag gshrbmbr n` lgi`hbøl gl dfre`sbemønbh` y gn v`nfr ag vgra`a qug tfe` n` prfpfsbhbøl. Ng a`efs ul gcgepnf.
;) t3 Nfs pgrrfs tbglgl pgnf.
0) a3 Nfs h`m`nnfs hfrrgl euy ràpbaf.
=) i3 Nfs iu`tge`ntghfs sfefs lfrtg`egrbh`lfs.
8) y3 Wl` p`n`mr` tbglg ngtr`s.
1) p3 Nfs `lbe`ngs sfl sgrgs vbvfs.
7) q3 Gn agp`rt`egltf ag Cutb`p` pgrtglghg ` Jflaur`s.
4) z3 9 gs ul d`htfr ag 74.
Gn rgnfc blabh` n` jfr`. T
~t3 Nfs pgrrfs lf tbglgl pgnf. D
T
=0E`tgeàtbh` ∝ Xge`l` 0
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[`r` i`l`r `ibnba`a eglt`n y r`pbagz, tbglg qug pr`htbh`r hflst`ltgegltg. Ng prfpflgefs rgsfnvgrnfs gcgrhbhbfs ag gstg `p`rt`af. Xu gsdugrzf y hflst`lhb` ng i`r`ltbz`ràl muglfs rgsunt`afs.
@. Xue`s y rgst`s.0) 0= + 77 5
=) 84 ‘ =7 5
8) =0 + 7= 5
1) 4: ‘ =8 5
7) 97 + =7 5
4) 7= + 10 5
9) 84 + 10 5
:) 97 ‘ 07 5
2) 7; ‘ =7 5
0;) 01 + 88 5
00) 02 ‘ 09 5
0=) =0 + 89 5
08) == + 88 5
01) 1= ‘ 8= 5
07) 19 ‘ 88 5
M. Euntbpnbh`hbflgs.0) = Ù : 5
=) 8 Ù = 5
8) 9 Ù 7 5
1) 4 Ù 1 5
7) : Ù 9 5
4) 7 Ù = 5
9) 8 Ù 7 5
:) 0; Ù = 5
2) : Ù 4 5
0;) 2 Ù 1 5
00) 9 Ù 9 5
0=) 7 Ù 4 5
08) ; Ù 9 5
01) 1 Ù = 5
07) 8 Ù 4 5
04) : Ù 7 5
09) 2 Ù 0; 5
0:) 7 Ù 2 5
02) : Ù = 5
=;) 0 Ù 9 5
=0) 2 Ù 8 5
==) 7 Ù 1 5
=8) ; Ù = 5
=1) 4 Ù 7 5
H. Abvbsbflgs.
0) 0 ù 0 5
=) 88 ù 00 5
8) ; ù 2 5
1) : ù 1 5
7) 2 ù 8 5
4) 22 ù 8 5
9) 22 ù 00 5
:) 71 ù 4 5
2) 84 ù 2 5
0;) 87 ù 7 5
00) 48 ù 2 5
0=) 0; ù 0; 5
08) 0;; ù 0 5
01) 9= ù 2 5
07) 1= ù 4 5
04) 17 ù 2 5
09) 84 ù 1 5
0:) ; ù 4 5
02) 2 ù 8 5
=;) =9 ù 2 5
=0) 74 ù : 5
==) 84 ù 8 5
=8) 84 ù 4 5
=1) 1: ù : 5
==BIG] ∝ [fnfhjbh
@ibnba`a ag hànhunf eglt`n
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[fli` ` tr`m`c`r sus lgurfl`s. ]gsugnv` nfs sbiubgltgs prfmnge`s ag nøibh`.
0) N`s afs pugrt`s. Wstga gstà glhgrr`af gl ul` j`mbt`hbøl gl n` qug j`y afs pugrt`s vbibn`a`s pfr
afs iu`rab`s. Wl` nngv` ` n` nbmgrt`a y n` ftr` ` sgiubr glhgrr`af. [ugag gngibr ul` pugrt` y `ltgspugag j`hgr ul` prgiult` ` ulf ag nfs iu`rab`s.
J`y ul prfmnge`3 ulf ag gnnfs sbgeprg abhg n` vgra`a, pgrf gn ftrf sbgeprg ebgltg.
¶Puâ prgiult` j`rå` p`r` s`nbr n` nbmgrt`a6 [bglsg su rgspugst` y gshråm`n` gl n`s nålg`s sbiubgltgs.
=) ¶Ag quâ hfnfr gs gn sfemrgrf6 Be`iblg qug j`y hu`trf rgfs gl ul` prbsbøl y qug tfafs gnnfsgstàl gltgrr`afs gl gn sugnf j`st` n` ntur` agn hugnnf, hfl nf qug sfnf ngs sfmrgs`ng n` h`mgz` p`r`pfagr rgspbr`r. J`y ul eurf qug sgp`r` ` trgs ag nfs prbsbflgrfs hfl rgspghtf `n hu`rtf, hfefeugstr` n` bnustr`hbøl.
@ M H A
Gn iu`rab` qug nfs gstà vbibn`laf ngs abhg3
‑ "H`a` ulf ag ustgags nngv` pugstf ul sfemrgrf. Ag nfs hu`trf sfemrgrfs afs sfl mn`lhfs yafs sfl lgirfs. Xbl j`mn`r gltrg så, tbglgl abgz eblutfs p`r` qug ulf ag ustgags eg abi` ag quâhfnfr gs su sfemrgrf. Xb nf `hbgrt`, nfs nbmgr`râ? y sb lf, sg quga`l gltgrr`afs m`cf gn `rabgltg sfn".
Dbl`negltg, n h`mf ag ul eblutf, ulf ag nfs prbsbflgrfs hflsbiug vgrbiu`r n` rgspugst` hfrrght`.
¶Puâ prbsbflgrf gs gn qug abhg gn hfnfr ag su sfemrgrf6 ¶Høef nf j` `vgrbiu`af6
=8E`tgeàtbh` ∝ Xge`l` 0
]`zfl`ebgltf nøibhf
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E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfir`af gl
prfhgsflf
nfir`af
A g s p u â s a g g s t u a b
` r . . .
Hflfzhf y hfeprglaf gn h`epf ag gstuabf ag n` nøibh`.
Bagltbhf y hflstruyf prfpfsbhbflgs sbepngs.Agtgreblf gn v`nfr ag vgra`a ag prfpfsbhbflgs sbepngs.
Gcgrhbtf n` lgi`hbøl ag prfpfsbhbflgs sbepngs.
Hfeprglaf y pr`htbhf gn usf ag hu`ltbh`afrgs.
]gsugnvf fpgr`hbflgs `rbteâtbh`s màsbh`s hfl `ibnba`a.
Fmsgrvg `tglt`egltg h`a` luegr`n y rfagg n` fphbøl qug hfepngt` n` sgrbghfrrght`egltg.
0) . . .
`. m. h. a.
=) ...
`. m. h. a.
8) ...
`. m. h. a.
1) ...
`. m. h. a.
7) . . .
`. m. h. a.
=1BIG] ∝ [fnfhjbh
]gvbsg su `prglabz`cg
Ags`rrfnng lugv`s j`mbnba`ags
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¡[`r` hfeglz`r!Nfs hflghtbvfs
Gl gn babfe` gsp`ðfn, nfs hflghtbvfs sfl p`n`mr`s utbnbz`a`s hfef lgxfs qug
sbrvgl p`r` ulbr ftr`s p`n`mr`s, dr`sgs f bag`s. @ulqug lf lfs agefs huglt`,
nfs gepng`efs h`a` vgz qug j`mn`efs f gshrbmbefs.
J`y ir`l lóegrf ag hflghtbvfs, niulfs ag gnnfs qug pfagefs eglhbfl`r sfl3
y, gltflhgs, `ageàs, f, u, f sg`, t`embâl.
Tg`efs `niul`s fr`hbflgs hfef gcgepnf3
J`hgr `htbvba`a dåsbh` y hfegr s`nua`mng lfs `yua` ` pgre`lghgr s`lfs.
Xb gstuab`efs abshbpnbl`a`egltg, gltflhgs pfargefs `prfm`r gn ir`af.
Dubefs ` n` pn`y`, `ageàs vbsbt`efs ` ulfs `ebifs.
Hfef pfagefs bagltbdbh`r, nfs hflghtbvfs `p`rghgl rgs`nt`afs y ulgl n`s bag`squg sg gxpflgl gl h`a` fr`hbøl.
Gl gn hu`arf sbiubgltg efstr`efs `niulfs ag nfs hflghtbvfs eàs hfeulgs3
¡@ tr`m`c`r!
Ng` n`s fr`hbflgs sbiubgltgs y sumr`yg gn hflghtbvf qug `p`rghg gl h`a` ul`. Ng a`efs ul gcgepnf.
;) Xb sfefs tfngr`ltgs, gltflhgs hflvbvbrgefs gl `reflå`.
0) Nfs c`iu`rgs s`nt`l ntf y sfl euy àibngs.
=) N`s pn`lt`s lghgsbt`l iu` p`r` vbvbr, gltflhgs lghgsbt`l sgr rgi`a`s.
8) Xb hfef euhj`s hjuhjgrå`s, gltflhgs tglarâ prfmnge`s ag s`nua.
1) Grlgstf pugag cui`r y hfrrgr euy ràpbaf.
]gn`hbøl Hflghtbvf
@abhbøl y, ageàs, t`embâl, så ebsef, pfr ð`abaur`, eàs ul
Hfltr`stgpgrf, sbl gem`rif, lf fmst`ltg, gl hbgrtf efaf, gl
hbgrt` egaba`, `ulqug, gl h`embf, `n hfltr`rbf
H`us` ‘ gdghtf
pfr t`ltf, ag `jå qug, gl hflsghuglhb`, `så pugs, pfr
hflsbiubgltg, pfr nf t`ltf, pfr gsf, pfr nf qug sbiug, pfr
gst` r`zøl, gltflhgs, gltflhgs rgsunt` qug, ag e`lgr`
qug, nugif, `så qug
Hfep`r`tbvfsagn ebsef efaf, biu`negltg, ag efaf sbebn`r, hfef,
`så hfef
=4BIG] ∝ [fnfhjbh
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0. [rfpfsbhbflgs hfepugst`sN`s prfpfsbhbflgs hfepugst`s sfl qugnn`s qug sg dfre`l pfr afs f eàs prf-
pfsbhbflgs sbepngs ulba`s ` tr`vâs ag hflghtbvfs nøibhfs, pgrf ¶quâ sfl gstfsóntbefs6 [fli`efs `tglhbøl.
Hfef ngåefs gl n` pàibl` `ltgrbfr, gl gn babfe` gsp`ðfn j`y p`n`mr`s, nn`e`a`shflghtbvfs, qug sbrvgl ag gln`hg gltrg bag`s u ftr`s p`n`mr`s. Gl n` nøibh`, t`e-mbâl gxbstgl gstfs lgxfs y sg nn`e`l hflghtbvfs nøibhfs. Nfs eàs hfeulgs sflhu`trf3 y , f , sb… gltflhgs y sb y sfnf sb.
Tg`efs ulfs gcgepnfs3
Gn qugtz`n tbglg pnue`s y pugag vfn`r.
Gl gst` prfpfsbhbøl hfepugst` sg ulgl afs sbepngs3 Gn qugtz`n tbglg pnue`s? gn
qugtz`n pugag vfn`r . Gst`s sg ulgl ` tr`vâs agn lgxf "y".
@arb`l` pugag br `n p`rtbaf ag dutmfn f `n hblg.
N` prfpfsbhbøl `ltgrbfr sg hfepflg ag afs sbepngs3 @arb`l` pugag br `n p`rtbaf
ag dutmfn, @arb`l` pugag br `n hblg. Gst`s sg ulgl pfr egabf agn lgxf "f".
[`r` agtgrebl`r gn v`nfr ag vgra`a ag ul` prfpfsbhbøl hfepugst` gs lghgs`rbfhflfhgr nfs hflghtbvfs nøibhfs y sus rgin`s.
=. Hflghtbvfs nøibhfsNfs hflghtbvfs nøibhfs sfl p`n`mr`s y såemfnfs qug sbrvgl hfef gln`hg p`r`ulbr afs f eàs prfpfsbhbflgs sbepngs y `så dfre`r ul` prfpfsbhbøl hfepugst`.
Gstfs hflghtbvfs, cultf ` nfs hu`ltbh`afrgs qug vbefs n` sge`l` p`s`a`, hfls-tbtuygl afs àrg`s dula`eglt`ngs gl n` nøibh`.
Lfsftrfs gstuab`rgefs nfs eàs utbnbz`afs3 hflculhbøl, absyulhbøl, hflabhbfl`ny mbhflabhbfl`n.
=.0 Hflculhbøl (∩)
Gl nøibh`, n` hflculhbøl hfrrgspflag n` hflculhbøl hfpun`tbv` " y". Qbglg hfefdulhbøl ulbr afs prfpfsbhbflgs y su såemfnf gs ∩.
Gcgepnf Xg gshrbmg Xg ngg
@ndrgaf gs `rqubtghtf y Jâhtfr gs eâabhf. p ∩ q p y q
=9E`tgeàtbh` ∝ Xge`l` =
Gn eulaf ag n` e`tgeàtbh`
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=.= Absyulhbøl (∪)
N` absyulhbøl sg hfepfrt` hfef n` hflculhbøl "f" qug sg utbnbz` gl gn babfe`gsp`ðfn. Xu dulhbøl gs ulbr prfpfsbhbflgs p`r` a`rngs ul sgltbaf ag gnghhbølgltrg afs f eàs gngegltfs y su såemfnf gs ∪.
Gcgepnf Xg gshrbmg Xg ngg
I`mrbgn` pugag sgr bliglbgr` f ghflfebst`. p ∪ q p f q
=.8 Hflabhbfl`n ( )
Q`embâl hflfhbaf hfef bepnbh`hbøl, gstà hfldfre`a` pfr n`s p`n`mr`s "Xb...gltflhgs". Xg hfepflg ag afs prfpfsbhbflgs. N` prbegr` gs rgqubsbtf blabspgl-s`mng p`r` qug sg huepn` n` sgiula`. Xu såemfnf gs ul` ghj` `pult`laf ` n`agrghj`3
Gcgepnf Xg gshrbmg Xg nggXb ân gstuab` euhjf, gltflhgs pfarà rgsfn-vgr gn gx`egl sbl abdbhunt`a.
p q Xb p gltflhgs q
=.1 Mbhflabhbfl`n ( )
Gstg hflghtbvf gxprgs` n` rgn`hbøl ag gqubv`nglhb` gltrg afs prfpfsbhbflgs. N`prbegr` sg huepng sfnf sb sg huepng n` sgiula` y vbhgvgrs`. N` dfre`l n`s p`n`-mr`s "sb y sfnf sb" y su såemfnf gs ul` afmng ghj`3
Gcgepnf Xg gshrbmg Xg ngg
Nfs `lbe`ngs vbvbràl s`lfs sb y sfnf sb nfshuba`l mbgl.
p q p sb y sfnf sb q
Gcgrhbhbf 0]gprgsgltg sbemønbh`egltg n`s prfpfsbhbflgs sbiubgltgs, jài`nf hfl n`s ngtr`s p y q. Wtbnbhg gn såemfnfagn hflghtbvf qug hfrrgspfla`. Ng a`efs ul gcgepnf.
;) A`lbgn m`rrg y Ibna` s`huag.
0) Xb hfrrg ràpbaf, gltflhgs i`l`rà n` hfepgtglhb`.=) [ugag nn`e`rnf f glvb`rng ul egls`cg.
8) Gn dugif glhglagrà sb y sfnf sb n` ngð` gstà sgh`.
1) [fargefs huntbv`r e`åz y drbcfn.
7) N` h`s` sgrà dbreg sb y sfnf sb tbglg muglfs hbebgltfs.
4) Ngflgn pugag gstuab`r pgrbtf f m`hjbnngr`tf.
p ∩ q
=:BIG] ∝ [fnfhjbh
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8. ]gin`s ag nfs hflghtbvfs nøibhfs[`r` gv`nu`r y agtgrebl`r gn v`nfr ag vgra`a ag ul` prfpfsbhbøl hfepugst`,agmgefs m`s`rlfs gl n`s rgin`s ag nfs hflghtbvfs nøibhfs. [`r` gstg `lànbsbs,utbnbz`efs n`s t`mn`s ag vgra`a. Tg`efs.
8.0 ]gin` ag n` hflculhbøl
Gn v`nfr ag vgra`a ag ul` hflculhbøl gs vgra`agrf sfnf sb `em`s prfpfsbhbflgssfl vgra`agr`s, y gs d`nsf ag hu`nqubgr ftr` dfre`.
p q p ∩ q
v v vv d d d v d d d d
Hfeprfmgefs n` rgin` hfl ul gcgepnf3
p ∩ q3 [`r` n` hfemustbøl sg lghgsbt` hfemustbmng y fxåiglf.
Gxtr`gefs afs prfpfsbhbflgs sbepngs3
p3 [`r` n` hfemustbøl sg lghgsbt` hfemustbmng.
q3 [`r` n` hfemustbøl sg lghgsbt` fxåiglf.
p3 Xg lghgsbt` hfemustbmng. (T)q3 Xg lghgsbt` fxåiglf. (T)
N` prfpfsbhbøl hfepugst` gs vgra`agr` pfrqug sg huepng hfl `emfs hrbtgrbfs.J`y hfemustbmng y j`y fxåiglf.
p3 Xg lghgsbt` hfemustbmng. (T)
~q3 Lf sg lghgsbt` fxåiglf. (D)N` prfpfsbhbøl gs d`ns`, pfrqug sbl fxå-iglf lf pugag j`mgr hfemustbøl.
~ p3 Lf sg lghgsbt` hfemustbmng. (D)
q3 Xg lghgsbt` fxåiglf. (T)N` prfpfsbhbøl gs d`ns`, pfrqug sblhfemustbmng lf pugag j`mgr dugif.
~ p3 Lf sg lghgsbt` hfemustbmng. (D)
~q3 Lf sg lghgsbt` fxåiglf. (D)
N` prfpfsbhbøl gs d`ns`. Lf sg huepnglbliulf ag nfs hrbtgrbfs, lb j`y hfemus-tbmng lb j`y fxåiglf.
Lftg høef ul` prfpfsbhbøl hfepugst` ulba` pfr n` hflculhbøl tfe` v`nfr agvgra`agrf ólbh`egltg sb n`s afs prfpfsbhbflgs sbepngs sfl hbgrt`s.
Q`mn` ag vgra`a3 gsul` rgprgsglt`hbøliràh` ag nfs v`nfrgsag vgra`a f d`nsga`aag ul` prfpfsbhbøl.
=2E`tgeàtbh` ∝ Xge`l` =
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8.= ]gin` ag n` absyulhbøl
Gl ul` absyulhbøl, gn v`nfr ag n` vgra`a rgsunt` d`nsf sfnf sb em`s prfpfsbhbflgssfl d`ns`s, y vgra`agrf ag hu`nqubgr ftr` dfre`.
p q p ∪ qv v vv d vd v vd d d
Hfeprfmgefs n` rgin` hfl ul gcgepnf3
p ∪ q3 [`r` hfeulbh`rsg pugag nn`e`rng f glvb`rng ul egls`cg.
Gxtr`gefs afs prfpfsbhbflgs sbepngs3
p3 [`r` hfeulbh`rsg pugag nn`e`rng.
q3 [`r` hfeulbh`rsg pugag glvb`rng ul egls`cg.
p3 [ugag nn`e`rng. (T)
q3 [ugag glvb`rng ul egls`cg. (T)
N` prfpfsbhbøl hfepugst` gs vgr-a`agr` pfrqug sg pugag hfeulbh`rpfr `em`s vå`s.
p3 [ugag nn`e`rng. (T)~q3 Lf pugag glvb`rng ul egls`cg. (D)
N` prfpfsbhbøl gs vgra`agr , pfrqug`ulqug lf sg pugag hfeulbh`r pfregls`cg, pugag j`hgrnf nn`e`laf.
~p3 Lf pugag nn`e`rng. (D)
q3 [ugag glvb`rng ul egls`cg. (T)
N` prfpfsbhbøl gs vgra`agr , pfrqug`ulqug lf sg pugag hfeulbh`r pfrnn`e`a`, pugag j`hgrnf glvb`laf ulegls`cg.
~p3 Lf pugag nn`e`rng. (D)~q3 Lf pugag glvb`rng ul egls`cg. (D)
N` prfpfsbhbøl gs d`ns`. Lf pugag
j`hgr ul` nn`e`a` lb glvb`r ul egl-s`cg, gltflhgs lf j`y hfeulbh`hbøl.
Lftg hfef ul` prfpfsbhbøl hfepugst` ulba` pfr n` absyulhbøl tfe` v`nfr agd`nsf ólbh`egltg sb n`s afs prfpfsbhbflgs sbepngs sfl d`ns`s.
8;BIG] ∝ [fnfhjbh
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8.8 ]gin` agn hflabhbfl`n
Hfl gstg hflghtbvf, ul` prfpfsbhbøl gs d`ns` sb p gs vgra`agr` y q gs d`ns`. [`r`hu`nqubgr ftrf h`sf rgsunt` vgra`agr`.
p q p q
v v vv d d d v vd d v
Hfeprfmgefs n` rgin` hfl ul gcgepnf3
p q3 Xb jfy gs vbgrlgs, gltflhgs e`ð`l` sgrà sàm`af.
Gxtr`gefs afs prfpfsbhbflgs sbepngs3
p3 Jfy gs vbgrlgs.
q3 E`ð`l` sgrà sàm`af.
p3 Jfy gs vbgrlgs. (T)
q3 E`ð`l` sgrà sàm`af. (T)
N` prfpfsbhbøl hfepugst` gs vgra`-agr` pfrqug nugif ag vbgrlgs sbiuggn sàm`af.
p3 Jfy gs vbgrlgs. (T)
~q3 E`ð`l` lf sgrà sàm`af. (D)
N` prfpfsbhbøl gs d`ns`. Xb gn aå``htu`n gs vbgrlgs, blaua`mngegltge`ð`l` sgrà sàm`af.
~ p3 Jfy lf gs vbgrlgs. (D)
q3 E`ð`l` sgrà sàm`af. (T)
Gn glulhb`af gs vgra`agrf.N` prbegr` prfpfsbhbøl gs d`ns`, nfqug blabh`rå` qug så gs vbgrlgs, y pfrt`ltf, e`ð`l` så sgrà sàm`af.
~ p3 Jfy lf gs vbgrlgs. (D)
~q3 E`ð`l` lf sgrà sàm`af. (D)N` prfpfsbhbøl gs vgra`agr`, pugagsgr hu`nqubgr ftrf aå` ag n` sge`l`.
Hfl gn hflabhbfl`n gxbstg ul prfmnge` aglfebl`af hfef p`r`afc`s ag n`
bepnbh`hbøl. Gst`s p`r`afc`s sg a`l hu`laf ul` prfpfsbhbøl gs vgra`agr` ag`hugraf n` nøibh`, pgrf gxprgs` ul pn`ltg`ebgltf qug lf tbglg sgltbaf hfeól.[fr gcgepnf3
Xb n` Nul` gstà jghj` ag qugsf, gltflhgs = + = 5 1.
N` prfpfsbhbøl `ltgrbfr gs vgra`agr`, ` pgs`r ag qug lf tbglg sgltbaf gl gnngliu`cg hfeól. Hfef lfsftrfs gst`efs `l`nbz`laf n`s prfpfsbhbflgs agsaggn pultf ag vbst` nøibhf, lf sg prgfhupg sb `niulfs glulhb`afs ng rgsunt`l hfl-tr`rbfs ` n` r`zøl.
80E`tgeàtbh` ∝ Xge`l` =
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8.1 ]gin` agn mbhflabhbfl`n
Gn v`nfr ag vgra`a ag ul mbhflabhbfl`n gs vgra`agrf hu`laf `em`s prfpfsbhbf-lgs tbglgl gn ebsef v`nfr ag vgra`a, gs aghbr, `em`s sfl vgra`agr`s f d`ns`ssbeuntàlg`egltg? gl h`sf hfltr`rbf, gs d`nsf.
p q p qv v vv d d d v d d d v
Fmsgrvgefs egcfr n` rgin` hfl ul gcgepnf3
p q3 @eånh`r pugag sgr prgsbagltg sb y sfnf sb gs hbua`a`lf iu`tge`ntghf.
Gxtr`gefs afs prfpfsbhbflgs sbepngs3
p3 @eånh`r pugag sgr prgsbagltg.
q3 @eånh`r gs hbua`a`lf iu`tge`ntghf.
p3 @eånh`r pugag sgr prgsbagltg. (T)
q3 @eånh`r gs hbua`a`lf iu`tge`ntghf. (T)
N` prfpfsbhbøl hfepugst` gsvgra`agr , pfrqug @eånh`rgs hbua`a`lf iu`tge`ntghfy pfr t`ltf pugag fpt`r sgrprgsbagltg.
p3 @eånh`r pugag sgr prgsbagltg. (T)
~q3 @eånh`r lf gs hbua`a`lf iu`tge`ntghf. (D)
N` prfpfsbhbøl gs d`ns`. @n lfsgr hbua`a`lf iu`tge`ntghf,@eånh`r lf pugag fpt`r `npugstf ag prgsbagltg.
~ p3 @eånh`r lf pugag sgr prgsbagltg. (D)
q3 @eånh`r gs hbua`a`lf iu`tge`ntghf. (T)
N` prfpfsbhbøl gs d`ns`.@eånh`r så pugag fpt`r `nh`rif ag prgsbagltg pfrquggs hbua`a`lf iu`tge`ntghf.
~ p3 @eånh`r lf pugag sgr prgsbagltg. (D)
~q3 @eånh`r lf gs hbua`a`lf iu`tge`ntghf. (D)
N` prfpfsbhbøl gs vgra`agr .@eånh`r lf gs iu`tge`ntghfy pfr t`ltf lf puag pfstu-n`rsg p`r` prgsbagltg.
Agmgefs tglgr huba`af ag lf hfldulabr n`s rgn`hbflgs mbhflabhbfl`ngs hfl n`sólbh`egltg hflabhbfl`ngs.
8=BIG] ∝ [fnfhjbh
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Gcgrhbhbf =@. Ng` nfs glulhb`afs sbiubgltgs, pfli` `tglhbøl `n hflghtbvf nøibhf qug nfs ulg. Gshrbm` gn v`nfr
nøibhf ag p y q gl gn hu`arf hfrrgspflabgltg. Nugif, tr`lsdfreg n` prfpfsbhbøl hfepugst` `ngliu`cg sbemønbhf y agtgreblg su v`nfr ag vgra`a. Fmsgrvg y iuågsg pfr gn gcgepnf.
;) Xb Ànv`rf pugag vft`r, gltflhgs gs e`yfr ag ga`a. p3 T q3 T
0) Xb H`rnfs l`hbø gl Bz`m`n, gltflhgs gs jflaurgðf. p3 q3
=) Gn : gs ul lóegrf bep`r y gs eóntbpnf ag =.
p3 q3
8) Gn Xfn s`ng pfr gn Gstg y n` Nul` gs eàs ir`lag qug n` Qbgrr`. p3 q3
1) N` Qbgrr` gs gsdârbh` sb y sfnf sb gn Xfn gs ul` gstrgnn`. p3 q3
7) Xb gn `iu` e`rbl` gs s`n`a`, gltflhgs gs rghfegla`mng mgmgr `iu` s`n`a`. p3 q3
4) Iu`tge`n` tbglg =8 agp`rt`egltfs f su h`pbt`n gs E`z`tgl`lif. p3 q3
9) Gn = lf gs abvbsfr ag 0= f gn 0= gs eóntbpnf ag 8. p3 q3
:) Nfs h`l`rbfs lf j`mn`l sb y sfnf nfs nfrfs vugn`l. p3 q3
p q3 T
88E`tgeàtbh` ∝ Xge`l` =
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M. [bglsg y gshrbm` fhjf prfpfsbhbflgs hfepugst`s. Nugif, agtgreblg su v`nfr ag vgra`a. Iuågsgpfr gn gcgepnf.
;)
0)
=)
8)
1)
7)
4)
9)
:)
H. ]g`nbhg nfs prfhgabebgltfs qug sg pbagl gl h`a` hfnuel` ag n`s t`mn`s ag vgra`a. ]ghugrag nfqug `prglabø n` sge`l` `ltgrbfr sfmrg n`s lgi`hbflgs. Iuågsg pfr gn gcgepnf.
;) p ~ p q p ∪ q ~ p q
v d v v v
v d d v v
d v v v v
d v d d d
8) p q ~q p ~q p ∩ q
v v
v d
d v
d d
0) p ~ p q ~q ~ p ∪ ~q ~ p ∩ q
v v
v d
d v
d d
1) p q ~q p ∪ ~q p ∩ ~q
v v
v d
d v
d d
=) p ~ p q ~ p q q ∪ ~ p
v v
v d
d v
d d
7) p ~ p q ~q ~q ∩ ~ p p ~q
v v
v d
d v
d d
Gn lóegrf 8 gs ul lóegrf bep`r y gs eóntbpnf ag 8. T
81BIG] ∝ [fnfhjbh
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0. N`s prfpfsbhbflgs hfepugst`s sfl `qugnn`s dfre`a`s pfr afs f eàs prfpfsbhbflgs sbepngs qugsg ulgl ` tr`vâs ag hflghtbvfs nøibhfs.
=. Nfs hflghtbvfs nøibhfs sfl p`n`mr`s y såemfnfs qug sbrvgl hfef lgxf p`r` ulbr afs f eàsprfpfsbhbflgs sbepngs.
Nfs eàs hfeulgs sfl n` hflculhbøl, n` absyulhbøl, gn hflabhbfl`n y gn mbhflabhbfl`n.
=.0 N` hflculhbøl (∩) hfrrgspflag ` n` "y".
=.= N` absyulhbøl (∪) hfrrgspflag ` n` "f" y sg utbnbz` p`r` a`r sgltbaf ag gnghhbøl gltrg afs feàs gngegltfs.
=.8 Gn hflabhbfl`n ( ) f bepnbh`hbøl tfe` dfre` gl n`s p`n`mr`s "Xb… gltflhgs". Gstg hflghtbvfblabh` qug n` prbegr` prfpfsbhbøl gs rgqubsbtf p`r` qug sg huepn` n` sgiula`.
N`s p`r`afc`s ag n` bepnbh`hbøl sfl sbtu`hbflgs gl n`s qug ul` prfpfsbhbøl tbglg v`nfrnøibhf vgra`agrf, pgrf lf gxprgs` ul` bag` hfjgrgltg sgiól gn sgltbaf hfeól.
=.1 Gn mbhflabhbfl`n ( ) sg dfre` hfl n`s p`n`mr`s "sb y sfnf sb" g blabh` ul` rgn`hbøl ag agpgl-aglhb` eutu`. Gs aghbr, n` prbegr` sg huepng sfnf sb sg huepng n` sgiula` y vbhgvgrs`.
8. [`r` agtgrebl`r gn v`nfr ag vgra`a ag ul` prfpfsbhbøl hfepugst` sg gepng`l n`s rgin`s ag nfshflghtbvfs nøibhfs. Xg prgsglt`l gl n`s t`mn`s ag vgra`a sbiubgltgs3
]gsuegl
p q p ∩ q
v v vv d d d v d d d d
Hflculhbøl
p q p ∪ qv v vv d vd v vd d d
Absyulhbøl
p q p qv v vv d d d v vd d v
Hflabhbfl`n
p q p qv v vv d d d v d d d v
Mbhflabhbfl`n
Ng blvbt`efs ` vgr nfs vbagfs qug sg eugstr`l gl n`s abrghhbflgs sbiubgltgs, gstfs ng `yua`ràl ` prfdul-
abz`r gl `niulfs `spghtfs sfmrg nfs hflghtbvfs nøibhfs.• Hflghtfrgs nøibhfs y t`mn`s ag vgra`a3 iff.in/njucRy
• ¶Høef hflvgrtbr ul` prfpfsbhbøl gl ngliu`cg sbemønbhf6 iff.in/07OjGC
• Q mn`s ag vgra`a p`r` eàs ag afs hflghtfrgs nøibhfs3 iff.in/lszj`m
Q embâl ng prfpflgefs pr`htbh`r ul pfhf eàs y rgsfnvgr `niulfs gcgrhbhbfs gxtr`, qug pfarà glhfltr`rgl gn gln`hg sbiubgltg.
• Gcgrhbhbfs prfpugstfs ag hflghtbvfs nøibhfs y t`mn`s ag vgra`a3 iff.in/YXAMeN
87E`tgeàtbh` ∝ Xge`l` =
Blvgstbiug gl n` rga...
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@htbvba`a 0. Ageugstrg nf prglabaf
@. Ng` h`a` agdblbhbøl ` n` bzqubgra`. Nugif, rgn`hbølgn` hfl gn hflghtbvf nøibhf qug ng hfrrgspflaghfnfh`laf gn lóegrf gl gn p`râltgsbs hfrrghtf.
0) Blabh` rgn`hbøl ag agpglaglhb` eutu`. Gs aghbr, n` prbegr` prfpfsb-hbøl sg huepng sfnf sb sg huepng n` sgiula` y vbhgvgrs`.
=) A` sgltbaf ag gnghhbøl gltrg afs f eàs gngegltfs. Xu v nfr ag vgra`argsunt` d`nsf sfnf sb `em`s prfpfsbhbflgs sfl d`ns`s.
8) Gqubv`ng ` n` p`n`mr` "y" gl gn babfe` gsp`ðfn. Xu v`nfr nøibhf gs vgr-a`agrf sfnf sb `em`s prfpfsbhbflgs sfl hbgrt`s.
1) Q embâl nn`e`a` bepnbh`hbøl? blabh` hflabhbøl agn tbpf "Xb… gltflhgs".]gsunt` gl d`nsf hu`laf p gs vgra`agr` y q gs d`ns`, y gs vgra`agrf gl
hu`nqubgr ftrf h`sf.
M. Ng` nfs glulhb`afs sbiubgltgs y sumr`yg gn hflghtbvf qug `p`rghg gl h`a` ulf. Nugif, tr`lsdfregn`s prfpfsbhbflgs ` ngliu`cg sbemønbhf y gshråm`n`s gl n` nålg` ag n` agrghj`. Iuågsg pfr gn gcgepnf.
;) Xb gn `iu` agn fhâ`lf gs s`n`a` gltflhgs lf sg pugag mgmgr.
0) Gn h`htus tbglg gspbl`s y lghgsbt` pfh` `iu`.
=) N` nnuvb` h`g agn hbgnf y E`rtg gs ul pn`lgt`.
8) Xb gn c`iu`r tbglg e`lhj`s, gltflhgs sg pugag h`eudn`r.
1) Xb gn iu` gs blhfnfr`, gltflhgs sg pugag mgmgr.
7) Xb jfy gs sàm`af, gltflhgs e`ð`l` sgrà afeblif.
4) Gn r`tøl hfrrg ràpbaf f gn i`tf lf pugag s`nt`r.
9) Hfnfemb` gstà gl Xur`eârbh` y [frtui`n gl Gurfp`.
:) Gl Zgn` j`y dråf sb y sfnf sb gl S`h`p` j`y h`nfr.
H. @jfr` gs su turlf. Gshrbm` hu`trf prfpfsbhbflgs hfepugst`s y gst`mngzh` su v`nfr ag vgra`a.]ghugrag nf `prglabaf sfmrg n`s rgin`s ag nfs hflghtbvfs nøibhfs.
0)
=)
8)
1)
@utfhfltrfn
( ) hflculhbøl
( ) mbhflabhbfl`n
( ) hflabhbfl`n
( ) absyulhbøl
r s
84BIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
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@htbvba`a =. [r`htbqug nf prglabaf
@. ]g`nbhg nfs prfhgabebgltfs qug sg pbagl gl h`a` hfnuel` ag n`s t`mn`s ag vgra`a. ]ghugrag nf qug`prglabø sfmrg n`s lgi`hbflgs.
0) r ~r s ~s ~r ~s r ∪ s ~r s
v vv d
d v
d d
8) ` ~` m ~` ∪ m ` ∩ m
v vv d
d v
d d
=) p ~ p q ~q ~ p ~q ~q ∩ ~ p
v v
v d
d v
d d
1) r s ~s r ~s r s
v
v
d
d
M. Qr`lsdfreg h`a` prfpfsbhbøl sbiubgltg ` ngliu`cg sbemønbhf y agtgreblg su v`nfr ag vgra`a.
0) Gn ngøl gs jgrmåvfrf y n` fvgc` gs h`rlåvfr`. p3 q3
=) Glgrf gs gn prbegr egs agn `ðf y abhbgemrg gs gn óntbef.
p3 q3
8) Nfs h`m`nnfs sfl `lbe`ngs sb y sfnf sb Tglus lf gs ul pn`lgt`. p3 q3
1) Xb gn e`rtgs v` nugif agn nulgs, gltflhgs gn vbgrlgs v` `ltgs agn cugvgs. p3 q3
7) Gn `iu`h`tg vbglg ag n` e`zfrh` f n` Qbgrr` gs ul pn`lgt`. p3 q3
4) 02 gs eóntbpnf ag 7 sb y sfnf sb = gs abvbsfr ag :. p3 q3
89E`tgeàtbh` ∝ Xge`l` =
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@. ]gsugnv` eglt`negltg n`s euntbpnbh`hbflgs sbiubgltgs. Gshrbm` n` rgspugst` hfrrght` sfmrg n` nålg`hfrrgspflabgltg.
0) = Ù 8 Ù 0 5
=) 7 Ù 1 Ù : 5
8) 4 Ù 8 Ù = 5
1) 7 Ù 8 Ù 9 5
7) 0 Ù 7 Ù 2 5
4) = Ù 8 Ù ; 5
9) 7 Ù 4 Ù : 5
:) = Ù 8 Ù : 5
2) 0 Ù = Ù 7 5
0;) 7 Ù 1 Ù 9 5
00) : Ù 7 Ù ; 5
0=) 7 Ù 4 Ù 7 5
08) 7 Ù 1 Ù 4 5
01) 2 Ù : Ù 9 5
07) 8 Ù 7 Ù 1 5
04) 7 Ù 8 Ù 1 5
09) : Ù 9 Ù 1 5
0:) 4 Ù 2 Ù 7 5
02) 2 Ù 7 Ù 7 5
=;) : Ù 2 Ù 1 5
=0) : Ù 2 Ù 7 5
==) 8 Ù = Ù ; 5
=8) 9 Ù : Ù 0 5
=1) : Ù 7 Ù = 5
=7) 1 Ù 0 Ù 8 5
=4) : Ù 1 Ù : 5
=9) : Ù : Ù 9 5
=:) 2 Ù 4 Ù 2 5
=2) 9 Ù 1 Ù 0 5
8;) ; Ù 0 Ù ; 5
M. ]gsugnv` eglt`negltg n`s euntbpnbh`hbflgs y abvbsbflgs sbiubgltgs. Gshrbm` n` rgspugst` hfrrght`sfmrg n` nålg` hfrrgspflabgltg.
0) 7 Ù = ù = 5
=) = Ù : ù 1 5
8) : Ù 9 ù 1 5
1) 2 Ù 2 ù 8 5
7) 9 Ù 9 ù 0 5
4) 9 Ù 1 ù = 5
9) 0 Ù 0 ù 0 5
:) 4 Ù 2 ù 8 5
2) 7 Ù 4 ù 0; 5
0;) : Ù 8 ù = 5
00) 1 Ù 0 ù 1 5
0=) 9 Ù : ù : 5
08) 2 Ù 4 ù = 5
01) 1 Ù 4 ù 8 5
07) 8 Ù : ù = 5
04) 9 Ù 0 ù 0 5
09) : Ù 2 ù 4 5
0:) ; Ù 0 ù 8 5
02) : Ù : ù 1 5
=;) 9 Ù 2 ù 9 5
=0) : Ù 0 ù 1 5
==) 1 Ù 7 ù 0; 5
=8) 7 Ù : ù =; 5
=1) 0 Ù = ù 0 5
=7) 8 Ù = ù 0 5
=4) 1 Ù 8 ù 4 5
=9) 4 Ù 0 ù 8 5
=:) 8 Ù 2 ù 0 5
=2) ; Ù 0 ù 0 5
8;) 8 Ù 8 ù 2 5
8:BIG] ∝ [fnfhjbh
@ibnba`a ag hànhunf eglt`n
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@. H`nhung gn v`nfr ag h`a` ul` ag n`s ngtr`s sbiubgltgs, m`sàlafsg gl n`s fpgr`hbflgs blabh`a`s.Qgli` gl huglt` qug n`s fpgr`hbflgs gstàl rgn`hbfl`a`s, pgrf pugagl lf tglgr ul fragl nblg`n.
P 5 N 5 Q 5 C 5 @ 5
J 5 A 5 D 5 E 5 ] 5
0) 7 + A 5 7
=) D ‘ A 5 0
8) 8C + =D 5 :
1) =A + D + 0;C 5 ]
7) ] ù 8 5 J
4) ‘=J + ] + 9A + 4 5 N
9) P 5 D + C + D
:) 0;J ‘ (D + P) + 8N ‘ 8P 5 E
2) Q 5 8;C + N + (@ + D) ‘ J
0;) E ‘ 7P 5 Q
M. Hflsbagr`laf nfs lóegrfs qug fmtuvf `ltgrbfregltg, h`nhung gn v nfr qug tbglgl n`s ngtr`ssbiubgltgs.
I 5 O 5 T 5 B 5 G 5
L 5 M 5 [ 5 H 5 X 5
0) (P ù =) Ù 1 5 H
=) N ‘ 8D + A 5 T
8) J + M 5 0=
1) (8T) ù = + GM 5 4G
7) Q ù = ‘ L + H + T 5 1=
4) G + 7M 5 B
9) 4B + 00] ‘ (G + =D) 5 [
:) B ‘ =G + (P Ù H) 5 I
2) 4@ + (P Ù E) ‘ [ + (C Ù X) + B 5 ;
0;) 4J + (G Ù T) + (=M + N) ‘ (8P + D) 5 O
82E`tgeàtbh` ∝ Xge`l` =
]`zfl`ebgltf nøibhf
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E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfir`af gl
prfhgsflf
nfir`af
A g s p u â s a g g s t u a b `
r . . .
Hfeprglaf g bagltbhf n`s prfpfsbhbflgs hfepugst`s.
H`tgifrbzf n`s rgin`s ag nfs hflghtbvfs nøibhfs eàs utbnbz`afs.
Agtgreblf gn v`nfr ag vgra`a ag prfpfsbhbflgs hfepugst`s.
Hflstruyf t`mn`s ag vgra`a.
]gsugnvf fpgr`hbflgs hfembl`a`s hfl `ibnba`a.
@jfr` pflargefs ` prugm` lugstr`s lgurfl`s. Hfl nfs afebløs agmgefs ag-tght`r gn fragl nøibhf qug sbiugl n`s hj`s prgsglt`a`s. Gstg prfhgabebgltf lfs`yua` ` ags`rrfnn`r lugstr` j`mbnba`a ag agauhhbøl.
[`r` nfs gcgrhbhbfs sbiubgltgs, e`rqug hfl ul` "x" n` fphbøl qug hfepngt` n`sgrbg3
=)
`. h.m. a.
`. m. h. a.
0)
1)
h.m. a.`.
8)
`. h.m. a.
1;BIG] ∝ [fnfhjbh
]gvbsg su `prglabz`cg
Ags`rrfnng lugv`s j`mbnba`ags
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80
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
Lóegrfs rg`ngs¶Puâ glhfltr`rà gst` sge`l`6
Nfs pbt`iørbhfs y nfs lóegrfs brr`hbfl`ngs
Fpgr`hbflgs hfembl`a`s
Nfs lóegrfs rg`ngs, sus prfpbga`ags, rgin`s ag n` pftglhb`hbøl y ag n` r`abh`hbøl [rfmnge`s ag `pnbh`hbøl hfl lóegrfs rg`ngs
Euntbpnbh`hbøl hfl gn eâtfaf Qzgnt`n
Gst` sge`l` nfir`rà3
@pnbh`r n`s prfpbga`ags ag nfs lóegrfs rg`ngs.
]gsfnvgr fpgr`hbflgs hfembl`a`s.
]gp`s`r n`s ngygs ag n` pftglhb`hbøl y ag n` r`abh`hbøl ag nfs lóegrfs rg`ngs. @pnbh`r nfs lóegrfs rg`ngs gl n` rgsfnuhbøl ag prfmnge`s.
[r`htbh`r n` euntbpnbh`hbøl pfr gn eâtfaf Qzgnt`n.
Ags`rrfnn`r `ibnba`a ag hànhunf hfl fpgr`hbflgs hfembl`a`s.
10E`tgeàtbh` ∝ Xge`l` 8
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0. Lóegrfs rg`ngs (] )
]
B
P
S
L
...‘1, ‘8, ‘=, ‘0 ;, 0, =, 8, 1 ...0.=0=0=...
;.7
78
81
= ϋ
Gst` sge`l` hfl n` jbstfrb` ag nfs pbt`iørbhfs blbhb`efs gn gstuabf agn hflcultfag nfs lóegrfs brr`hbfl`ngs.
]ghfragefs qug ul lóegrf brr`hbfl`n gs `qugn qug lf pugag sgr gxprgs`afhfef dr`hhbøl y su p`rtg aghbe`n lf gs pgrbøabh`, gs aghbr, sus aghbe`ngs lf sb-iugl ul p`trøl p`r` rgpgtbrsg. Nfs lóegrfs brr`hbfl`ngs (B), cultf nfs l`tur`ngs(L), nfs gltgrfs (S) y nfs r`hbfl`ngs (P) dfre`l gn hflcultf ag nfs lóegrfs rg`ngs(] ), hfef sg bnustr` gl n` be`igl ag `rrbm`.
Nfs lóegrfs rg`ngs sfl nfs qug pugagl gxprgs`rsg pfr ul lóegrf gltgrf f pfrul aghbe`n, sg` pgrbøabhf f lf. Gstg hflcultf sg rgprgsglt` hfl n` ngtr` ] y sgaglg sbemønbh`egltg hfef3 ] 5 P ∣ B
0.0 H`r`htgråstbh`s ag nfs lóegrfs rg`ngs
• Nfs lóegrfs rg`ngs hflstbtuygl ul hflcultf bldblbtf pfrqug gstà dfre`afpfr n` ulbøl ag nfs hflcultfs ag lóegrfs r`hbfl`ngs (P) g brr`hbfl`ngs (B).
• ] gs ul hflcultf aglsf, gstf qubgrg aghbr qug gltrg afs lóegrfs r`hbfl`ngsj`y bldblbtfs brr`hbfl`ngs.
• ] gs ul hflcultf hfepngtf y hfltbluf, agmbaf qug tfaf pultf gl n` rght`lueârbh` sg pugag rgprgsglt`r hfl ul lóegrf rg`n. Nfs pultfs nbmrgs qugj`y gltrg afs lóegrfs r`hbfl`ngs sfl nngl`afs pfr nfs brr`hbfl`ngs, hfef
vgefs gl n` rght` qug gstà `m`cf.
N` hfltbluba`a ag nfs lóegrfs rg`ngs pgrebtg qug j`y` lóegrfs h`a` vgz eàshgrh`lfs ulfs ag ftrfs.
[fr gcgepnf3 =.2, =.22, =.222, =.2222… f 8.;;;0, 8.;;0, 8.;0, 8.0...
= 18
=.22
8.0
8.;0
=.2
18E`tgeàtbh` ∝ Xge`l` 8
Gn eulaf ag n` e`tgeàtbh`
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0.= T`nfr `msfnutf ag ul lóegrf rg`n
Gl e`tgeàtbh` sg ng nn`e` v`nfr msfnutf ag ul lóegrf rg`n, ` n` abst`lhb` qugj`y gltrg gn hgrf u frbigl y gstg lóegrf sbl bepfrt`r su sbilf, sg` pfsbtbvf (+)f lgi`tbvf (‘). Nf bagltbh`efs gshrbmbglaf gn lóegrf gltrg m`rr`s |`| y sg ngg"v`nfr `msfnutf ag `".
[fr gcgepnf3
`. Xb gn lóegrf gs pfsbtbvf f hgrf, su v`nfr `msfnutf gs gn ebsef lóegrf.
|81| 5 81
| 0
1| 5 0
1
m. Xb gn lóegrf gs lgi`tbvf, su v`nfr `msfnutf gs gn fpugstf agn lóegrf.
|‘8| 5 ‘(‘8) 5 8
|‘;.=7| 5 ‘(‘;.=7) 5 ;.=7
Ag gstfs rgsunt`afs pfagefs hflhnubr qug gn v`nfr `msfnutf ag ul lóegrfsbgeprg gs pfsbtbvf, pugs nf qug sg tfe` agn lóegrf gs n` abst`lhb` gltrg ân ygn hgrf.
|‘7| 5 7 |7| 5 7
‘4 ‘7 ‘1 ‘8 ‘= ‘0 ; 0 = 8 1 7 4
@quå lft`efs qug ‘7 y 7 tbglgl gn ebsef v`nfr `msfnutf pfrqug `emfs sgglhugltr`l ` n` ebse` abst`lhb` agn frbigl f hgrf.
Gcgrhbhbf 0Gshrbm` gn v`nfr `msfnutf ag nfs lóegrfs sbiubgltgs. Qbglg ul gcgepnf.
;) |‘ 1 | 5 9) |1.84=1| 5
0) |‘:| 5 :) |;| 5
=) |‘=7| 5 2) |‘8.=:41| 5
8) | 7
2| 5 0;) |
8 :| 5
1) |‘04| 5 00) |‘1.897| 5
7) |‘ =2| 5 0=) |‘ =7| 5
4) | ‘ :
09| 5 08) | ‘
8
7| 5
1
11BIG] ∝ [fnfhjbh
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=. Fpgr`hbflgs hfembl`a`sHfl nfs lóegrfs rg`ngs sg pugagl rg`nbz`r abdgrgltgs fpgr`hbflgs y n`s qugrgólgl v`rb`s fpgr`hbflgs gl ul` sfn` sg ngs nn`e` fpgr`hbflgs hfembl`a`s.
Gl n`s fpgr`hbflgs hfembl`a`s pugagl p`rghgr sue`s, rgst`s, euntbpnbh`hbflgs,
abvbsbflgs, pftglhb`s, r`åhgs y sbilfs ag irup`hbøl. [`r` rg`nbz`rn`s, gs lghgs`rbfrgspgt`r n` cgr`rquå` ag n`s fpgr`hbflgs. [rgstg `tglhbøl ` nfs h`sfs sbiubgltgs3
`. Xb n`s fpgr`hbflgs lf blhnuygl sbilfs ag `irup`hbøl, prbegrf sg rgsugnvgln`s euntbpnbh`hbflgs y abvbsbflgs gl gn fragl qug p`rghgl, nugif sg gdghtó`ln`s `abhbflgs y sustr`hhbflgs hfrrgspflabgltgs.
Gcgepnf3
• [rbegrf, rg`nbz`efs n`s euntbpnbh`hbflgsy abvbsbflgs.
• Nugif, n`s `abhbflgs y sustr`hhbflgs.
Ftrf gcgepnf3
• [rbegrf, rg`nbz`efs n` euntbpnbh`hbøl yabvbsbøl.
• Nugif, n` `abhbøl y sustr`hhbøl.
m. Xb n`s fpgr`hbflgs blhnuygl sbilfs ag `irup`hbøl, gstfs sg gnbebl`l ag
`agltrf j`hb` dugr` y sg gdghtó`l n`s fpgr`hbflgs blabh`a`s agltrf ag gnnfs.Gcgepnf3
• [rbegrf, rg`nbz`efs n` fpgr`hbøl quggstà agltrf ag nfs p`râltgsbs.
• Nugif, n` sustr`hhbøl qug gstà gl nfshfrhjgtgs.
• R pfr óntbef n` `abhbøl.
Ftrf gcgepnf3
• [rbegrf, rg`nbz`efs n` fpgr`hbøl quggstà agltrf ag nfs p`râltgsbs.
• Nugif, n` qug gstà agltrf ag nfs hfrhjg-tgs.
• Agspuâs n`s fpgr`hbflgs qug gstàlagltrf ag n`s nn`vgs.
• [fr óntbef, n` sustr`hhbøl.
7 • = + 8 ‘ = • 8 + 9 ‘ 1 ù = 5
0; + 8 ‘ 4 + 9 ‘ = 5 0=
0
1 •
7
8 +
7
2 ù
0
= ‘
7
4 5
7
0= +
0;
2 ‘
7
4 5
07 + 1; ‘ 8;
84 5
=7
84
07 + _: ‘ (= • 8)U 5
07 + _: ‘ 4U 5
07 + = 5 09
7 ‘ {0; ‘ _4 ‘ (07 ù 8)U + :} 5
7 ‘ {0; ‘ _4 ‘ 7U + :} 5
7 ‘ {0; ‘ 0 + :} 5
7 ‘ 09 5 ‘0=
]ghugrag gn prfhgab-ebgltf p`r` sue`r frgst`r dr`hhbflgs hflabdgrgltg aglfebl`afr.
17E`tgeàtbh` ∝ Xge`l` 8
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8. [rfpbga`ags ag nfs lóegrfs rg`ngsNfs lóegrfs rg`ngs huepngl hfl hbgrt`s h`r`htgråstbh`s f rgin`s gl sus abdgrgl-tgs fpgr`hbflgs nn`e`a`s prfpbga`ags. [`r` hfeprglagr gst`s prfpbga`ags, gslghgs`rbf hflsbagr`r qug n`s ngtr`s `, m y h pugagl rgprgsglt`r hu`nqubgr lóegrfrg`n, gs aghbr, tfe`r hu`nqubgr v`nfr. Tg`efs n`s prblhbp`ngs prfpbga`ags ag nfs
lóegrfs rg`ngs.
Hn`usur`tbv`
Gn rgsunt`af ag n` fpgr`hbøl gltrg lóegrfs rg`ngs a`hfef rgsunt`af ftrf lóegrf rg`n.
` + m 5 h
=7 + 7 5 8;
Gngegltf lgutrf abtbvf
@n sue`r ul lóegrf rg`n hfl gn ; a` hfef rgsunt`af
gn ebsef lóegrf.
` + m 5 `
=.44 + ; 5 =.44
Blvgrsf `abtbvf u fpugstf
[`r` tfaf lóegrf ` gxbstg ftrf lóegrf ‘ nn`e`afblvgrsf `abtbvf u fpugstf t`n qug3
` + (‘ ) 5 ;
8
1 + ( ‘
8
1 ) 5 ;
N` `abhbøl gs hfleut`tbv`
Gn fragl gl qug sg fpgrgl nfs sue`lafs lf `ntgr` gnrgsunt`af.
` + m 5 m + `
0= + 8 5 8 + 0=
07 5 07
N` euntbpnbh`hbøl gs hfleut`tbv`
Gn fragl gl qug sg fpgrgl nfs d`htfrgs lf `dght` gn
prfauhtf.
` • m 5 m • `
2 • 4 5 4 • 271 5 71
N` euntbpnbh`hbøl gs `sfhb`tbv`
N` e`lgr` gl qug sg `irup`l nfs d`htfrgs lf `ntgr`gn prfauhtf.
(m • h) 5 ( • m)h
1(= • :) 5 (1 • =):
1(04) 5 (:):
41 5 41
N` `abhbøl gs `sfhb`tbv`
Gn fragl gl qug sg `irup`l nfs sue`lafslf `ntgr` gn rgsunt`af.
( + m) + h 5 ` + (m + h)
(=.7 + 4) + 0; 5 =.7 + (4 + 0;)
:.7 + 0; 5 =.7 + 04
0:.7 5 0:.7
14BIG] ∝ [fnfhjbh
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Blvgrsf euntbpnbh`tbvf
[`r` tfaf lóegrf ` gxbstg ftrf lóegrf0
` nn`e`af
blvgrsf euntbpnbh`tbvf f sbeâtrbhf euntbpnbh`tbvf t`nqug3
` •0
` 5 0
7 •0
7 5 0
7
7 5 0
0 5 0
Abstrbmutbv` agn prfauhtf rgspghtf ` n` sue`
Gn prfauhtf ag ul lóegrf pfr ul` sue`
gs biu`n ` euntbpnbh`r gn lóegrf pfr h`a`ulf ag nfs sue`lafs.
`(m + h) 5 `m + `h
7(1 + 4) 5 (7 • 1) + (7 • 4)7(0;) 5 =; + 8;
7; 5 7;
Abstrbmutbv` agn prfauhtf rgspghtf ` n` rgst`
Gn prfauhtf ag ul lóegrf pfr ul` rgst`gs biu`n ` euntbpnbh`r gn lóegrf pfr gnebluglaf y pfr gn sustr`glaf.
`(m ‘ h) 5 `m ‘ `h
8(: ‘ =) 5 (8 • :) ‘ (8 • =)
8(4) 5 =1 ‘ 4
0: 5 0:
Gngegltf lgutrf ag n` euntbpnbh`hbøl
@n euntbpnbh`r ul lóegrf rg`n pfr 0 lf `ntgr` gnrgsunt`af.
` • 0 5 `
‘;.=7 • 0 5 ‘;.=7
Gcgrhbhbf =Fmsgrvg n` prfpbga`a qug gcgepnbh` h`a` gcgrhbhbf. Nugif, gshrbm` gn lfemrg sfmrg n` nålg` ag n`agrghj`. Qbglg ul gcgepnf.
;) : • 7 5 7 • :
0) 0; + 17 5 77
=)8
1 + ( ‘
8
1 ) 5 ;
8) 0= • 4 5 4 • 0=
1) 04 •0
04 5 0
7) ‘8.0104 • 0 5 ‘8.0104
4) (0
1 + =) + 7 5
0
1 + (= + 7)
9) 1 (8 + 0) 5 ( 1 • 8) + ( 1 • 0)
hfleut`tbv`
19E`tgeàtbh` ∝ Xge`l` 8
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1. [ftglhb`hbøl ag lóegrfs rg`ngsN` pftglhb`hbøl ag nfs lóegrfs rg`ngs sbrvg p`r` gshrbmbr ag dfre` ràpba` ullóegrf qug sg euntbpnbh` v`rb`s vghgs pfr så ebsef. Wl` pftglhb`, hfef sg nghflfhg, gstà dfre`a` pfr nfs gngegltfs sbiubgltgs3
`lm`sg gxpflgltg
• N` m`sg gs gn lóegrf qug sg euntbpnbh` pfr så ebsef.
• Gn gxpflgltg blabh` gn lóegrf ag vghgs qug n` m`sg sg euntbpnbh` pfr såebse`. Ag dfre` sbemønbh` sg rgprgsglt` `så3
`l 5 ` • ` • ` . . . • `
1.0 ]gin`s ag nfs gxpflgltgs p`r` lóegrfsrg`ngs
[`r` `pnbh`r n` pftglhb`hbøl ` nfs lóegrfs rg`ngs, gs lghgs`rbf hflfhgr n`s prf-pbga`ags qug nfs rbigl. Gst`s sfl3
Gn gxpflgltg hgrf
Qfaf lóegrf gngv`af n gxpflgltg hgrf a` hfefrgsunt`af 0.
`; 5 0
7; 5 0
(‘1); 5 0
Gn gxpflgltg lgi`tbvf
[`r` gngv`r ul lóegrf gltgrf ` ul` pftglhb`lgi`tbv`, `pnbh`efs su blvgrsf, pgrf hflgxpflgltg pfsbtbvf.
`‘l 50
`l
2 ‘8 50
28
(‘=) ‘7 50
(‘=)7
[`r` gngv`r ul` dr`hhbøl ` ul` pftglhb` lgi`tbv`,blvgrtbefs n` dr`hhbøl y gn gxpflgltg sg hflvbgrtggl pfsbtbvf.
( `
m )‘l
5 ( m
` )l
( =
1 ) ‘8
5 ( 1
= )8
( :
8 ) ‘=
5 ( 8
: )=
1:BIG] ∝ [fnfhjbh
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N` euntbpnbh`hbøl ag pftglhb`s hfl biu`n m`sg
[`r` euntbpnbh`r pftglhb`s ag biu`n m`sg, sghfpb` n` m`sg y sg sue`l nfs gxpflgltgs.
`l • `e 5 `l + e
1= • 18 5 1= + 8 5 17
(‘=)7 • (‘=)8 5 (‘=)7 + 8 5 (‘=):
[ftglhb` ag ul` pftglhb`
[`r` gngv`r ul` pftglhb` ` ftr` pftglhb`,hfpb`efs n` m`sg y euntbpnbh`efs nfs gxpf-lgltgs.
( l)e 5 ( )l • e
(7=)1 5 (7)= • 1 5 (7):
(0;8)7 5 (0;)8 • 7 5 (0;)07
[ftglhb` ag ul prfauhtf
[`r` gngv`r ul prfauhtf ` ul` pftglhb`,gngv`efs h`a` d`htfr `n gxpflgltg a`af.
( • m)l 5 `l • ml
(= • :)4 5 =4 • :4
(4 • 1)8 5 48 • 18
[ftglhb` ag ul hfhbgltg hfl m`sgs biu`ngs
[`r` rgauhbr ul` dr`hhbøl hfl biu`n m`sg, sghfpb` n` m`sg y sg rgst`l nfs gxpflgltgs.
`l
`e 5 `l ‘ e
47
48 5 47 ‘ 8 5 4=
29
24 5 29 ‘ 4 5 2
[ftglhb` ag ul` dr`hhbøl
[`r` gngv`r ul hfhbgltg ` ul` pftglhb`, sggngv` gn luegr`afr y gn aglfebl`afr n gxpf-lgltg blabh`af.
( `
m )l
5`l
ml
( 7
= ):
57:
=:
( 2
9 )1
521
91
Gcgrhbhbf 8@pnbqug n`s rgin`s ag n` pftglhb`hbøl p`r` rgsfnvgr nfs gcgrhbhbfs sbiubgltgs. Qbglg ul gcgepnf.
;) :8 • :4 5 7) (‘8); 5
0) 4; 5 4) 29
24 5
=) ( 0
= )=
( 0
= )=
5 9) ( =
7 )8
5
8) ( 8
9 )1
5 :) : ‘= 5
1)7:
78 5 2) (‘=) ‘1 5
:8 + 4 5 :2
12E`tgeàtbh` ∝ Xge`l` 8
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7. ]`abh`hbøl ag lóegrfs rg`ngsN` r`abh`hbøl gs n` fpgr`hbøl blvgrs` ` n` pftglhb`hbøl, hflsbstg gl j`nn`r n`r`åz ag ul lóegrf a`af, t`n qug, `n gngv`rnf ` n` pftglhb` blabh`a` sg fmtgli`lugv`egltg n` ebse` h`ltba`a. Nfs gngegltfs ag n` r`abh`hbøl sfl3
l
` 5 m r`åz
r`abh`laf
sbilf r`abh`n
ålabhg
• Gn ålabhg blabh` gn ir`af ag n` r`åz qug sg agmg gxtr`gr.
• Gn sbilf r`abh`n blabh` n` h`ltba`a ` n` qug sg ng gxtr`g n` r`åz.
• Gn r`abh`laf gs gn lóegrf `n qug sg ng gxtr`g n` r`åz.
• N` r`åz gs gn rgsunt`af ag n` r`abh`hbøl.
7.0 [rfpbga`ags ag nfs r`abh`ngs p`r` lóegrfsrg`ngs
N` r`abh`hbøl t`embâl huepng hfl hbgrt`s prfpbga`ags qug sfl m`st`ltg p`rg-hba`s ` n`s prfpbga`ags ag n` pftglhb`hbøl, y` qug ul` r`åz gs ul` pftglhb` hflgxpflgltg r`hbfl`n.
]`åz ag ul prfauhtf
N` r`åz ag ul prfauhtf gs biu`n `n prfauhtf
ag n`s r`åhgs ag nfs r`abh`lafs hfl gn ebsefålabhg.
` • m 5 ` • m
1 • =7 5 1 • =78 : • : 5
8 : •
8 :
]`åz ag ul hfhbgltg
N` r`åz ag ul` dr`hhbøl gs biu`n `n hfhbgltg agn` r`åz agn luegr`afr gltrg n` r`åz agn aglf-ebl`afr.
l`
m 5
l `
l m
41
1 5 41
1
8
:
=9
5
8 :
8 =9
]`åz ag ftr` r`åz
Hu`laf ul` r`åz gstà `dght`a` pfr ftrf r`ab-h`n, sg euntbpnbh`l nfs ålabhgs ag n`s r`åhgs, sghflsgrv` gn r`abh`laf y sg agc` hfef ul sfnfr`abh`n.
l e 5
l • e `
8 9 5
= • 89 5
4 9
1 7 0; 5
1 • 70; 5
=; 0;
7;BIG] ∝ [fnfhjbh
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]gsuegl
]`åz ag ul` pftglhb`Hu`laf gn r`abh`laf hfltbglg ul` pftglhb` hfl lóegrfs rg`ngs sg pugaglprgsglt`r afs h`sfs.
`. Xb gn ålabhg agn r`abh`n gs biu`n qug gn gxpflgltg agnr`abh`laf, `emfs sg gnbebl`l.
8
78
58
78
5 7
4= 5 4= 5 4
l`l 5
l`l 5 `
m. Xb gn ålabhg gs abdgrgltg `n gxpflgltg agn r`abh`laf,sg pugag tr`lsdfre`r gl ul` dr`hhbøl y sb gs pfsbmngsg pugag sbepnbdbh`r.
e l 5 `l/e
81 5 81/= 5 8=
1 20= 5 20=/1 5 28
Gcgrhbhbf 1@pnbqug n`s prfpbga`ags ag n` r`abh`hbøl p`r` rgsfnvgr nfs gcgrhbhbfs sbiubgltgs. Aflag sg` pfsbmng
gxprgsg n` rgspugst` hfef ul lóegrf gltgrf. Qbglg ul gcgepnf.
;)8 44 5 1)
88 5
0)41
1 5 7)
04
1 5
=) 0;= 5 4)7 17 5
8)1 04 • 04 5 9) :4 5
4 4/8 5 4 = 5 84
0. Gn hflcultf ag nfs lóegrfs rg`ngs sg rgprgsglt` hfl n` ngtr` ] y sg agdblg sbemønbh`egltg hfef] 5 P ∣ B. Gstfs lóegrfs sg pugagl gxprgs`r pfr ul gltgrf f ul aghbe`n, sg` pgrbøabhf f lf.
0.0 Nfs lóegrfs rg`ngs sg h`r`htgrbz`l pfr sgr ul hflcultf dblbtf, aglsf, hfepngtf y hfltbluf.
0.= Gn v`nfr `msfnutf ag ul lóegrf rg`n gs n` abst`lhb` qug j`y gltrg gn hgrf u frbigl y gstglóegrf. Xu rgsunt`af sbgeprg gs pfsbtbvf.
=. N`s fpgr`hbflgs hfembl`a`s sfl v`rb`s fpgr`hbflgs qug sg rgólgl gl ul` sfn`. [`r` rgsfnvgrn`s,gs lghgs`rbf rgspgt`r n` cgr`rquå` ag n`s fpgr`hbflgs.
8. Gn hflcultf ag nfs lóegrfs rg`ngs huepng hfl abdgrgltgs prfpbga`ags.
1. N` pftglhb`hbøl ag lóegrfs rg`ngs gs n` dfre` ràpba` ag gshrbmbr ul lóegrf qug sg euntbpnbh`v`rb`s vghgs pfr så ebsef.
7. N` r`abh`hbøl gs n` fpgr`hbøl blvgrs` ` n` pftglhb`hbøl, hflsbstg gl j`nn`r n` r`åz ag ul lóegrfa`af.
70E`tgeàtbh` ∝ Xge`l` 8
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@htbvba`a =. [r`htbqug nf prglabaf
@. @pnbqug n`s prfpbga`ags ag n` pftglhb`hbøl gl nfs gcgrhbhbfs sbiubgltgs y gxprgsg su rgspugst` hfeful lóegrf rg`n, gl `niulfs h`sfs gs lghgs`rbf `pnbh`r eàs ag ul` rgin`. Qbglg ul gcgepnf.
;) ( 7
= ) ‘8
5 00) ‘94
‘9: 5
0) 8; 5 0=):0=
:0; 5
=) 2 ‘= 5 08)0;4
0;1 5
8)1:
14 5 01) ( =
1 ) ‘=
5
1) 0;; 5 07) ( ‘ 0
7 )8
5
7) =; ‘= 5 04) ‘0=8
‘0== 5
4) (7=)= 5 09) 8= • 8= 5
9) ( 8
= )1
5 0:) (8 • 7)= 5
:) (‘8) ‘8 5 02) (;.=7)8 5
2) (‘07); 5 =;) (‘7 • 1)8 5
0;) (‘88)= 5 =0) ( 8
= )=
( 8
= )8
5
M. @pnbqug n`s prfpbga`ags ag n` r abh`hbøl gl nfs gcgrhbhbfs sbiubgltgs. Xbepnbdbqug n` rgspugst` aflagsg` pfsbmng. Qbglg ul gcgepnf.
;)8 : • 41 5 2)
7 07 5
0)4 88 5 0;)
=7
04 5
=): 8: 5 00)
4 1 0; 5
8)1
8 5 0=) 8=04
: 5
1) 0=1 5 08) ( 81 )
;
5
7):0
2 5 01)
12
0;; 5
4)2
04 5 07)
8 1 0; 5
9) (0=.7)= 5 04)8 :4 5
:) 0;; • 1 5 09)8 =9 • 0=7 5
8 : •
8 41 5 = • 1 5 :
( = 7 )8
5=8
78 5
:
0=7
78E`tgeàtbh` ∝ Xge`l` 8
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]gsugnv` n`s fpgr`hbflgs rgspgt`laf n` cgr`rquå` ag n`s fpgr`hbflgs. Qbglg ul gcgepnf p`r` h`a`h`sf.
@.;) 7 Ù : ù 1 5
0) 2 Ù = ù 4 5
=) 8 Ù 1 ù 8 5
8) 4 Ù 8 ù 2 5
1) 2 Ù 4 ù 8 5
7) 1 Ù 4 ù : 5
4) 8 Ù 1 ù 4 5
9) = Ù : ù = 5
:) 9 Ù 1 ù 1 5
2) 0= ù 4 Ù 8 5
0;) =1 ù : Ù 7 5
00) 4; ù 4 Ù 1 5
0=) 07 ù 8 Ù : 5
08) =1 ù 4 Ù ; 5
01) 1: ù = Ù = 5
07) 0: ù 4 Ù 2 5
04) 07 ù 7 Ù 9 5
09) =7 ù 7 Ù ; 5
M.
;) 1 + = Ù 8 5
0) : + 7 Ù = 5
=) 4 + 1 Ù 1 5
8) 7 + 9 Ù 8 5
1) 0 + : Ù 9 5
7) 4 + 9 Ù = 5
4) 8 + = Ù 8 5
9) 9 + : Ù 7 5
:) 2 + 8 Ù ; 5
2) : + 7 Ù 1 5
0;) 4 Ù 7 + 2 5
00) 1 Ù 9 + 0 5
0=) 9 Ù 8 + 9 5
08) 0; Ù = + : 5
01) 07 Ù 0 + 4 5
07) =7 Ù 1 + = 5
04) 7; Ù 0 + 2 5
09) =; Ù 8 + 7 5
0:) :; Ù = + : 5
02) 8; Ù 8 + 1 5
=;) 4; Ù 7 + 2 5
H.
;) 1 + 1 ù = 5
0) : + 2 ù 8 5
=) 2 + 7 ù 7 5
8) 1 + : ù 1 5
1) 0 + 4 ù 8 5
7) 7 + 8 ù 8 5
4) 1 + 0= ù 4 5
9) 9 + =; ù 1 5
:) 2 + 0: ù 2 5
2) 0 + 8; ù 8 5
0;) 0; ‘ 1 ù = 5
00) 0= ‘ 4 ù 8 5
0=) 07 ‘ 2 ù 8 5
08) 04 ‘ : ù : 5
01) =; ‘ 4 ù = 5
07) 0: ‘ 0= ù 1 5
04) 8; ‘ 0: ù 2 5
09) 0; ‘ =7 ù 7 5
0:) :; ‘ 07 ù 8 5
02) 7; ‘ =7 ù 7 5
=;) 14 ‘ 0: ù 4 5
0;
0;
4
71BIG] ∝ [fnfhjbh
@ibnba`a ag hànhunf eglt`n
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]gsugnv` gl su hu`agrlf nfs prfmnge`s sbiubgltgs. ]g`nbhg n`s fpgr`hbflgs `rbteâtbh`s qug lghgsbtg.
0) Wl `utfmós vb`c` ` ul` vgnfhba`a ag 92 oe pfr jfr`. @n h`mf ag 1 jfr`s, ¶huàltfs obnøegtrfs ng
d`nt`ràl p`r` dbl`nbz`r ul vb`cg ag 104 oe6 =) @vgrbiþg huàltfs eblutfs j`l tr`lshurrbaf agsag n`s 0; jfr`s y 07 eblutfs j`st` n`s 04 jfr`s y
=; eblutfs agn ebsef aå`.
8) Nubs vglabø gn nulgs =9 nbmr`s ag e`lz`l`s, gn e`rtgs gn afmng qug gn nulgs, y gn ebârhfngs n`tgrhgr` p`rtg qug gn nulgs y gn e`rtgs cultfs. ¶Huàlt`s nbmr`s ag e`lz`l`s vglabø6
1) Wl` pgrsfl` qug tr`m c` gl ul supgregrh`af gltr` gl gn hu`rtf ag n`s vgraur`s qug gstà ` 1 »H,nugif p`s` `n hu`rtf ag nfs jgn`afs qug sg glhugltr` ` ‘=; »H. ¶Huàn gs n` v`rb`hbøl gl n`tgepgr`tur`6
7) Gl gn t`lqug ag `iu` ag n` hfeulba`a j`y 0:;; nbtrfs ag `iu`. Xb pfr ul tumf blirgs`l 7; nbtrfspfr eblutf y `n ebsef tbgepf pfr ftrf tumf s`ngl 8; nbtrfs pfr eblutf, ¶huàltfs nbtrfs ag `iu`j`mrà gl gn agpøsbtf agspuâs ag 8; eblutfs6
4) Wl prgebf ag P=,4;;.;; dug rgp`rtbaf gltrg trgs pgrsfl`s ag n` e`lgr` sbiubgltg3 ` n` prbegr` nghfrrgspflabø 1/2 ag gs` h`ltba`a, ` n` sbiubgltg 0/8 y ` n` óntbe` gn rgstf. ¶Puâ h`ltba`a rghbmbøh`a` ul` y quâ dr`hhbøl agn ablgrf rghbmbø n` tgrhgr`6
9) [`r` n` blst`n`hbøl gnâhtrbh` ag ul` h`s` sg hfeprø ul rfnnf ag `n`emrg ag 7; egtrfs ag nflibtua.Xb sg hfrt`rfl 0;, 7.4, =.7 y :.97 egtrfs, ¶quâ h`ltba`a ag `n`emrg qugaø6
:) Wl lóegrf ag nftgrå` gstà dfre`af pfr 0; h`hjbtfs. Xb gn prgebf e`yfr ag n` nftgrå` gs P7;,;;;.;;y dug i`l`af pfr ]utj qug tbglg 1, C`hbltf 7 y Qfe`s` 0 h`hjbtf, ¶huàltf ablgrf ng hfrrgspflag` h`a` ulf, sgiól gn lóegrf ag h`hjbtfs qug tbglgl6
2) Wl` mfem` gxtr`g `iu` ag ul pfzf ` 7; e ag prfdulaba`a y n` gngv j`st` ul agpøsbtf qug sgglhugltr` ` =; e ag `ntf. ¶Huàltfs egtrfs gs gngv`a` gn `iu`6
0;) N` d`ebnb` [ârgz rg`nbz` ul `lànbsbs ag su ghflfeå` hfl nfs blirgsfs sbiubgltgs3 afð` H`regnbl`tr`m`c` trgs jfr`s ab`rb`s nulgs, ebârhfngs y vbgrlgs? nfs e`rtgs y cugvgs sfnf tr`m`c` afs jfr`s.[fr h`a` jfr` tr`m`c`a` ng p`i`l P=;.;;.
Gn gspfsf ag afð` H`regnbl` tbglg ul sugnaf ag P8,4;;.;; n egs y su jbcf Cfsâ, qug sfnf tr`m`c`trgs aå`s ` n` sge`l`, i`l` n` ebt`a qug su p`arg. Nfs trgs `pfrt`l n` tft`nba`a ag sus s`n`rbfs.Hfl n` bldfre`hbøl prfpfrhbfl`a` rgspfla` n`s prgiult`s sbiubgltgs3
`. ¶Huàltf ablgrf blirgs` ` n` d`ebnb` gl afs sge`l`s6
m. N` d`ebnb` agstbl` 0/1 ag nfs blirgsfs ag 07 aå`s p`r` p`i`r gn `nqubngr ag n` h`s`? y p`r`gn i`stf ag n` hfeba` agstbl`l P=7;.;; sge`l`ngs. ¶Huàltf j`mràl `jfrr`af h`a` afssge`l`s6
77E`tgeàtbh` ∝ Xge`l` 8
]`zfl`ebgltf nøibhf
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1 0
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
Lóegrfs hfepngcfs
¶Puâ glhfltr`rà gst` sge`l`6
]`åhgs p`rgs g bep`rgs
Lóegrfs hfepngcfs y fpgr`hbflgs `rbteâtbh`s
Hànhunf eglt`n ag fpgr`hbflgs hfembl`a`s
[rfmnge`s ag sghuglhb`s nøibh`s
Gst` sge`l` nfir`rà3 Bagltbh`r gn hflcultf ag nfs lóegrfs hfepngcfs.
Fmtglgr r`åhgs p`rgs g bep`rgs ag lóegrfs pfsbtbvfs y lgi`tbvfs.
Gshrbmbr lóegrfs hfepngcfs purfs y gl dfre` mblfeb`n.
Agtgrebl`r gn hflcui`af ag lóegrfs hfepngcfs.
]g`nbz`r fpgr`hbflgs `rbteâtbh`s hfl lóegrfs hfepngcfs.
]gsfnvgr fpgr`hbflgs hfembl`a`s hfl `ibnba`a.
79E`tgeàtbh` ∝ Xge`l` 1
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¡[`r` hfeglz`r!]`åhgs p`rgs g bep`rgs
l
`
N` sge`l` `ltgrbfr `prglabefs sfmrg nfs lóegrfs rg`ngs y n` r`abh`hbøl. Gs-
tuab`efs n`s r`åhgs hu`ar`a`s y hómbh`s, pgrf nf hbgrtf gs qug n`s gxprgsbflgs
r`abh`ngs pugagl tglgr hfef ålabhg hu`nqubgr lóegrf gltgrf pfsbtbvf.
N`s gxprgsbflgs r`abh`ngs qug tbglgl hfef ålabhg =, 1, 4,… f hu`nqubgr lóegrf
gltgrf p`r pfsbtbvf rghbmgl gn lfemrg ag r`åhgs p`rgs.
Gcgepnfs ag r`åhgs p`rgs
=7 5 7 pfrqug 7= 5 7 • 7 5 =7
1 =74 5 1 pfrqug 11 5 1 • 1 • 1 • 1 5 =74
40
41 5 0
= pfrqug (0
= )4
5 (0
= )(0= )(0= )(0= )(0= )(0= ) 5 0
41
Gstf lfs ageugstr` qug n`s r`åhgs p`rgs ag hu`nqubgr lóegrf rg`n pfsbtbvf a`
hfef rgsunt`af ul lóegrf rg`n lf lgi`tbvf.
[fr ftr` p`rtg, n`s gxprgsbflgs r`abh`ngs qug tbglgl hfef ålabhg 8, 7, 9,… f
hu`nqubgr lóegrf gltgrf bep`r pfsbtbvf, sg ng nn`e`l r`åhgs bep`rgs.
Gcgepnfs ag r`åhgs bep`rgs
8 : 5 = pfrqug =8 5 = • = • = 5 :
8 ‘: 5 ‘= pfrqug (‘=)8 5 (‘=)(‘=)(‘=) 5 ‘:
7 =18 5 8 pfrqug (8)7 5 8 • 8 • 8 • 8 • 8 5 =18
7 ‘=18 5 ‘8 pfrqug (‘8)7 5 (‘8)(‘8)(‘8)(‘8)(‘8) 5 ‘=18
[fagefs lft`r qug n` r`åz bep`r ag ul lóegrf pfsbtbvf gs ul lóegrf pfsbtbvf
y n` r`åz bep`r ag ul lóegrf lgi`tbvf a` hfef rgsunt`af ul lóegrf lgi`tbvf.
¡@ tr`m`c`r!]gspfla`3 ¶Puâ p`s`rå` sb ` ul lóegrf lgi`tbvf sg bltglt`r` gxtr`grng ul` r`åz p`r6 [rugmg hflhu`nqubgr h`ltba`a y gshrbm` su rgsunt`af.
7:BIG] ∝ [fnfhjbh
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Gcgrhbhbf 0@. ]gsugnv` n`s r`åhgs p`rgs ag lóegrfs pfsbtbvfs y j`i` n` hfep`r`hbøl hfl n` pftglhb`hbøl.
Qbglg ul gcgepnf.
;) =74 5 ? 5 4) ==7 5 ? 5
0) 2 5 ? 5 9) 1;; 5 ? 5
=) 04 5 ? 5 :) 2;; 5 ? 5
8) :0 5 ? 5 2) 0=0 5 ? 5
1) 011 5 ? 5 0;) 84;; 5 ? 5
7) 8=1 5 ? 5 00) =7;; 5 ? 5
M. ]gsugnv` n`s r`åhgs bep`rgs ag lóegrfs pfsbtbvfs y lgi`tbvfs. J`i` n` hfep`r`hbøl hfl n`
pftglhb`hbøl. Qbglg ul gcgepnf.
;) 8=9 5 ? 5 4) 8
‘=9 5 ? 5
0) 841 5 ? 5 9) 8
‘41 5 ? 5
=) 78= 5 ? 5 :) 7
‘8= 5 ? 5
8) 80=7 5 ? 5 2) 8
‘0=7 5 ? 5
1) 80;;; 5 ? 5 0;) 8
‘0;;; 5 ? 5
7) 9=0:9 5 ? 5 00) 9
‘=0:9 5 ? 5
H. @pnbqug sus hflfhbebgltfs sfmrg r`åhgs p`rgs lgi`tbv`s gl nfs h`sfs sbiubgltgs.Iuågsg pfr gn gcgepnf.
;) ‘1;; 5
0) ‘04 5
=) ‘=7 5
8) ‘84 5
1) ‘41 5
7) ‘042 5
4) ‘0;; 5
9) ‘024 5
04 04 = =74
8 88 =9
1;;(‘0) 5 1;;b= 5 1;; • b= 5 =;b
4;BIG] ∝ [fnfhjbh
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=. Xue` y rgst` ag lóegrfshfepngcfs
Hfl nfs lóegrfs hfepngcfs, `n biu`n qug hfl nfs lóegrfs rg`ngs, t`embâl sgpugagl rg`nbz`r n`s fpgr`hbflgs `rbteâtbh`s màsbh`s. @prgla`efs n` sue` y n`
rgst`. [rgstg `tglhbøl ` nfs p`sfs y ` nfs gcgepnfs.
0. Gshrbm` tfafs nfs lóegrfs be`ibl`rbfs gl n` dfre` mb.
=. Xueg f rgstg n`s p`rtgs rg`ngs ag nfs lóegrfs hfepngcfs.
8. Xueg f rgstg n`s p`rtgs be`ibl`rb`s ag nfs lóegrfs hfepngcfs.
1. Gshrbm` gn rgsunt`af ag n` dfre` ` + mb.
Xuegefs (8 + =b) + (: + 1b) 5
• Gnbebl`efs nfs p`râltgsbs. 8 + =b + : + 1b 5
• Fragl`efs nfs târeblfs. 8 + : + =b + 1b 5
• ]gauhbefs târeblfs sgegc`ltgs. 00 + 4b
Xuegefs (: + 7b) + (‘= ‘ 0;b) 5
• Gnbebl`efs nfs p`râltgsbs. : + 7b ‘ = ‘ 0;b 5
• Fragl`efs nfs târeblfs. : ‘ = + 7b ‘ 0;b 5
• ]gauhbefs târeblfs sgegc`ltgs. 4 ‘ 7b
]gstgefs (‘2 + 8b) ‘ (4 ‘ :b) 5
• Gnbebl`efs nfs p`râltgsbs. ‘2 + 8b ‘ 4 + :b 5
• Fragl`efs nfs târeblfs. ‘2 ‘ 4 + 8b + :b 5
• ]gauhbefs târeblfs sgegc`ltgs. ‘07 + 00b
]gstgefs (0
1 ‘0
8b ) ‘ (0
4 ‘0
=b ) 5• Gnbebl`efs nfs p`râltgsbs. 0
1 ‘
0
8b ‘
0
4 +
0
=b 5
• Fragl`efs nfs târeblfs. 0
1 ‘
0
4 ‘
0
8b +
0
=b 5
• ]gauhbefs târeblfs sgegc`ltgs. 0
0= +
0
4b
4=BIG] ∝ [fnfhjbh
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8. Euntbpnbh`hbøl ag lóegrfshfepngcfs
@jfr` `prgla`efs ` euntbpnbh`r lóegrfs hfepngcfs. @n biu`n qug gl n` sue` ygl n` rgst`, agmg sgiubr `niulfs p`sfs3
0. Gshrbm` tfafs nfs lóegrfs be`ibl`rbfs gl n` dfre` mb.
=. @pnbqug n` prfpbga`a abstrbmutbv` ag n` euntbpnbh`hbøl.
8. Xustbtuy` b= pfr ‘0 aflag `p`rgzh`.
1. ]gauzh` nfs târeblfs sgegc`ltgs y gn rgsunt`af gshråm`nf ag n`dfre` ` + mb.
Tg`efs nfs gcgepnfs3Euntbpnbqugefs 9b(1 + =b) 5
• @pnbh`efs n` prfpbga`a abstrbmutbv` ag 9b(1) + 9b(=b) 5
n` euntbpnbh`hbøl. =:b + 01b= 5
• Xustbtubefs b= pfr ‘0. =:b + 01(‘0) 5
• ]g`nbz`efs n` fpgr`hbøl. =:b ‘ 01 5
• Gshrbmbefs gn rgsunt`af ag n` dfre` ` + mb. ‘01 + =:b
Euntbpnbqugefs ‘2(7 ‘ 8b) 5
• [rbegrf, fmtglgefs n` r`åz ag ‘2 . 8b(7 ‘ 8b) 5
• @pnbh`efs n` prfpbga`a abstrbmutbv` ag07b ‘ 2b= 5 n` euntbpnbh`hbøl.
• Xustbtubefs b= pfr ‘0. 07b ‘ 2(‘0) 5
• ]g`nbz`efs n` fpgr`hbøl. 07b + 2 5
• Gshrbmbefs gn rgsunt`af ag n` dfre` ` + mb. 2 + 07b
Euntbpnbqugefs(= + 7b) (1 ‘ :b) 5
• @pnbh`efs n` prfpbga`a abstrbmutbv` ag : ‘ 04b + =;b ‘ 1;b= 5
n` euntbpnbh`hbøl.
• ]gauhbefs târeblfs sgegc`ltgs. : + 1b ‘ 1;b= 5
• Xustbtubefs b= pfr ‘0. : + 1b ‘ 1;(‘0) 5
: + 1b + 1; 5
• Gshrbmbefs gn rgsunt`af ag n` dfre` ` + mb. 1: + 1b
@tglhbøl3 gl n`s r`åhgsp`rgs ag lóegrfslgi`tbvfs lf sghuepng n` prfpbga`aagn prfauhtf.
‘2 • ‘1 ≫ ‘2 • ‘1
48E`tgeàtbh` ∝ Xge`l` 1
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Euntbpnbqugefs
• @pnbh`efs n` prfpbga`a abstrbmutbv` ag n`euntbpnbh`hbøl.
• ]gauhbefs târeblfs sgegc`ltgs.
• Xustbtubefs b= pfr ‘0.
• Gshrbmbefs gn rgsunt`af ag n` dfre` ` + mb.
( 7
1 + =b)( 0
= ‘ 8b) 5
7
: ‘
07b
1 +
=b
= ‘ 4b= 5
7
: ‘
00b
1 ‘ 4b= 5
7
: ‘
00b
1 ‘ 4(‘0) 5
7
: ‘
00b
1 + 4 5
78
: ‘
00b
1
Gcgrhbhbf 8]g`nbhg n`s fpgr`hbflgs blabh`a`s hfl lóegrfs hfepngcfs y gxprgsg su rgspugst` gl dfre` mblfeb`n.
0) (: + 8b) + (= + 9b) 5 =) (8 + 7b) ‘ (2 ‘ =b) 5
8) (‘1 + 9b) ‘ (0 ‘ 8b) 5 1) 7b(8 ‘ :b) 5
7) (1 ‘ 4b)(: + 0;b) 5 4) ( 0=
+ =b)( 18
‘ 7b) 5
41BIG] ∝ [fnfhjbh
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1. Abvbsbøl ag lóegrfs hfepngcfsNfs lóegrfs hfepngcfs t`embâl sg pugagl abvbabr, pgrf `ltgs ag gepgz`r `ags`rrfnn`r gst` fpgr`hbøl gs lghgs`rbf qug hflfzh`efs ul p`sf bepfrt`ltg3`prglagr ` r`hbfl`nbz`r gn aglfebl`afr egab`ltg gn hflcui`af. Dbcâeflfs glquâ hflsbstg gstg prfhgabebgltf.
1.0 ]`hbfl`nbz`hbøl agn aglfebl`afr egab`ltggn hflcui`af
]`hbfl`nbz`r ul` gxprgsbøl dr`hhbfl`rb` gs ul` tâhlbh` e`tgeàtbh` qug tfe` gnaglfebl`afr ag n` dr`hhbøl, sg glhugltr` su hflcui`af y sg euntbpnbh` pfr gnluegr`afr y aglfebl`afr ag n` gxprgsbøl blbhb`n.
Gl gn h`sf ag nfs lóegrfs hfepngcfs, gn hflcui`af sg fmtbglg h`emb`laf gnsbilf ag n` p`rtg be`ibl`rb`. Tg`efs `niulfs gcgepnfs3
Lóegrf hfepngcf Hflcui`af
9b ‘9b
8 + =b 8 ‘ =b
‘1 ‘ 7b ‘1 + 7b
R s`mbafs ag høef sg fmtbglg gn hflcui`af ag lóegrfs hfepngcfs, vg`efs nfsp`sfs p`r` rg`nbz`r n` abvbsbøl.
0. H`embg nfs lóegrfs be`ibl`rbfs ` n` dfre` mb.
=. ]`hbfl`nbhg gn aglfebl`afr, euntbpnbh`laf gn luegr`afr yaglfebl`afr pfr gn hflcui`af.
8. Xustbtuy` b= pfr ‘0 aflag `p`rgzh`.
1. ]g`nbhg n`s fpgr`hbflgs lghgs`rb`s y gshrbm` n` rgspugst` ag n`dfre` ` + mb.
Abvba`efs
• Euntbpnbh`efs gn luegr`afr y aglfebl`afrpfr gn hflcui`af.
• Xustbtubefs b= pfr ‘0.
• Gshrbmbefs gn rgsunt`af ag n` dfre` ` + mb.
0
8 + =b
08 + =b
( 8 ‘ =b8 ‘ =b ) 5 8 ‘ =b
2 ‘ 1b=
8 ‘ =b
2 ‘ 1( ‘0) 5
8 ‘ =b
2 + 1 5
8 ‘ =b
08 5
8
08 ‘
=b
08
]ghugrag qug3
(8 + =b)(8 ‘ =b) 5 2 ‘ 1b=
47E`tgeàtbh` ∝ Xge`l` 1
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@htbvba`a 0. Ageugstrg nf prglabaf
@. Fmsgrvg nfs lóegrfs hfepngcfs. Nugif, hfepngtg n` t`mn` gshrbmbglaf n` p`rtg rg`n y n` p`rtgbe`ibl`rb` gl h`a` h`sbnn`.
Lóegrf hfepngcf [`rtg rg`n [`rtg be`ibl`rb`
‘01 + :b
0.7 ‘ 0=b
‘7 + 0:b
‘;.=7 ‘ 9b
M. Gshrbm` gn hflcui`af p`r` nfs lóegrfs hfepngcfs sbiubgltgs. Qbglg ul gcgepnf.;) 7 ‘ :b 5
0) 1b 5
=) 4b 5
8) ‘1b 5
1) ‘0;b 5
7) ‘=8b 5
4) 8 + 4b 5
9) ‘7 ‘ 1b 5
:) 04 ‘ 2b 5
2) ‘07 ‘ 4b 5
0;) ‘02 + 01b 5
00) ‘=1 ‘ 0=b 5
@htbvba`a =. [r`htbqug nf `prglabaf
@. Gshrbm` h`a` gxprgsbøl hfef ul lóegrf hfepngcf ag n` dfre` ` + mb.
0) = + ‘1 5 :) 12 + ‘12 5
=) ‘: + ‘2 5 2) 7
9 + ‘2 5
8) 0
1 + ‘:0 5 0;) ‘ 12 + ‘12 5
1) ;.=7 ‘ ‘1 5 00) ‘07 ‘ ‘042 5
7) ‘07 + ‘0;; 5 0=) 8: + ‘0=0 5
4) 7
= ‘ ‘=74 5 08) 8
‘=9 + ‘=7 5
9) 2 + ‘41 5 01) 78= ‘ ‘04 5
@utfhfltrfn
7 + :b
4:BIG] ∝ [fnfhjbh
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]gsugnv` n`s fpgr`hbflgs nf eàs ràpbaf qug puga`. ]ghugrag qug p`r` nngi`r ` n` rgspugst` hfrrght`gs lghgs`rbf qug rgspgtg n` cgr`rquå` ag n`s fpgr`hbflgs. J`y ul gcgepnf p`r` h`a` h`sf.
@. ;) =(1 + 8) 5
0) 4(7 + 7) 5
=) :(= + 9) 5
8) 8(9 + :) 5
1) 9(1 + 0) 5
7) 1(: + 1) 5
4) 4(2 + ;) 5 9) 9(= + 8) 5
:) 2(1 + =) 5
2) 7(9 + 7) 5
0;) 0;(4 + 1) 5
00) ‘1(: ‘ =) 5
0=) ‘=(07 ‘ 1) 5
08) ‘7(0; ‘ 8) 5
01) ‘2(=; ‘ ;) 5
07) ‘4(0: ‘ 8) 5
04) ‘:(01 ‘ 9) 5
09) ‘=(=7 ‘ 7) 5 0:) ‘1(04 ‘ 1) 5
02) ‘=;(: ‘ 8) 5
=;) ‘=7(0: ‘ 04) 5
=0) ‘1;(== ‘ 0:) 5
M.
;) =1 ù (8 + 7) 5
0) 7; ù (= + 8) 5
=) 12 ù (1 + 8) 5
8) 8; ù (= + 0) 5
1) 41 ù (1 + 1) 5
7) 44 ù (0 + =) 5
4) :0 ù (7 + 1) 5
9) 48 ù (1 + 8) 5 :) 1; ù (8 + 7) 5
2) 0: ù (4 + 8) 5
0;) 74 ù (1 + 8) 5
00) 84 ù (8 + 0) 5
0=) :: ù (4 + =) 5
08) 0= ‘ (84 ù 4) 5
01) 0; ‘ (=1 ù :) 5
07) 02 ‘ (71 ù 4) 5
04) 0: ‘ (8; ù 7) 5
09) =; ‘ (1: ù 4) 5
0:) 7; ‘ (7; ù 7) 5
02) =7 ‘ (12 ù 9) 5
=;) 07 ‘ (:0 ù 2) 5 =0) =2 ‘ (48 ù 9) 5
==) :; ‘ (2; ù 2) 5
=8) 47 ‘ (17 ù 7) 5
=1) 27 ‘ (1; ù :) 5
=7) 07 ‘ (=0 ù 8) 5
01
8
9;BIG] ∝ [fnfhjbh
@ibnba`a ag hànhunf eglt`n
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E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfir`af gl
prfhgsflf
nfir`af
A g s p u â s a g g s t u a b ` r . . .
Bagltbhf gn hflcultf ag nfs lóegrfs hfepngcfs.
Fmtglif r`åhgs p`rgs g bep`rgs ag lóegrfs pfsbtbvfs y lgi`tbvfs.Gshrbmf lóegrfs hfepngcfs purfs y ag n` dfre` mblfeb`n.
Agtgreblf gn hflcui`af ag lóegrfs hfepngcfs.
]g`nbzf fpgr`hbflgs ag sue`, rgst`, euntbpnbh`hbøl y abvbsbøl aglóegrfs hfepngcfs.
]gsugnvf fpgr`hbflgs hfembl`a`s hfl `ibnba`a.
Hu`arf eàibhf
[`r` sgiubr gcgrhbt`laf su j`mbnba`a ag hànhunf, ng prfpflgefs hfepngt`r gnsbiubgltg hu`arf eàibhf.
Agmg rgnngl`r n`s h`sbnn`s hfl nfs lóegrfs agn 0 n 2, sbl rgpgtbrsg y ag t`n e`lgr`qug n`s fpgr`hbflgs hfembl`a`s huepn`l hfl n` biu`na`a. Ng `yua`efs hfl n`prbegr` n`.
: Ù = + 1 5 =;
+ Ù Ù
Ù ‘ 5 =;+ + Ù
+ + 5 =;
5 =; 5 =; 5 =;
9=BIG] ∝ [fnfhjbh
]gvbsg su `prglabz`cg
Ags`rrfnng lugv`s j`mbnba`ags
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70
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
Xue` y rgst` ag e`trbhgs
¶Puâ glhfltr`rà gst` sge`l`6
Xuafou3 ul cugif ag bliglbf
E`trbhgs y su `abhbøl
Euntbpnbh`hbflgs y sbepnbh`hbøl ag dr`hhbflgs
Cugifs ag suafou
Gst` sge`l` nfir`rà3
Hflhgptu`nbz`r quâ gs ul` e`trbz.
Hflfhgr g bagltbh`r n`s abeglsbflgs ag ul` e`trbz.
Fpgr`r sue`s y rgst`s ag e`trbhgs.
Hflfhgr n`s prfpbga`ags ag n` sue` ag e`trbhgs.
]gsfnvgr euntbpnbh`hbflgs y sbepnbh`r dr`hhbflgs hfl `ibnba`a.
98E`tgeàtbh` ∝ Xge`l` 7
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¡[`r` hfeglz`r!Xuafou3 ul cugif ag bliglbf
Pubzàs nf j`y` vbstf `n gst`r jfcg`laf ul pgrbøabhf, qubzàs y` nf j`y` rgsugntf
`niul` vgz, lfs rgdgrbefs `n suafou, gn pfpun`r cugif qug j`hg dulabr lugstr`s
lgurfl`s.
Gn suafou gs ul glbie` e`tgeàtbhf pumnbh`af pfr prbegr` vgz ` dbl`ngs ag
nfs `ðfs sgtglt`, sg vfnvbø pfpun`r gl C`pøl j`hb` 02:4 y sg abf ` hflfhgr
bltgrl`hbfl`negltg hgrh` ag =;;7.
Gn fmcgtbvf agn suafou gs rgnngl`r ul` hu`aråhun` ag 2 Ù 2 hgna`s, abvbaba` gl
sumhu`aråhun`s ag 8 Ù 8, nn`e`a`s "h`c`s" f "rgibflgs", hfl nfs lóegrfs agn 0 n
2, p`rtbglaf ag lóegrfs qug y` gstàl hfnfh`afs gl `niul`s hgna`s.
Gn truhf gstà gl qug ul ebsef lóegrf lf sg agmg rgpgtbr gl ul` ebse` dbn`,
hfnuel` f h`c`. Tg`efs ul gcgepnf y su sfnuhbøl3
¡@ tr`m`c`r!
@jfr` ng tfh` ` ustga. ]gsugnv` gn suafou sbiubgltg. Ng rghfegla`efs gshrbmbr hfl nàpbz pfr sb lghg-sbt` hfrrgibr `niól a`tf.
8 1
4 9 7
7
: 7 4 =
9 1 7 :
8 = 1
4 2 9 8 1
1 = 81 : 4 0 9
8 = 1 7 :
4 9 : 2 = 8
1 4 8 9 74 = 2 8 : 9
8 9 1 : 2 4
1 : 2 9 0 8
7 4 : 2
1 7
2 9 8 4 =
Blbhbf
0 9 : 8 2 7 = 1 4
8 = 1 0 : 4 9 2 7
2 7 4 = 9 1 8 : 0
: 2 7 4 8 9 0 = 1
9 1 0 7 = : 2 4 8
4 8 = 1 0 2 : 7 9
= 4 2 9 7 8 1 0 :
7 0 9 : 1 = 4 8 21 : 8 2 4 0 7 9 =
Xfnuhbøl
91BIG] ∝ [fnfhjbh
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0. E`trbhgsXg lfemr` e`trbz ` tfaf hflcultf ag lóegrfs f gxprgsbflgs abspugstfs gl
dfre` rght`liun`r, dfre`laf n`s y hfnuel`s? gstf qubgrg aghbr qug gn suafouhfl qug lfs gltrgtglå`efs gl n` sghhbøl `ltgrbfr gs ul` e`trbz.
Gn fragl f abeglsbøl ag ul` e`trbz sg agtgrebl` pfr su lóegrf ag n`s yhfnuel`s. Wl` e`trbz ag abeglsbøl e Ù l tbglg e n`s y l hfnuel`s.
Ag gst` dfre`, ul` e`trbz pugag sgr ag fragl3 = Ù 8 (= n`s y 8 hfnuel`s), 1 Ù 4 (1 n`s y 4 hfnuel`s), gth. Xb n` e`trbz tbglg gn ebsef lóegrf ag n`s qug aghfnuel`s, aghbefs qug gs ul` e`trbz hu`ar`a` y sg aglfebl` ag fragl3=, 8, 1...
H`a` ulf ag nfs lóegrfs qug hfldfre` n` e`trbz sg aglfebl` gngegltf. Wl
gngegltf sg abdgrglhb` ag ftrf pfr n` pfsbhbøl qug fhup`, gs aghbr, n` n` yn` hfnuel` ` n` qug pgrtglghg. Afs e`trbhgs sfl biu`ngs sb tbglgl n` ebse`abeglsbøl y nfs gngegltfs qug fhup`l gn ebsef nui`r gl `em`s, sfl biu`ngs.
Wl gngegltf hu`nqubgr` ag n` e`trbz, qug sg glhugltr` gl n` n` b y gl n` hfnuel` c, sg aglft` pfr `bc. Tgàefsnf iràh`egltg p`r` gltglagr egcfr3
`00 `0= ... `0c ... `0l
`=0 `== ... `=c ... `=l
... ... ... ... ... ...`b0 `b= ... `bc ... `bl
... ... ... ... ... ...`e0 `e= ... `ec ... `el
@ 5
Gl n` e`trbz @ lft`efs su abeglsbøl ag e Ù l y sus gngegltfs gl n`s b yhfnuel`s c.
Wl` e`trbz sglfemr` hfl ul`ngtr` e`yóshun`.
Gcgrhbhbf 0Gst`mngzh` n` abeglsbøl ag n`s e`trbhgs a`a`s y gshråm`n` gl n` nålg` hfrrgspflabgltg. Iuågsg pfr gngcgepnf.
;) ‘0 ; =
9 ‘0 2 ‘4 1 =
=) 0 ‘0
: ;
1) 9 =
0 ‘7= ‘8
0) 7 =
9 :
1 9
8) 2 9 7 1 7) ‘9 0 1 7
8 4 = ‘7
: ‘8 = ‘=
1 8 ‘0 ;
8 Ù 8
97E`tgeàtbh` ∝ Xge`l` 7
Gn eulaf ag n` e`tgeàtbh`
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8. Fpgr`hbflgs gltrg e`trbhgs
N`s fpgr`hbflgs qug sg pugagl rg`nbz`r gltrg e`trbhgs sfl n` sue`, n` rgst` y n`euntbpnbh`hbøl. Gst` sge`l` prglagrgefs gn prfhgabebgltf p`r` sue`r y rgst`rafs f eàs e`trbhgs.
8.0 Xue` y rgst`
Xfn`egltg n`s e`trbhgs qug tbglgl n` ebse` abeglsbøl sg pugagl sue`r frgst`r. N` e`trbz sue` sg fmtbglg sue`laf nfs gngegltfs ag n`s e`trbhgs qugfhup`l n` ebse` pfsbhbøl? p`r` rgst`r sg sbiug gn ebsef prfhgsf.
Tg`efs hfl agtglbebgltf3
Xb @ 5 (`bc)eÙl y M 5 (mbc)eÙl gltflhgs su sue` gs @ + M 5 (` bc + mbc)eÙl
@ 5= ; 08 ; ;
7 0 0
M 50 ; 00 = 0
0 0 ;
@ + M 5= + 0 ; + ; 0 + 0
8 + 0 ; + = ; + 0
7 + 0 0 + 0 0 + ;
58 ; =
1 = 0
4 = 0
Gn prfhgabebgltf p`r` rgst`r gs euy sbebn`r, fmsgrvgefs3
@ ‘ M 5
= ‘ 0 ; ‘ ; 0 ‘ 0
8 ‘ 0 ; ‘ = ; ‘ 07 ‘ 0 0 ‘ 0 0 ‘ ;
5
0 ; ;
= ‘= ‘01 ; 0
Lf gs lghgs`rbf qug n`s e`trbhgs sg`l hu`ar`a`s p`r` pfagr sue`rn`s frgst`rn`s, n` ólbh` hflabhbøl gs qug tgli`l n`s ebse`s abeglsbflgs.
Q embâl pfagefs fpgr`r eàs ag afs e`trbhgs, vg`efs ftrf gcgepnf3
Z 58 = 7
7 ‘0 ‘= R 5
; ‘9 0
8 ; 0 S 5
‘0 8 0
: 4 ;
Z + R + S 58 + ; + (‘0) = + (‘9) + 8 7 + 0 + 0
7 + 8 + : (‘0) + ; + 4 (‘=) + 0 + ; 5
= ‘= 9
04 7 ‘0
Z ‘ R ‘ S 58 ‘ ; ‘ (‘0) = ‘ (‘9) ‘ 8 7 ‘ 0 ‘ 0
7 ‘ 8 ‘ : (‘0) ‘ ; ‘ 4 (‘=) ‘ 0 ‘ ; 5
1 4 8
‘4 ‘9 ‘8
99E`tgeàtbh` ∝ Xge`l` 7
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Gcgrhbhbf =Xbglaf n`s e`trbhgs3
@ 5= 7
8 ‘1 M 5
‘8 ‘2
4 = H 5
7 ‘9
‘0 ; A 5
4 1
; ‘2
G 5 ‘7 4 =
0 ‘0 ;
8 ; 1
D 54 ; ;
‘2 ‘= 1
0 1 =
I 57 0 09 ‘0 1
8 ‘= =
J 59 ‘9 24 8 ;
= 0 ;
]gsugnv` n`s sbiubgltgs sue`s y rgst`s ag e`trbhgs. [ugag us`r jfc`s `p`rtg sb nf lghgsbt`. ]ghugragtr`m`c`r gl fragl.
0) @ + M 4) D + I
=) @ + A 9) G + J + I
8) @ + H ‘ A :) I + J
1) H + A + @ 2) G + D + I ‘ J
7) G + D 0;) G + I ‘ J
9:BIG] ∝ [fnfhjbh
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8.= [rfpbga`ags ag n` sue` ag e`trbhgs
N` sue` gltrg e`trbhgs tbglg prfpbga`ags qug sfl euy sbebn`rgs ` n`s qugpfsggl nfs lóegrfs rg`ngs. [fli`efs `tglhbøl.
Bltgrl`
N` sue` f rgst` ag afs f eàs e`trbhgs ag fragl e Ù l gs ftr` e`trbz ag fragle Ù l. Gs aghbr, n` e`trbz rgsunt`af sgrà ag n`s ebse`s abeglsbflgs qug n`se`trbhgs qug sg fpgr`rfl.
(e Ù l) ° (e Ù l)5 e Ù l
9 ‘=
; 0
= ‘1
+; ‘=
8 ‘0
= =
5
9 ‘1
8 ;
1 ‘=
8 n`s 8 n`s
= hfnuel`s = hfnuel`s
8 Ù = 5 8 Ù =
@sfhb`tbv`
N` `sfhb`tbvba`a lfs abhg qug gn rgsunt`af ag n` fpgr`hbøl sgrà gn ebsef sblbepfrt`r gn `irup`ebgltf ag nfs târeblfs.
@ + (M + H) 5 (@ + M) + H
: 40 0
0 ‘9
+ ‘8 9 ‘1 ‘=
0 ;
+
= 1
; 0
0 0
5
9 09
‘8 ;
8 ‘4
: 4
0 0
0 ‘9
+
‘8 9
‘1 ‘=
0 ;
+
= 1
; 0
0 0
5
9 09
‘8 ;
8 ‘4
Gngegltf lgutrf
N` e`trbz hgrf, aglft`a` hfl n` ngtr` F, gs ul` e`trbz gl n` qug h`a` ulf ag sus
gngegltfs gs hgrf. Gst` e`trbz gs gn gngegltf lgutrf.
@ ° F 5 @
0 =
‘8 0
1 ‘=
+; ;
; ;
; ;
5
0 =
‘8 0
1 ‘=
92E`tgeàtbh` ∝ Xge`l` 7
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Gngegltf fpugstf
N` e`trbz fpugst` gs `qugnn` gl n` qug tfafs nfs gngegltfs tbglgl gn sbilfhfltr`rbf rgspghtf ` ul` e`trbz frbibl`n. Xg rgprgsglt` `ltghgabglaf ul sbilfeglfs `n lfemrg ag n` e`trbz (‘M, ‘@, ‘S, gth).
@ + (‘@) 5 F
9 ‘1
‘1 0
4 ‘=
+ ‘9 1
1 ‘0
‘4 =
5
; ;
; ;
; ;
Hfleut`tbv`
Gst` prfpbga`a gst`mnghg qug `n h`emb`r gn fragl ag nfs sue`lafs, gn rgsunt`afpgre`lghg blv`rb`mng, gs aghbr, lf sg efabh`.
@ + M 5 M + @
= 4
0 ‘8
0 0
+7 ‘:
2 9
= ‘2
5
9 ‘=
0; 1
8 ‘:
57 ‘:
2 9
= ‘2
+= 4
0 ‘8
0 0
Gcgrhbhbf 8]gsugnv` n`s fpgr`hbflgs sbiubgltgs. Nugif, gshrbm` gl gn gsp`hbf hfrrgspflabgltg quâ prfpbga`a sggvbaglhb`. Iuågsg pfr gn gcgepnf.
[rfpbga`a
;) ; 0 ;
8 7 ‘4
7 : ‘9
+
= ‘1 :
; 1 ‘8
‘8 0 ‘4
+
2 ; ‘2
9 ‘= :
0 0 0
5
00 ‘8 ‘0
0; 9 ‘0
8 0; ‘0=
`sfhb`tbv`
0) = ‘1 :
; 1 ‘8
‘8 0 ‘4
+
‘= 1 ‘:
; ‘1 8
8 ‘0 4
5
=) 9 : ‘1
0 ; 7
8 4 9
+
; ; ;
; ; ;
; ; ;
5
:;BIG] ∝ [fnfhjbh
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]gsuegl
0. Wl` e`trbz gs ul hflcultf ag lóegrfs f gxprgsbflgs abspugstfs gl dfre` rght`liun`r, dfre`lafdbn`s y hfnuel`s. Gn fragl f abeglsbøl ag ul` e`trbz sg agtgrebl` pfr gn lóegrf ag dbn`s yhfnuel`s qug n` hfepflgl.
Wl` e`trbz e Ù l tbglg e dbn`s y l hfnuel`s. Xb tbglg gn ebsef lóegrf ag dbn`s qug ag hfnuel`s,aghbefs qug gs ul` e`trbz hu`ar`a`. @ h`a` lóegrf f gxprgsbøl qug dfre` n` e`trbz ng aglfeb-l`efs gngegltf.
=. J`y `niulfs tbpfs ag e`trbhgs hfl h`r`htgråstbh`s p`rtbhun`rgs3
=.0 Wl` e`trbz tr`lspugst` (@t ) gs n` qug sg dfre` `n h`emb`r fragl`a`egltg n`s dbn`s pfr n`shfnuel`s ag n` e`trbz a`a`.
=.= Wl` e`trbz gs sbeâtrbh` sb gs hu`ar`a` y gs biu`n ` su tr`lspugst`.
=.8 Wl` e`trbz bagltba`a gs ul` e`trbz gl huy` ab`ifl`n prblhbp`n tfafs sus gngegltfs sflbiu`ngs ` 0 y nfs ageàs gngegltfs sfl ; (hgrf). Xg lfemr` hfl n` ngtr` B.
=.1 Wl` e`trbz lf sbliun`r f blvgrtbmng gs `qugnn` qug tbglg e`trbz blvgrs`.=.7 Wl` e`trbz sbliun`r lf tbglg blvgrs`.
8. N`s fpgr`hbflgs qug sg pugagl j`hgr hfl e`trbhgs sfl n` `abhbøl, n` rgst` y n` euntbpnbh`hbøl3
8.0 [`r` sue`r f rgst`r e`trbhgs gs lghgs`rbf qug tgli`l n` ebse` abeglsbøl. [`r` fpgr`r, sgagmgl sue`r nfs gngegltfs ag n`s e`trbhgs qug fhup`l n` ebse` pfsbhbøl . Xg pugaglfpgr`r afs f eàs e`trbhgs.
8.= N` sue` ag e`trbhgs tbglg n`s prfpbga`ags sbiubgltgs3
• Gs bltgrl` pfrqug n` e`trbz rgsunt`af sgrà ag n`s ebse`s abeglsbflgs qug n`s e`trbhgsqug sg fpgr`rfl.
• Gs `sfhb`tbv pfrqug gn rgsunt`af gs gn ebsef sbl bepfrt`r høef irupgefs nfs târeblfs.
• Qbglg gngegltf lgutrf, hflfhba` hfef e`trbz hgrf (F).
• Qbglg gngegltf fpugstf, gn hu`n gs n` e`trbz hfl nfs ebsefs gngegltfs qug n` e`trbzfrbibl`n, pgrf hfl nfs sbilfs hfltr`rbfs.
• Gs hfleut`tbv pfrqug gn rgsunt`af pgre`lghg blv`rb`mng sbl bepfrt`r gn fragl gl qugfpgr`efs nfs târeblfs.
Ng blvbt`efs ` qug vbsbtg nfs sbiubgltgs sbtbfs ag bltgrlgt. Gl gn prbegrf, pfarà ags`rrfnn`r su pgls`-ebgltf `l`nåtbhf `n ebsef tbgepf qug sg gltrgtbglg cui`laf suafou. Gl n` sgiula` y tgrhgr` abrghhbøl,glhfltr`rà gcgrhbhbfs bltgr`htbvfs ag n` sue` ag e`trbhgs.
• Xuafou gl nålg`3 iff.in/oS=RDx
• Gcgrhbhbfs. @abhbøl y sustr`hhbøl ag e`trbhgs3 iff.in/w;7jZJ
• Xue` y rgst` ag e`trbhgs3 iff.in/H8rxjZ
Blvgstbiug gl n` rga...
:0E`tgeàtbh` ∝ Xge`l` 7
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@htbvba`a 0. Ageugstrg nf prglabaf
@. Bagltbdbqug n` abeglsbøl ag n`s e`trbhgs sbiubgltgs. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.
0) 8 =; ‘2
8 ‘:
8) 8 1 4 ‘9 ‘7 9 = ;
0 ; ; ‘8
=) 0 ‘9
0 7
1) 8 1 :
‘9 ; 0
M. Dfreung y gshrbm` ul` e`trbz hfl gn fragl blabh`af.0) 8 Ù 8 1) 0 Ù 8
=) = Ù 1 7) 8 Ù 0
8) 8 Ù = 4) = Ù 0
H. Dfreung, gshrbm` g bagltbdbqug n`s e`trbhgs qug sg pbagl gl h`a` blhbsf. Iuågsg pfr gn gcgepnf.
@utfhfltrfn
E`trbz tr`lspugst` E`trbz hu`ar`a` E`trbz bagltba`a
Mt 5 ‘0 8 ‘:
; ‘0 :
2 8 ;
:=BIG] ∝ [fnfhjbh
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@htbvba`a =. [r`htbqug nf prglabaf
]gsugnv` n`s sbiubgltgs sue`s y rgst`s ag e`trbhgs. [ugag us`r jfc`s `p`rtg sb n`s lghgsbt`. ]ghugragtr`m`c`r fragl`a`egltg. Xu frbglt`afr pugag sfnbhbt`rng hflst`lhb` ag su prfhgabebgltf.
A`a`s n`s e`trbhgs sbiubgltgs3
Y 5 ‘1 = 0 :; 0 0 ‘1
R 5 7 ‘: 9 =0 ‘0 ; :
S 5 ‘1 : ‘: 02 9 ‘= ‘0
Fpgrg3
0) Y + R =) (Y + S) ‘ R
8) R + S 1) Y ‘ R + S
A`a`s n`s e`trbhgs3
A 5 ‘7 : 4
; 0 1
‘0 = 0
X 5 ‘= 1 ‘0
9 ‘0 :
= 8 ;
G 59 ‘7 ;7 = ;
‘9 : 2
7) A + G 4) A + X + G
9) X ‘ G :) G ‘ A
:8E`tgeàtbh` ∝ Xge`l` 7
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@. Gl gn gsp`hbf hfrrgspflabgltg, gshrbm` gn târeblf d`nt`ltg p`r` qug n`s euntbpnbh`hbflgs sg`lhfrrght`s.
0) = Ù 5 :
=) 8 Ù 5 2
8) : Ù 5 1;
1) 8 Ù 5 =9
7) 7 Ù 5 07
4) 2 Ù 5 0:
9) 1 Ù 5 0=
:) 4 Ù 5 71
2) 2 Ù 5 84
0;) 8 Ù 5 0=
00) Ù 9 5 1=
0=) Ù 2 5 17
08) Ù = 5 4
01) Ù 9 5 9
07) Ù 4 5 1=
04) Ù = 5 1
09) Ù : 5 9=
0:) Ù 1 5 ;
02) Ù 9 5 48
=;) Ù 0 5 0
=0) 1 Ù 5 04
==) 8 Ù 5 2
=8) : Ù 5 41
=1) 0 Ù 5 ;
=7) 9 Ù 5 01
=4) 8 Ù 5 =0
=9) ; Ù 5 ;
=:) 7 Ù 5 =7
=2) = Ù 5 0;
8;) 4 Ù 5 0:
M. Xbepnbdbqug n`s dr`hhbflgs sbiubgltgs j`st` su eålbe` gxprgsbøl. Wsg dr`hhbflgs beprfpb`s ygltgrfs ag sgr lghgs`rbf.
0) 4
:
5
=) 0;
=; 5
8) ==
88 5
1) =;
:; 5
7) 01
9
5
4) 01
=; 5
9) 7
=7 5
:) 0=
84 5
2) 0;
0;;
5
0;) =
4 5
00) 4
2 5
0=) 2
07 5
08) 07
0=
5
01) 9
12 5
07) 1
0; 5
04) :
9= 5
09) 84
4
5
0:) 4
:= 5
02) 07
9= 5
=;) =
: 5
=0) 1
0:
5
==) 2;
0=; 5
=8) 00
22 5
=1) 0;
:7 5
:1BIG] ∝ [fnfhjbh
@ibnba`a ag hànhunf eglt`n
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@n blbhbf ag gst` sge`l` `prglabefs n`s rgin`s p`r` cui`r suafou, cugif qug sg j` hflvgrtbaf gl ulgltrgtglbebgltf euy pfpun`r gl nfs óntbefs `ðfs.
Gstg cugif lf sfnf gstbeun` n` nøibh`, n` egefrb` y n` h`p`hba`a ag hflhgltr`hbøl? rghbgltgegltg, ulgstuabf ag n` Wlbvgrsba`a ag Gabemurif, gl Blin`tgrr`, j` agefstr`af qug gn suafou t`embâl gcgrhbt`lugstr`s lgurfl`s, prfnfli` su tbgepf ag vba` hgnun`r y n`s j`hg eàs rgsbstgltgs ` gldgrega`agshfef gn @nzjgbegr.
Hfl tfa`s gst`s vglt`c`s, ¶quâ ng p`rghg sb rgsfnvgefs ulfs cugifs eàs6 ¡Àlbef!, qug gstàl dàhbngs.
7 9 = 1
4 1 7
1 8 : 7
: = 4 9 8 1 2 7
8 1 = 7 2 4
: 8 =
9 4 = 7 1
4 8 1 :
0 9
9 : 7 = 4 2
4 2 9 1 : 8
2 = 0 : 4
0 4 1 :
9 8 2 : 7 17 : 4 0 9
4 0 2 =
1 0 = 4 :
= 0 9 7
:7E`tgeàtbh` ∝ Xge`l` 7
]`zfl`ebgltf nøibhf
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E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfir`af gl
prfhgsflf
nfir`af
A g s p u â s a g g s t u a b ` r . . .
Hflhgptu`nbzf quâ gs ul` e`trbz.Hflfzhf g bagltbhf n`s abeglsbflgs ag ul` e`trbz.
Fpgrf sue`s y rgst`s ag e`trbhgs.
Hflfzhf n`s prfpbga`ags ag n` sue` ag e`trbhgs.
]gsugnvf euntbpnbh`hbflgs y abvbsbflgs hfl `ibnba`a.
Gn ags`rrfnnf ag n` bltgnbiglhb` gsp`hb`n gs gsglhb`n p`r` pfagr e`lgc`r y e`-lbpun`r hfl agstrgz` nf rgn`hbfl`af `n hfnfr, dfre`, nålg`, iur`, gsp`hbf, y n`rgn`hbøl qug gxbstg gltrg tfafs nfs `ltgrbfrgs.
@jfr` pflargefs ` tr`m`c`r lugstr`s j`mbnba`ags gsp`hb`ngs. Fmsgrvg n`bnustr`hbøl sbiubgltg, nugif, hfepnâtgn` abmuc`laf n` ebt`a qug d`nt` ag ul`dfre` sbeâtrbh`.
:4BIG] ∝ [fnfhjbh
]gvbsg su `prglabz`cg
Ags`rrfnng lugv`s j`mbnba`ags
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40
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
Euntbpnbh`hbøl ag e`trbhgs
¶Puâ glhfltr`rà gst` sge`l`6
Gn `cgargz y n` ngygla` ag Xbs`
Euntbpnbh`hbøl ag e`trbhgs
Ghu`hbflgs ag prbegr ir`af
Xgrbgs ag iur`s rgn`hbfl`a`s
Gst` sge`l` nfir`rà3
Hflfhgr y `pnbh`r n`s rgin`s p`r` euntbpnbh`r ul` e`trbz pfr ul lóegrf rg`n.
[r`htbh`r n` euntbpnbh`hbøl ag ul` e`trbz pfr ul lóegrf rg`n.
Hflfhgr y `pnbh`r n`s rgin`s p`r` euntbpnbh`r e`trbhgs gltrg så.
[r`htbh`r n` euntbpnbh`hbøl gltrg e`trbhgs.
:9E`tgeàtbh` ∝ Xge`l` 4
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¡[`r` hfeglz`r!Gn `cgargz y n` ngygla` ag Xbs`
Gn `cgargz gs ul cugif gltrg afs pgrsfl`s, h`a` ul` huglt` hfl 04
pbgz`s qug sg eugvgl sfmrg ul t`mngrf gl dfre` e`trbhb`n f ag e`trbz
ag :Ù:, hfl 41 hu`arfs. Xg tr`t` ag ul cugif ag gstr`tgib` gl gn qug
gn fmcgtbvf gs vglhgr f "agrrfh`r" `n "rgy" agn fpflgltg.
Xfmrg gn frbigl agn `cgargz j`l suribaf euhj`s ngygla`s, ul` ag gnn`s
gs n` agn s`mbf Xbs`, ng`efs3
Gl tbgepfs rgeftfs, `n lfrfgstg ag n` `ltbiu` Blab`, gl nf qug jfy sgrå`
[`obstàl f @di`lbstàl, vbvå` ul pfagrfsf y rbhf rgy nn`e`af ]`b Mj`nbt.
@ pgs`r ag su hflabhbøl, gn rgy j`må` pgrabaf tfa` dgnbhba`a agmbaf ` n`
eugrtg ag su jbcf gl n` iugrr`.
¡@ tr`m`c`r!
]gspfla`3 ¶Puâ ng j`mrå` pgabaf ustga `n rgy ]`b Mj`nbt hfef rghfepgls`6 ¶[fr quâ6
Agv`st`af pfr gst` tr`igab`, gn rgy sg `m`laflø ` så ebsef y agshubaø su rgblf. Xus eàs hgrh`-
lfs hflsgcgrfs y sbrvbgltgs sg gsdfrz`m`l pfr `lbe`rnf, blvbt`m`l mudflgs, m`bn`rblgs y h`lt`ltgs,
pgrf l`a` dulhbfl`m`.
Wl vbgcf s`mbf nn`e`af Xbs` aghbabø hrg`r ul cugif p`r` agvfnvgr n` dgnbhba`a ` su rgy y pfagr gl-
trgtglgrng. Nugif ag euhjf tr`m`cf y rgdngxbøl, Xbs` sg prgsgltø drgltg `n sfmgr`lf p`r` efstr`rng
gn lugvf cugif. @mrbø ul` h`c` y `p`rghbø ul t`mngrf ag e`agr` hfl 41 h`sbnn`s y 8= dbiur`s agn
ebsef e`tgrb`n. Qr`s gxpnbh`r n`s rgin`s `n rgy, sg pusbgrfl ` cui`r.
@n ifmgrl`ltg ng glh`ltø gn p`s`tbgepf y cuiø aur`ltg jfr`s y aå`s hfltr` tfaf `qugn qug nf
rgt`r`.
@ir`aghbaf hfl Xbs`, gn rgy ng fdrghbø n` rghfepgls` qug gn s`mbf qubsbgsg. Gl ul prblhbpbf, gn s`mbf
sg lgi`m` ` `hgpt`r rgi`nfs, pgrf nugif ag euhj` blsbstglhb`, abcf3 ‑"Xgðfr, eg hfldfre`rå`
hfl qug eg p`iuâbs ul ir`lf ag trbif pfr gn prbegr hu`ar`af, afs pfr gn sgiulaf, hu`trf gl gn
tgrhgrf, gth."
Gn rgy `hhgabø ag blegab`tf, y e`laf ` sus `yua`ltgs ` h`nhun`r gn tft`n ag trbif y ` gltrgiàrsgnf `
Xbs`. Xbl gem`rif, nugif ag v`rbfs hànhunfs, n` hbdr` tft`n gr` ag ¡0:.1 trbnnflgs ag ir`lfs ag trbif!
Gn rgy qugaø sfrprglabaf pfrqug lb hfl tfa`s n`s hfsghj`s agn rgblf pfarå` rgulbr t`n h`ltba`a.
Gn s`mbf, s`tbsdghjf pfrqug gn rgy gr` dgnbz ag lugvf, hfl ul` sflrbs` rglulhbø `n rgi`nf.
::BIG] ∝ [fnfhjbh
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0. Euntbpnbh`hbøl ag e`trbhgsGl "N` ngygla` ag Xbs`" sg j`hg gvbagltg qug n` dfre` e`trbhb`n tbglg v`rbga`a
ag `pnbh`hbflgs y usfs, ulf ag gstfs gs gn t`mngrf ag `cgargz.N` sge`l` p`s`a` `prglabefs sfmrg n`s e`trbhgs, høef sue`rn`s y rgst`rn`s?`jfr` vgrgefs gn prfhgabebgltf p`r` euntbpnbh`r ul` e`trbz pfr ul lóegrf rg`ny pfr ftr` e`trbz. T`efs, qug gs dàhbn, pfli`efs `tglhbøl.
0.0 [rfauhtf ag ul lóegrf rg`n pfr ul` e`trbz
Xb tglgefs ul` e`trbz @ 5 (`bc) y n` euntbpnbh`efs pfr ul lóegrf rg`n o , n`e`trbz rgsunt`ltg sgrà ul` e`trbz ag n` ebse` abeglsbøl qug @ y gl n` qug h`a`gngegltf gstà euntbpnbh`af pfr o .
Gs aghbr, sbepngegltg agmgefs euntbpnbh`r gn lóegrf o pfr h`a` ulf ag nfsgngegltfs ag n` e`trbz.
[`r` gst` fpgr`hbøl, sg pugagl euntbpnbh`r e`trbhgs ag hu`nqubgr fragl.
o @ 5 o ` m h
a g d
i j b
5`o mo ho
ao go do
io jo bo
¶R lftø nf sglhbnnf qug gs6 Tg`efs ulfs gcgepnfs3
=J 5 = = ; 08 ; ;
7 0 0
5= • = = • ; = • 0= • 8 = • ; = • ;
= • 7 = • 0 = • 0
51 ; =4 ; ;
0; = =
‘:Y 5 ‘: ‘= 8 ;
: 2 ‘0
0 ; 9
5(‘:) • (‘=) (‘:) • 8 (‘:) • ;
(‘:) • : (‘:) • 2 (‘:) • (‘0)
(‘:) • 0 (‘:) • ; (‘:) • 9
504 ‘=1 ;
‘41 ‘9= :
‘: ; ‘74
7M 5 7 4 ‘7 0
; 1 ‘=
0 0 0
57 • 4 7 • (‘7) 7 • 0
7 • ; 7 • 1 7 • (‘=)
7 • 0 7 • 0 7 • 0
58; ‘=7 7
; =; ‘0;
7 7 7
Wl óntbef gcgepnf `ltgs ag p`s`r ` n` pràhtbh`3
‘8Q 5 ‘8 1 7
8 1
‘8 ‘0
5(‘8) • 1 (‘8) • 7
(‘8) • 8 (‘8) • 1
(‘8) • (‘8) (‘8) • (‘0)
5 ‘0= ‘07
‘2 ‘0=
2 8
:2E`tgeàtbh` ∝ Xge`l` 4
Gn eulaf ag n` e`tgeàtbh`
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Gcgrhbhbf 0Gs su turlf, rgsugnv` n`s euntbpnbh`hbflgs sbiubgltgs.
0) 7 ‘= 2 ‘0 ; =)1 ‘= 2
0 1
8) ‘1
0 7
‘9 ;
1 8
1) ‘0
02 0;
; ‘0
‘01 =
7)= 1 0 ; =
‘= 8 0 0
4)4 4 : ‘0; 8
0 9 ; 0
9) ‘4
‘0 7 ‘=
0 ; 4
9 1 2
:)2
2 ‘1 4
; 0 ‘:
0 ‘0; 9
2;BIG] ∝ [fnfhjbh
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0.= [rfauhtf ag e`trbhgs
[rbegrf, agmgefs tglgr hn`rf qug p`r` qug afs e`trbhgs (@ y M) puga`l euntb-pnbh`rsg, gn lóegrf ag hfnuel`s ag @ agmg hfblhbabr hfl gn lóegrf ag n`s
ag M. Xb sg huepng gst` rgstrbhhbøl, gst`efs nbstfs p`r` hfeglz`r.
Gn gngegltf hbc ag n` e`trbz prfauhtf sg fmtbglg euntbpnbh`laf h`a` gngegltfag n` n` b ag n` e`trbz @ pfr h`a` gngegltf ag n` hfnuel` c ag n` e`trbz M ysueàlafnfs. Lf sg prgfhupg, lf gs hfepnbh`af, vg`efs3
H 5 @ • M 5` m h
a g d
i j b
•r s t
u v w
x y z
5`r + mu + hx `s + mv + hy `t + mw + hz
ar + gu + dx as + gv + dy at + gw + dz
ir + ju + bx is + jv +by it + jw + bz
Hfef vgefs gl n`s e`trbhgs `ltgrbfrgs, sg euntbpnbh` h`a` gngegltf ag n` prb-egr` n` ag @ pfr h`a` gngegltf ag n` prbegr` hfnuel` ag M. @n tgrebl`rgstf, sg sue`l nfs prfauhtfs y `så sg fmtbglg gn gngegltf h00 ag n` e`trbz H.
Nugif, vfnvgefs ` tfe`r n` prbegr` n` ag @, pgrf `jfr` n` euntbpnbh`efs pfrn` sgiula` hfnuel` ag M ag n` ebse` dfre` qug nf jbhbefs ltgrbfregltg. Hflgstf fmtglgefs gn gngegltf h0=. Hfltblu`efs `så j`st` n` óntbe` hfnuel` ag M.
Hu`laf j`y`efs `h`m`af hfl n` prbegr` n` ag @, p`s`efs n` sbiubgltg n` y rgpgtbefs gn ebsef prfhgsf. @så j`st` tgrebl`r hfl tfa`s n`s n`s, fmtgl-argefs gltflhgs lugstr` e`trbz rgsunt`af. Fmsgrvgefs ul gcgepnf3
I • A 5= 8
0 7 •
0 7 8
1 = =
Hfeprfm`efs qug gn lóegrf ag hfnuel`s ag n` e`trbz I (= hfnuel`s) hfblhbaghfl gn lóegrf ag n`s ag n` e`trbz A (= n`s), pfr t`ltf, sfl euntbpnbh`mngs.
0. Euntbpnbh`efs n` prbegr` dbn` ag I pfr n` prbegr` hfnuel` ag A (gngegltfpfr gngegltf) y sue`efs nfs rgsunt`afs, hfl gstf fmtglargefs gn gngegltfh00 ag n` lugv` e`trbz.
I • A 5= 8
0 7 • 0 7 8
1 = = 5
(= • 0) + (8 • 1)
=. Xgiubefs gn prfhgabebgltf euntbpnbh`laf n` prbegr` dbn` ag I hfl n` sg-iula` hfnuel` ag A. Hfl gstf fmtglgefs gn gngegltf h0= ag n` e`trbzrgsunt`af.
= 8
0 7 •
0 7 8
1 = = 5
(= • 0) + (8 • 1) (= • 7) + (8 • =)
8. Euntbpnbh`efs, ag lugvf, n` prbegr` dbn` ag I jfr` hfl n` tgrhgr` hfnuel`ag A, n` óntbe`. @n fmtglgr h08 hfepngt`efs n` prbegr` dbn` ag n` lugv` e`trbz.
= 8
0 7 •
0 7 8
1 = = 5
(= • 0) + (8 • 1) (= • 7) + (8 • =) (= • 8) + (8 • =)
@ • M 5 H@e Ù l • Ml Ù p
5
20E`tgeàtbh` ∝ Xge`l` 4
8/20/2019 Libro Polochic Matemática 1er. Sem
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1. @jfr` p`s`efs ` n` sgiula` dbn` ag I, qug n` euntbpnbh`efs pfr n` prbegr`hfnuel` ag A, fmtglbglaf gn gngegltf h=0.
= 8
0 7 •
0 7 8
1 = = 5
(= • 0) + (8 • 1) (= • 7) + (8 • =) (= • 8) + (8 • =)
(0 • 0) + (7 • 1)
7. Euntbpnbh`efs n` sgiula` dbn` ag I pfr n` sgiula` hfnuel` ag A. Fmtglg-efs gn gngegltf h==.
= 8
0 7 •
0 7 8
1 = = 5
(= • 0) + (8 • 1) (= • 7) + (8 • =) (= • 8) + (8 • =)
(0 • 0) + (7 • 1) (0 • 7) + (7 • =)
4. ¡R h`sb tgrebl`efs! Euntbpnbh`efs n` sgiula` dbn` ag I pfr n` tgrhgr`hfnuel` ag A, n` óntbe`. Fmtglgefs gn gngegltf h=8.
= 8
0 7 • 0 7 8
1 = = 5
(= • 0) + (8 • 1) (= • 7) + (8 • =) (= • 8) + (8 • =)
(0 • 0) + (7 • 1) (0 • 7) + (7 • =) (0 • 8) + (7 • =)
9. [`r` dbl`nbz`r, fpgr`efs n`s euntbpnbh`hbflgs y n`s sue`s. Hfl gstf nngi`efs` lugstr` e`trbz rgsunt`af.
(= • 0) + (8 • 1) (= • 7) + (8 • =) (= • 8) + (8 • =)
(0 • 0) + (7 • 1) (0 • 7) + (7 • =) (0 • 8) + (7 • =) 5
01 04 0=
=0 07 08
I • A 5= 8
0 7 •
0 7 8
1 = = 5
01 04 0=
=0 07 08
Tg`efs ftrf gcgepnf, `jfr` hfl afs e`trbhgs ag fragl 8. Gs gx`ht`egltg gnebsef prfhgabebgltf3
D • ] 5; ‘= 0
8 1 ‘0
= ; =
•1 8 7
7 ‘7 ;
8 1 0
0. Euntbpnbh`efs n` prbegr` dbn` ag D pfr h`a` ul` ag n`s hfnuel`s ag ] ysue`efs sus prfauhtfs. Hfl gstf fmtglargefs n` prbegr` dbn` ag n` lugv`e`trbz.
D • ] 5; ‘= 0
8 1 ‘0
= ; =
•1 8 7
7 ‘7 ;
8 1 0
5
(; • 1) + (‘= • 7) + (0 • 8) (; • 8) + (‘= • ‘7) + (0 • 1) (; • 7) + (‘= • ;) + (0 • 0)
=. Euntbpnbh`efs n` sgiula` dbn` ag D pfr h`a` ul` ag n`s hfnuel`s ag ] ysue`efs sus prfauhtfs. ]g`nbz`laf nf `ltgrbfr, fmtglargefs n` sgiula`dbn` ag n` lugv` e`trbz.
D • ] 5; ‘= 0
8 1 ‘0
= ; =
•1 8 7
7 ‘7 ;
8 1 0
5
(; • 1) + (‘= • 7) + (0 • 8) (; • 8) + (‘= • ‘7) + (0 • 1) (; • 7) + (‘= • ;) + (0 • 0)
(8 • 1) + (1 • 7) + (‘0 • 8) (8 • 8) + (1 • ‘7) + (‘0 • 1) (8 • 7) + (1 • ;) + (‘0 • 0)
2=BIG] ∝ [fnfhjbh
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8. Fpgr`efs n` euntbpnbh`hbøl ag n` tgrhgr` y óntbe` dbn` ag D pfr h`a` ul`ag n`s hfnuel`s ag ] y sue`efs sus prfauhtfs. Hfl gstf fmtglarå`efs n`óntbe` dbn` ag n` e`trbz rgsunt`af.
D • ] 5; ‘= 0
8 1 ‘0
= ; =
•1 8 7
7 ‘7 ;
8 1 0
5
(; • 1) + (‘= • 7) + (0 • 8) (; • 8) + (‘= • ‘7) + (0 • 1) (; • 7) + (‘= • ;) + (0 • 0)
(8 • 1) + (1 • 7) + (‘0 • 8) (8 • 8) + (1 • ‘7) + (‘0 • 1) (8 • 7) + (1 • ;) + (‘0 • 0)
(= • 1) + (; • 7) + (= • 8) (= • 8) + (; • ‘7) + (= • 1) (= • 7) + (; • ;) + (= • 0)
1. [`r` dbl`nbz`r, rgsfnvgefs n`s fpgr`hbflgs qug quga`l y nngi`efs ` lugstrfrgsunt`af dbl`n.
(; • 1) + (‘= • 7) + (0 • 8) (; • 8) + (‘= • ‘7) + (0 • 1) (; • 7) + (‘= • ;) + (0 • 0)
(8 • 1) + (1 • 7) + (‘0 • 8) (8 • 8) + (1 • ‘7) + (‘0 • 1) (8 • 7) + (1 • ;) + (‘0 • 0)
(= • 1) + (; • 7) + (= • 8) (= • 8) + (; • ‘7) + (= • 1) (= • 7) + (; • ;) + (= • 0)
5 ‘9 01 0
=2 ‘07 01
01 01 0=
D • ] 5
; ‘= 0
8 1 ‘0
= ; = •
1 8 7
7 ‘7 ;
8 1 0 5
‘9 01 0
=2 ‘07 01
01 01 0=
¡Wl hflsgcf!
Xb `n blbhbf sg ng abhunt` umbh`rsg gl n`s n`s y hfnuel`s qug agmg fpgr`r p`r`h`nhun`r `niól gngegltf ag n` e`trbz rgsunt`af, pugag utbnbz`r n` tâhlbh` qug ngefstr`rgefs ` hfltblu`hbøl. Fmsgrvg gn gcgepnf.
@ • M 57 4
= 8
4 =
•7 4 2
0 = ‘=
Nf qug j`hgefs gs dfre`r 8 t`mn`s, ul` p`r`n` e`trbz @, ftr` p`r` n` e`trbz M y ul` p`r` n`e`trbz rgsunt`af (H). N`s dfre`efs hfl n`s n`sy hfnuel`s nblg`a`s, t`n hfef sg eugstr` gl n`iur` ag n` agrghj`.
[fr gcgepnf, sb qugrgefs j`nn`r gn gngegltf H00,agmgefs fpgr`r n` n` y hfnuel` qug sg nblg`lhfl gst` h`sbnn`. Gl gstg h`sf sgrå` n` prbegr` n`ag @ y n` prbegr` hfnuel` ag M.
Gst` iur` lfs `yua` ` iub`rlfs sfmrg quâ hfnuel`s y quâ n`s agmgefs fpg-r`r p`r` fmtglgr ul agtgrebl`af gngegltf ag H. Hfl n` pràhtbh`, sg vfnvgrà eàsàibn y y` lf lghgsbt`rà iub`rsg ag gst` dfre`.
7 4 2
0 = ‘=
M
H
@7 4
= 8
4 =
7 4 2
0 = ‘=
M
H
@80 1= 88
08 0: 0=
8= 1; 7;
7 4
= 8
4 =
28E`tgeàtbh` ∝ Xge`l` 4
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Gcgrhbhbf =@jfr` ng tfh` pr`htbh`r ` ustga. ]gsugnv` n`s euntbpnbh`hbflgs sbiubgltgs. Wtbnbhg jfc`s `p`rtg y gshrbm`su rgspugst` gl gn gsp`hbf hfrrgspflabgltg. Qr`m`cg gl fragl y nbepbgz`, su frbglt`afr pfarå` pgabrnghflst`lhb` ag sus prfhgabebgltfs.
0) 8 = ‘0 ;
• 8 70 7
=)= ‘8 1 •
‘1 ;0 =
‘= 0
8) ; 0
‘= 0
•
7 ‘0
0 0
1) ; 0
= 1
8 ‘=
•
1 ‘0
‘8 8
7) 8 1
4 ‘=
‘8 ;
• ‘1 ‘=
; 8
4)= 8 ‘=
‘7 ; ‘0
•
= ‘0
‘0 0
‘1 8
9) ; 0 7
= 4 0
; 8 ‘0
•
‘0 = 9
0 1 0
= ‘= 2
:) 8 ‘7 0
7 ; ‘0
8 ‘= 0
•
8 ‘7 0
1 ; 4
0 ‘1 =
2) 0 0 0
= ; 0
8 1 ‘0
•
‘= 8 ‘1
; 1 ‘7
9 ; ‘1
0;) = ‘= :
0 ; 9
; = ;
•
0 4 ‘8
‘= ‘0 ;
9 = 0
21BIG] ∝ [fnfhjbh
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]gsuegl
0. N` euntbpnbh`hbøl f prfauhtf ag e`trbhgs gs ul` fpgr`hbøl qug pugag gdghtu`rsg gltrg afse`trbhgs f mbgl gltrg ul` e`trbz y ul lóegrf rg`n f gsh`n`r. [`r` rgsfnvgr gstg prfhgabebgltfagmgefs iub`rlfs pfr agtgrebl`a`s rgin`s.
0.0 [`r` euntbpnbh`r ul` e`trbz pfr ul lóegrf rg`n sg agmg euntbpnbh`r gsg lóegrf pfr h`a`ulf ag nfs gngegltfs ag n` e`trbz. Xg pugag euntbpnbh`r ul` e`trbz ag hu`nqubgr fragl.
o @ 5 o ` m h
a g d
i j b
5`o mo ho
ao go do
io jo bo
0.= [`r` euntbpnbh`r afs e`trbhgs gltrg så , lfs `pfy`efs gl gn prfhgabebgltf sbiubgltg3
• Xg hfeprugm` qug gn lóegrf ag hfnuel`s ag n` prbegr` e`trbz (@) hfblhba` hfl gn ló-egrf ag dbn`s ag n` sgiula` (M). Ag nf hfltr`rbf, lf sg pugagl fpgr`r.
• Xg euntbpnbh` n` prbegr` dbn` ag @ pfr n` prbegr` hfnuel` ag M. [rbegr gngegltf pfrprbegrf, sgiulaf pfr sgiulaf, gth. y sue`efs nfs prfauhtfs. Hfl gstf hflsgiubefs gnprbegr gngegltf ag n` e`trbz rgsunt`af.
• Xgiubefs euntbpnbh`laf n` prbegr` dbn` ag @ pfr n`s sbiubgltgs hfnuel`s ag M.
• ]gpgtbefs gn prfhgabebgltf hfl n`s sbiubgltgs dbn`s ag @ y h`a` hfnuel` ag M.
• Nugif ag tgrebl`r hfl tfa`s n`s dbn`s ag @, nngi`efs `n rgsunt`af dbl`n.
H 5 @ • M 5` m h
a g d
i j b
•r s t
u v w
x y z
5`r + mu + hx `s + mv + hy `t + mw + hz
ar + gu + dx as + gv + dy at + gw + dz
ir + ju + bx is + jv +by it + jw + bz
Ng `lbe`efs ` qug fmsgrvg nfs vbagfs sbiubgltgs sfmrg n` euntbpnbh`hbøl ag e`trbhgs, gstfs ng pugagl
sgrvbr p`r` `hn`r`r `niul` aua`. ]ghugrag qug sb ng quga` `niul` blqubgtua, sbgeprg pugag hflsunt`rn`hfl su frbglt`afr gl gn hårhunf ag gstuabf.
• Euntbpnbh`hbøl ag ul` e`trbz pfr ul lóegrf rg`n3 iff.in/vZxT2]
• Euntbpnbh`hbøl ag e`trbhgs3 iff.in/7rp;vh iff.in/H=Ca7S
Q embâl ng prgsglt`efs `niulfs gcgrhbhbfs gxtr` p`r` qug pr`htbqug nf qug j` `prglabaf gst` sge`l`3
• [rfauhtf ag e`trbhgs ag fragl 83 iff.in/jxoy;X
Blvgstbiug gl n` rga...
27E`tgeàtbh` ∝ Xge`l` 4
8/20/2019 Libro Polochic Matemática 1er. Sem
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@htbvba`a 0. Ageugstrg nf prglabaf
Gshrbm` "d`nsf" f "vgra`agrf" sgiól hfrrgspfla` gl h`a` glulhb`af.
0) Wl lóegrf rg`n pugag sgr euntbpnbh`af pfr ul` e`trbz ag hu`nqubgr fragl.
=) Xb ul` e`trbz @ tbglg = hfnuel`s y ul` e`trbz M tbglg = dbn`s, sg pugagl euntbpnbh`r.
8) Xb ul` e`trbz ] tbglg 8 hfnuel`s y ul` e`trbz E tbglg 1 dbn`s, sg pugagl euntbpnbh`r.
1) Xb ul` e`trbz Q tbglg = hfnuel`s y ul` e`trbz J tbglg = dbn`s, lf sg pugagl euntbpnbh`r.
@htbvba`a =. [r`htbqug nf `prglabaf
@. ]gsugnv` n`s euntbpnbh`hbflgs sbiubgltgs. Qr`m`cg gl fragl.
0) ‘=
8 =
‘1 0
=) 7 0 = ‘7 1
8) ‘8
; 0
= 1
8 ‘=
1)9 ‘9 8 ; =
0 1 ‘= 8
7)1
‘0 ; =
1 ‘7 4
0 ; 8
4) ‘:
7 ; 0
‘: = ;
0 0 2
@utfhfltrfn
24BIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
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M. ]gsugnv` n`s euntbpnbh`hbflgs sbiubgltgs. Qr`m`cg gl fragl, pugag utbnbz`r jfc`s `p`rtg sb nf lghgsbt`]ghugrag qug su frbglt`afr pugag sfnbhbt`rng hflst`lhb` ag sus prfhgabebgltfs.
0) 8 =
‘0 ;
•
8 7
0 7
=) ‘= :
8 ;
•
0 =
; ‘=
8) ‘1 ; = •
; =
0 ‘8
8 ‘0
1) ; ‘0
1 :
•
0 ‘4
7 7
7) 8 ‘=
; ‘0
1 8
•
8 0 ‘=
; 0 =
4)8 7 ‘0
= 0 0
•
8 9
‘0 0
; ‘8
9) = 8 ‘=
; 0 ;
9 ‘1 ;
•
= 0
; ‘1
; =
:) ‘: ; 2
0 7 ‘9
8 8 =
•
1 = ‘=
0 0 0
0 8 ;
2) = ‘0 0
8 9 1
0 ‘= 0
•
; 9 =
‘0 0 =
= 8 ‘7
0;) 2 ‘0 0
‘1 : =
; ; 8
•
; ; 0
2 ‘9 :
‘= 8 ‘0
29E`tgeàtbh` ∝ Xge`l` 4
8/20/2019 Libro Polochic Matemática 1er. Sem
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@. ]gsugnv` n`s ghu`hbflgs nblg`ngs sbiubgltgs. Gshrbm` gn v`nfr ag n` blhøilbt` gl n` nålg`hfrrgspflabgltg.
0) x + 8 5 2 x 5
=) x ‘ : 5 7 x 5
8) x + 9 5 07 x 5
1) x + 0; 5 ‘= x 5
7) x ‘ : 5 00 x 5
4) x ‘ 9 5 ; x 5
9) x ‘ ; 5 0 x 5
:) x + 2 5 2 x 5
2) x + 1 5 00 x 5
0;) x + : 5 04 x 5
00) : ‘ x 5 = x 5
0=) 07 ‘ x 5 2 x 5
08) 0; + x 5 0= x 5
01) ‘= + x 5 = x 5
07) 0 + x 5 7 x 5
04) ‘4 ‘ x 5 ‘8 x 5
09) ; ‘ x 5 0 x 5
0:) 0; + x 5 0; x 5
02) = ‘ x 5 ; x 5
=;) ‘: ‘ x 5 1 x 5
=0) 1 + x 5 0 x 5
==) 2 ‘ x 5 ‘9 x 5
M. ]gsugnv` n`s ghu`hbflgs nblg`ngs sbiubgltgs. Gshrbm` gn v`nfr ag n` blhøilbt` gl n` nålg`hfrrgspflabgltg.
0) : y 5 04 y 5
=) 1 y 5 =; y 5
8) ‘2 y 5 71 y 5
1) 8 y 5 0= y 5
7) 7 y 5 ‘=7 y 5
4) ‘= y 5 = y 5
9) 9 y 5 87 y 5
:) ‘1 5 =; y 5
2) 00 y 5 ‘22 y 5
0;) ‘: y 5 ‘1; y 5
00) 4 y 5 0: y 5
0=) ‘9 y 5 12 y 5
08) : y 5 41 y 5
01) ‘= y 5 ‘: y 5
07) ‘ y 5 ‘0 y 5
04) 0; y 5 ; y 5
09) = y 5 ‘0= y 5
0:) ‘8 y 5 ‘2 y 5
02) 8 y 5 ‘2 y 5
=;) 9 y 5 ‘74 y 5
=0) ‘4 y 5 1: y 5
==) 7 y 5 87 y 5
2:BIG] ∝ [fnfhjbh
@ibnba`a ag hànhunf eglt`n
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@ hfltblu`hbøl glhfltr`rà flhg åtges. Gl h`a` ulf p`rghgl hu`trf iur`s, trgs ag gnn`s iu`ra`l ul`rgn`hbøl ag niól tbpf. Wstga agmg bagltbh`r n` iur` qug lf gstà rgn`hbfl`a` y hbrhun`rn`. Iuågsg pfrgn gcgepnf. Gst` prugm` ng `yua`rà ` ags`rrfnn`r su h`p`hba`a ag `lànbsbs y r`zfl`ebgltf.
;) `. m. h. a.
0) `. m. h. a.
=) `. m. h. a.
8) `. m. h. a.
1) `. m. h. a.
7) `. m. h. a.
4) `. m. h. a.
9) `. m. h. a.
:) `. m. h. a.
2) `. m. h. a.
0;) `. m. h. a.
22E`tgeàtbh` ∝ Xge`l` 4
]`zfl`ebgltf nøibhf
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E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfir`af gl
prfhgsflf
nfir`af
A g s p u â s a g g s t u a b ` r . . . Hflfzhf y pnbhf n`s rgin`s p`r` euntbpnbh`r ul` e`trbz pfr ul lóegrf
rg`n.
[r`htbhf n` euntbpnbh`hbøl ag ul` e`trbz pfr ul lóegrf rg`n.
Hflfzhf y `pnbhf n`s rgin`s p`r` euntbpnbh`r e`trbhgs gltrg så.
[r`htbhf n` euntbpnbh`hbøl gltrg e`trbhgs.
[`r` gn ags`rrfnnf agn r`zfl`ebgltf `mstr`htf y nøibhf sg gepng`l n`s prugm`shfl e`trbhgs, pugs rgqubgrgl gn hfrrghtf e`lgcf g bagltbh`hbøl ag nfs gngegl-tfs qug dfre`l n` e`trbz y gn `lànbsbs p`r` agshumrbr gn gngegltf qug sbiug gnp`trøl.
@ hfltblu`hbøl, ustga glhfltr`rà hu`trf blhbsfs hfl e`trbhgs ag 8 Ù 8? agmgràgngibr ag gltrg n`s fphbflgs gn gngegltf qug j`hg d`nt` gl n` h`sbnn` v`hå` yhbrhun`rnf.
0)0 1 0
8 0 8
1 0
=)
8)
1)9 04
: 7 08
1 1 :
`. 0 h. 7
m. 1 a. =
`. h.
m. a.
`. h.
m. a.
`. 4 h. 8
m. =0 a. 2
0;;BIG] ∝ [fnfhjbh
]gvbsg su `prglabz`cg
Ags`rrfnng lugv`s j`mbnba`ags
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90
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
Xuhgsbflgs e`tgeàtbh`s¶Puâ glhfltr`rà gst` sge`l`6
N` suhgsbøl ag Dbmfl`hhb
Xuhgsbflgs `rbteâtbh`s y igfeâtrbh`s
Ghu`hbflgs ag prbegr ir`af
Xgrbgs lueârbh`s
Gst` sge`l` nfir`rà3
Hfeprglagr n`s h`r`htgråstbh`s ag ul` suhgsbøl `rbteâtbh`.
H`nhun`r gn târeblf d`nt`ltg gl ul` suhgsbøl `rbteâtbh`.
Hfeprglagr n`s h`r`htgråstbh`s ag ul` suhgsbøl igfeâtrbh`. H`nhun`r gn târeblf d`nt`ltg gl ul` suhgsbøl igfeâtrbh`.
]gsfnvgr prfmnge`s gepng`laf n`s suhgsbflgs e`tgeàtbh`s.
]gsfnvgr ghu`hbflgs ag prbegr ir`af hfl `ibnba`a.
0;0E`tgeàtbh` ∝ Xge`l` 9
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¡[`r` hfeglz`r!N` suhgsbøl ag Dbmfl`hhb
Ngfl`raf ag [bs`, t`embâl hflfhbaf hfef Dbmfl`hhb, dug ul agst`h`af e`tg-
eàtbhf bt`nb`lf qug vbvbø gl gn sbinf ZBBB. Gltrg sus hfltrbmuhbflgs `n h`epf ag
nfs lóegrfs sg glhugltr` n` suhgsbøl ag Dbmfl`hhb, ul` sghuglhb` ag lóegrfs
qug blbhb` gl 0 y sg gxtbglag j`st` gn bldblbtf. Tg`efs `niulfs ag sus prbegrfs
gngegltfs3
0, 0, =, 8, 7, :, 08, =0, 81, 77, :2, 011, =88...
Hfef eglhbfl`efs, n` suhgsbøl hfebglz` hfl nfs lóegrfs 0 y 0, y ` p`rtbr ag
gstfs, h`a` târeblf gs n` sue` ag nfs afs `ltgrbfrgs.
0 + 0 5 =, 0 + = 5 8, = + 8 5 7, 8 + 7 5 :, 7 + : 5 08,
: + 08 5 =0, 08 + =0 5 81 ...
Nf sfrprglagltg ag gst` hflstruhhbøl e`tgeàtbh` gs qug sg prgsglt` rghurrgl-
tgegltg gl n` l`tur`ngz`. @p`rghg gl n` abstrbmuhbøl ag n`s r`e`s ag nfs àrmfngs,
gl n` dfre`hbøl ag n`s jfc`s ag ul t`nnf, gl n` rgprfauhhbøl ag nfs hflgcfs, gl
nfs drutfs ag n` pbð` trfpbh`n, gl n` abspfsbhbøl ag n`s sgebnn`s gl luegrfs`s
dnfrgs, gltrg euhjfs ftrfs h`sfs.
[fr gcgepnf, n`s jfc`s ag ul` pn`lt` lulh` hrghgl abrght`egltg ul` `m`cf ag
ftr`, mrft`l gl dfre` ag gspbr`n hfl ul` hfldbiur`hbøl qug sbiug n` sghuglhb`
Dbmfl`hhb.
Ftrf gcgepnf gs gn p`trøl gl qug sg rgprfauhgl nfs hflgcfs, ul` p`rgc` aggstfs `lbe`ngs t`ra` ul egs gl nngi`r ` su ga`a dârtbn, ` p`rtbr ag gsg efegltf
h`a` egs gliglar`rà ` ftr` p`rgc` ag hflgcfs, qug ` su vgz, nugif ag nngi`r `
n` ga`a ag dgrtbnba`a, tglarà ftr` p`rgc` eàs, y `så suhgsbv`egltg.
¡@ tr`m`c`r!
Gepng`laf gn bltgrlgt f `niól ftrf rghursf, blvgstbiug ftrf nfirf agn e`tgeàtbhf Ngfl`raf ag [bs`.Gshrbm` sus j`nn`zifs gl n`s nålg`s sbiubgltgs3
0;=BIG] ∝ [fnfhjbh
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0. Xuhgsbflgs e`tgeàtbh`s@så hfef puabefs lft`r gl n` sghhbøl ltgrbfr, ul` suhgsbøl e`tgeàtbh` gs ul
hflcultf fragl`af ag lóegrfs. H`a` ulf ag gstfs lóegrfs sg aglfebl` "târ-eblf", "gngegltf" f "ebgemrf", y `n lóegrf tft`n ag gngegltfs qug hfepflgn` suhgsbøl sg ng nn`e` "nflibtua", gst` pugag sgr lbt` f bllbt`.
0;
0.³ târeblf
=;
=.³ târeblf
8;
8.³ târeblf
1;
1.³ târeblf
7;
7.³ târeblf
J`y v`rbfs tbpfs ag suhgsbflgs e`tgeàtbh`s, qug sbiugl abdgrgltgs p`trflgs,lfsftrfs gstuab`rgefs afs3 n`s suhgsbflgs `rbteâtbh`s y n`s suhgsbflgsigfeâtrbh`s.
0.0 Xuhgsbflgs `rbteâtbh`sWl` suhgsbøl f prfirgsbøl rbteâtbh` gs ul` sgrbg ag lóegrfs t`ngs qug h`a`ulf ag gnnfs, gxhgptu`laf gn prbegr târeblf, sg fmtbglg sue`laf f rgst`laful` h`ltba`a c` `n târeblf `ltgrbfr, ` gst` h`ltba`a hflst`ltg sg ng hflfhghfef abdgrglhb` y sg rgprgsglt` hfl n` n` ngtr` a .
`0 `= `8 `1 `7 `l...
°a °a °a °a ...
Gl aflag n` ngtr` "`" rgprgsglt` h`a` târeblf y su sumålabhg blabh` n` pfsbhbøl
qug fhup` gl n` prfirgsbøl. Fmsgrvgefs gn gcgepnf sbiubgltg3
1 : 0= 04 =; =1
+1 +1 +1 +1 +1
[`r` h`nhun`r n` abdgrglhb` agmgefs rgst`r afs târeblfs hflsghutbvfs. Agpgl-abglaf sb n` abdgrglhb` (a ) ag ul` prfirgsbøl `rbteâtbh` sg` pfsbtbv , lun` flgi`tbv`, aghbefs3
• a < ;3 prfirgsbøl hrghbgltg. H`a` gngegltf gs e`yfr qug gn `ltgrbfr.
Gcgepnf3 0;, =;, 8;, 1;... (a 5 0;)
• a 5 ;3 prfirgsbøl hflst`ltg. Qfafs nfs târeblfs sfl biu`ngs.
Gcgepnf3 2, 2, 2, 2, 2... (a 5 ;)
• a > ;3 prfirgsbøl aghrghbgltg. H`a` gngegltf gs eglfr qug gn `ltgrbfr.
Gcgepnf3 0;, :, 4, 1, =, ;, ‘=, ‘1, ‘4... (a 5 ‘=)
0;8E`tgeàtbh` ∝ Xge`l` 9
Gn eulaf ag n` e`tgeàtbh`
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Qâreblf iglgr`n ag ul` suhgsbøl rbteâtbh`
[`r` h`nhun`r gn v`nfr ag ul gngegltf agshflfhbaf gl ul` suhgsbøl `rbteâtbh`,utbnbz`efs n` døreun` agn târeblf iglgr`n, n` hu`n lfs pgrebtg `vgrbiu`rhu`nqubgr târeblf ag ul` suhgsbøl sb hflfhgefs gn prbegrf ag gnnfs y suabdgrglhb`.
N` døreun` agn târeblf iglgr`n ag ul` prfirgsbøl `rbteâtbh` gs3`l 5 `0 + (l ‘ 0)a
Gl n` hu`n `l rgprgsglt` gn târeblf qug qugrgefs `vgrbiu`r ag `hugraf ` supfsbhbøl? l gs n` pfsbhbøl qug fhup` `l y `0 gs gn prbegr târeblf ag n` suhgsbøl.
Tg`efs ul gcgepnf.
Glhfltrgefs quâ lóegrf fhup` n` pfsbhbøl 12 gl n` suhgsbøl sbiubgltg3
02, 8;, 10, 7=, 48... `12
• [`r` j`nn`r gn târeblf iglgr`n, sustbtubefs n` abdgrgl-hb` (a 5 00) y gn prbegr târeblf ( 0 5 02) gl n` døreun`qug vbefs `rrbm`. Nugif, fpgr`efs.
• Glhfltr`efs `så gn târeblf iglgr`n p`r` gst` sgrbg?hfl gst` gxprgsbøl pfagefs `vgrbiu`r hu`nqubgrtâreblf hfl t`l sfnf rgepn`z`r l pfr gn lóegrf agpfsbhbøl qug fhup`.
• Hfef qugrgefs glhfltr`r gn gngegltf `12 , j`hgefsn`s sustbtuhbflgs lghgs`rb`s y fpgr`efs.
Gn gngegltf qug fhup` n` pfsbhbøl 12 gs 719.
Hfl n` døreun` agn târeblf iglgr`n t`embâl pfagefs h`nhun`r n` abdgrglhb` y gnprbegr târeblf ag ul` prfirgsbøl, sfn`egltg agmgefs agspgc`r n` v`rb`mng quglfs bltgrgs`.
Tg`efs ul óntbef gcgepnf.
Glhfltrgefs gn prbegr târeblf ag ul` suhgsbøl huyf târeblf `:2 gs 0;9: y suabdgrglhb` gs 0=.
• Agspgc`efs n` døreun` agn târeblf iglgr`n gl
dulhbøl ag `0.
• Xustbtubefs nfs a`tfs y fpgr`efs.
Glhfltr`efs qug gn prbegr târeblf gs ==.
`l 5 `0 + (l ‘ 0)a `l 5 02 + (l ‘ 0)00
`l 5 02 + 00l ‘ 00
`l 5 : + 00l
`12 5 : + 00(12)
`12 5 719
`0 5 `l ‘ (l ‘ 0)a
`0 5 `:2 ‘ (:2 ‘ 0)0=
`0 5 0;9: ‘ 0;74
`0 5 ==
0;1BIG] ∝ [fnfhjbh
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Xue` ag nfs târeblfs gl ul` prfirgsbøl `rbteâtbh`
[`r` d`hbnbt`r gn prfhgsf ag sue`r nfs gngegltfs ag ul` suhgsbøl `rbteâtbh`,gepng`efs n` døreun` sbiubgltg3
s 5 (`0 + `l )l
=
Gl aflag `0 gs gn prbegr târeblf ag n` sgrbg, l gs n` pfsbhbøl j`st` n` hu`nqugrgefs sue`r y `l gs gn v`nfr agn târeblf qug fhup` gs` pfsbhbøl.
[fli`efs `tglhbøl `n gcgepnf.
Xuegefs j`st` gn târeblf `7; ; ag ul` suhgsbøl huyf prbegr gngegltf gs 07; ysu abdgrglhb` gs =7.
• [rbegrf, glhfltr`efs gn v`nfr agn târeblf`7; ;
• Nugif, `pnbh`efs y fpgr`efs n` døreun`p`r` sue`r tfafs nfs târeblfs j`st` `7; ;
N` sue` tft`n j`st` gn gngegltf 7;; ag n` suhgsbøl gs 8 028 97;.
`7; ; 5 `0 + (l ‘ 0)a
`7; ; 5 07; + (7;; ‘ 0)(=7)
`7;; 5 0= 4=7
s 5(07; + 0= 4=7)(7;;)
=
s 5 8 028 97;
Gcgrhbhbf 0@. H`nhung n` abdgrglhb` ag n`s suhgsbflgs sbiubgltgs. Nugif, gshrbm` sb sfl hrghbgltgs, hflst`ltgs f
aghrghbgltgs. Iuågsg pfr gn gcgepnf.
;) 09, 00, 7, ‘0, ‘9, ‘08, ‘02...
0) 88, 14, 72, 9=, :7, 2:...
=) =.7, =, 0.7, 0, ;.7, ;, ‘;.7, ‘0...
8) :, :, :, :, :, :, :, :...
1) 087, 002, 0;8, :9, 90...
7) 0;;, 0;9, 001, 0=0, 0=:...
4) 18, 18, 18, 18, 18...
9) 84, 8;.7, =7, 02.7, 01...
:) 0:, =1, 8;, 84, 1=, 1:...
a 5 ‘4 , gs aghrghbgltg
0;7E`tgeàtbh` ∝ Xge`l` 9
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M. H`nhung gn târeblf qug sg ng pbag gl h`a` ul` ag n`s suhgsbflgs sbiubgltgs.
0) `18 ag n` suhgsbøl3 =1, =:, 8=, 84, 1;... `18 5
=) `22 ag n` suhgsbøl3 79, 19, 89, =9, 09... `22 5
8) `0;0 ag n` suhgsbøl3 07, 0;.7, 4, 0.7, ‘8... `0;0 5
1) `0 ag ul` suhgsbøl hfl a 5 =8 y `87 5 0;1 `0 5
7) a ag ul` suhgsbøl hfl `0 5 8 y `74 5 42;.7 a 5
4) `49 ag ul` suhgsbøl hfl `0 5 81 y a 5 9 `49 5
9) a ag ul` suhgsbøl hfl `0 5 74 y `024 5 ‘07;1 a 5
:) `0 ag ul` suhgsbøl hfl a 5 ‘2.7 y `07; 5 =27 `0 5
H. H`nhung n` sue` ag nfs gngegltfs gl n`s prfirgsbflgs sbiubgltgs. [ugag tr`m`c`r gl jfc`s `p`rtg
sb gs lghgs`rbf. ]ghugrag tr`m`c`r gl fragl y agc`r hflst`lhb` ag su prfhgabebgltf.
0) Xue`tfrb` j`st` gn gngegltf `=;;, `0 5 =7 y a 5 0=
`=;; 5 s 5
=) Xue`tfrb` j`st` gn gngegltf `07;, `0 5 7 y a 5 0;
`07; 5 s 5
8) Xue`tfrb` j`st` gn gngegltf `97, `0 5 08 y a 5 07
`97 5 s 5
1) Xue`tfrb` j`st` gn gngegltf `8;;, `0 5 =1 y a 5 :
`8;; 5 s 5
7) Xue`tfrb` j`st` gn gngegltf `0=7, `0 5 8= y a 5 =
`0=7 5 s 5
4) Xue`tfrb` j`st` gn gngegltf `7;, `0 5 7 y a 5 02
`7; 5 s 5
9) Xue`tfrb` j`st` gn gngegltf `==7, `0 5 8 y a 5 8
`==7 5 s 5
:) Xue`tfrb` j`st` gn gngegltf `0;;, `0 5 0; y a 5 9
`0;; 5 s 5
0;4BIG] ∝ [fnfhjbh
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0.= Xuhgsbflgs igfeâtrbh`s
Wl` suhgsbøl igfeâtrbh` gs ul` suhgsbøl gl n` qug h`a` târeblf sg fmtbglgeuntbpnbh`laf `n `ltgrbfr pfr ul` h`ltba`a c` aglfebl`a` r`zøl (r ).
= 4 0: 71 04= 1:4 ...
Ù8 Ù8 Ù8 Ù8 Ù8
N` r`zøl sg fmtbglg abvbabglaf afs gngegltfs hflsghutbvfs ag n` sgrbg. Gl gstgtbpf ag suhgsbøl, y` lf gs pfsbmng h`nbh`r ul` suhgsbøl igfeâtrbh` hfef hrg-hbgltg, aghrghbgltg f hflst`ltg ag `hugraf `n sbilf ag su r`zøl.
@jfr`, sb n` r`zøl gs ul` dr`hhbøl prfpb`, n` prfirgsbøl gs aghrghbgltg? sb sur`zøl tfe` gn v`nfr ag n` ulba`a, gs hflst`ltg.
Xb r gs ul lóegrf gltgrf pfsbtbvf, n` suhgsbøl sgrà hrghbgltg y sb gs ul lóegrfgltgrf lgi`tbvf, n` prfirgsbøl brà v`rb`laf ag sbilf hfl h`a` gngegltf.
Qâreblf iglgr`n ag ul` suhgsbøl igfeâtrbh`Ag dfre` sbebn`r qug hfl n`s prfirgsbflgs rbteâtbh`s, sg pugag fmtglgr gn v`nfrag ul gngegltf agtgrebl`af ag n` sghuglhb` egab`ltg n` døreun` agn târeblfiglgr`n3
`l 5 `0 • r (l ‘ 0)
Gl aflag `l gs gn gngegltf qug sg agsg` hflfhgr, l gs n` pfsbhbøl qug fhup`gstg târeblf? `0 gs gn prbegr gngegltf y r , n` r`zøl.
Nfs p`sfs p`r` h`nhun`r ul târeblf agshflfhbaf sfl euy p`rghbafs ` nfs qugsgiubefs hfl n`s prfirgsbflgs `rbteâtbh`s.
[fli`efs `tglhbøl ` ul gcgepnf.
Glhfltrgefs quâ lóegrf fhup` n` pfsbhbøl 08 gl n` suhgsbøl sbiubgltg3
=, 4, 0:, 71, 04=, 1:4...
• H`nhun`efs n` r`zøl (r ) p`r` gst` sgrbg abvb-abglaf afs târeblfs hflsghutbvfs.
• @jfr`, rgepn`z`efs gn prbegr târeblf (`0)
y n` r`zøl gl n` døreun`.Hfl gstf fmtglgefs gn târeblf iglgr`n.
• [fr óntbef, sustbtubefs l, gs aghbr, gn lóeg-rf ag pfsbhbøl qug fhup` gn târeblf qugqugrgefs j`nn`r y fpgr`efs.
Gn târeblf qug fhup` n` pfsbhbøl 08 gs 0 ;4= ::=.
4
= 5 8
r 5 8
`l 5 `0 • r (l ‘ 0)
`l 5 = • 8(l ‘ 0)
`l 5 = • 8(l ‘ 0)
`08 5 = • 8(08 ‘ 0)
`08 5 0 ;4= ::=
]ghugrag qug hfl n`døreun` agn târeblfiglgr`n t`embâl pfag-efs h`nhun`r n` r`zøly gn prbegr târeblf agul` prfirgsbøl, sfnfm`st` hfl agspgc`rn` v`rb`mng qug lfsbltgrgs`. [fr gcgepnf3
r 5 l ‘ 0`l
`0
0;9E`tgeàtbh` ∝ Xge`l` 9
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Gcgrhbhbf =@. H`nhung n`s r`zflgs p`r` n`s suhgsbflgs sbiubgltgs. Nugif, gshrbm` sb sfl hrghbgltgs, aghrghbgltgs f
hflst`ltgs. Iuågsg pfr gn gcgepnf.
;) 7;;, 0;;, =;, 1, ;.:...
0) =, :, 8=, 0=:, 70=, =;1:...
=) 8, 2, =9, :0, =18, 9=2...
8) 0;;;, 7;;, =7;, 0=7...
1) 0, 7, =7, 0=7, 4=7, 80=7...
7) 08, 08, 08, 08, 08, 08...
4) 0;;;, =;;, 1;, :...
9) :, 1:, =::, 09=:...
:) 7;, 7, ;.7, ;.;7, ;.;;7...
Xue` ag nfs târeblfs gl ul` prfirgsbøl igfeâtrbh`
@n biu`n qug hfef `prglabefs hfl n`s suhgsbflgs `rbteâtbh`s, sb agsg`efssue`r nfs târeblfs ag ul` prfirgsbøl igfeâtrbh`, pfagefs d`hbnbt`r lugstrftr`m`cf gepng`laf n` døreun` sbiubgltg3
s 5 (`l • r ) ‘ `0
r ‘ 0
Gl aflag `l gs gn v`nfr agn târeblf qug fhup` n` pfsbhbøl j`st` aflag qugrg-efs sue`r? r gs n` r`zøl igfeâtrbh` y `0
gs gn prbegr târeblf.
Hfeprglagrgefs egcfr hfl ul gcgepnf.
Xuegefs j`st` gn târeblf `07 ag ul` suhgsbøl huyf prbegr gngegltf gs 8 y sur`zøl gs =.
• [rbegrf agmgefs glhfltr`r gn v`nfr agngngegltf `07
• @jfr`, `pnbh`efs y fpgr`efs n` døreun`p`r` sue`r tfafs nfs gngegltfs j`st` `07
N` sue`tfrb` ag nfs gngegltfs j`st` `07 gs 2: 8;0. ¶R` vg qug gs euy dàhbn6
`07 5 `0 • r (l ‘ 0)
`07 5 8 • =(07 ‘ 0)
`07 5 12 07=
s 5 (`07 • r ) ‘ `0
r ‘ 0
s 5(12 07= • =) ‘ 8
= ‘ 0
s 5 2: 8;0
r 5 0
7 , aghrghbgltg
0;:BIG] ∝ [fnfhjbh
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M. H`nhung gn târeblf qug sg ng pbag gl h`a` ul` ag n`s suhgsbflgs sbiubgltgs. [ugag `prfxbe`r susrgspugst`s sgiól hrg` hflvglbgltg.
0) `0; ag n` suhgsbøl3 =, 1, :, 04, 8=... `0; 5
=) `07 ag n` suhgsbøl3 0, 8, 2, =9, :0... `07 5
8) `9 ag n` suhgsbøl3 7;;, 0;;;, =;;;... `9 5
1) `7 ag n` suhgsbøl3 2;, 8;, 0;... `7 5
7) `08 ag n` suhgsbøl3 8, ‘2, =9, ‘:0, =18... `08 5
4) `0 ag ul` suhgsbøl hfl r 5 7 y `9 5 0;2 897 `0 5
9) N` r`zøl ag ul` suhgsbøl hfl `0 5 1 y `7 5 8=1 r 5
:) N` r`zøl ag ul` suhgsbøl hfl `0 5 : y `7 5 0; 84: r 5
2) `9 ag n` suhgsbøl3 9, ‘74, 11:, ‘87:1... `9 5
0;) `0 ag ul` suhgsbøl hfl r 5 4 y `: 5 0 822 4:; `0 5
H. H`nhung n` sue` ag nfs gngegltfs gl n`s prfirgsbflgs sbiubgltgs. [ugag tr`m`c`r gl jfc`s `p`rtgsb gs lghgs`rbf. ]ghugrag tr`m`c`r gl fragl y agc`r hflst`lhb` ag su prfhgabebgltf.
0) Xue`tfrb` j`st` gn gngegltf `2, `0 5 9 y r 5 8
`2 5 s 5
=) Xue`tfrb` j`st` gn gngegltf `:, `0 5 0;; ;;; y r 5 0
=
`: 5 s 5
8) Xue`tfrb` j`st` gn gngegltf `07, `0 5 2 y r 5 =
`07 5 s 5
1) Xue`tfrb` j`st` gn gngegltf `7, `0 5 =7 y r 5 9
`7 5 s 5
7) Xue`tfrb` j`st` gn gngegltf `4, `0 5 9 y r 5 :
`4 5 s 5
4) Xue`tfrb` j`st` gn gngegltf `0=, `0 5 1 y r 5 8
`0= 5 s 5
0;2E`tgeàtbh` ∝ Xge`l` 9
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@pnbh`hbflgs ag n`s suhgsbflgs `rbteâtbh`s y igfeâtrbh`s
N`s àrg`s gl n`s qug pfagefs `pnbh`r lugstrfs hflfhbebgltfs sfmrg suhgsbflgse`tgeàtbh` sfl euy v`rb`a`s. [fr gcgepnf3
Gl n` m`lh`, nfs blvgrsbflbst`s y n`s pgrsfl`s qug tbglgl huglt`s m`lh`rb`s,pugagl h`nhun`r n`s i`l`lhb`s pftglhb`ngs sfmrg ul lgifhbf gepng`laf n`s
prfirgsbflgs `rbteâtbh`s.Gl nfs agpfrtgs, gl abvgrs`s abshbpnbl`s, hfef gn tglbs, sg gepng`l n`s suhgsbflgsigfeâtrbh`s y `rbteâtbh`s p`r` `l`nbz`r n`s pfsbmngs gstr`tgib`s qug ul cui`afrpugag tfe`r gl ul p`rtbaf.
Gl n` bliglbgrå`, nfs lóhngfs ag ul e`tgrb`n r`ab`htbvf sg agsbltgir`l gl dfre`ulbdfreg gl ul agtgrebl`af pgråfaf ag tbgepf, gstg hànhunf pugag sgr agshrbtfpfr ul` prfirgsbøl igfeâtrbh`.
[fr t`ltf, pfagefs rgsfnvgr euhjfs prfmnge`s hfl t`l sfnf pfagr bagltbh`rhfrrght`egltg nfs a`tfs ag n` prfirgsbøl y pnbh`r n`s døreun`s qug pr`htbh`efs.
Gcgrhbhbf 8]gsugnv` nfs prfmnge`s sbiubgltgs gepng`laf nf prglabaf sfmrg prfirgsbflgs rbteâtbh`s y igfeâ-trbh`s. Ng `hflsgc`efs prbegrf bagltbh`r y hn`sbh`r nfs a`tfs agn prfmnge`.
0) N` afsbs ag ul egabh`egltf gs ag 7;; ei gn prbegr aå` y 07 ei eglfs h`a` ulf ag nfs aå`ssbiubgltgs.
`. Xb gn tr`t`ebgltf aur` =7 aå`s, ¶huàltfs ebnbir`efs tfe`rà gn gldgref gl gn óntbef aå`6
m. ¶Huàltfs ebnbir`efs j`mrà tfe`af gn gldgref ` nf n`rif ag tfaf gn tr`t`ebgltf6
=) Wl tbpf ag m`htgrb` sg rgprfauhg aupnbhàlafsg h`a` egab` jfr`. Xb gl n` prbegr` egab` jfr` sgfmsgrv`l 0;;; m`htgrb`s, ¶huàlt`s j`mràl nugif ag 7 jfr`s6
8) Nfs vghblfs ag ul` hfeulba`a aghbagl `jfrr`r p`r` hflstrubr ul s`nøl ag usfs eóntbpngs. Gnprbegr aå` agpfsbt`l P07;.;; gl ul` `nh`lhå` y h`a` aå` aupnbh`l n` h`ltba`a agpfsbt`a` gn aå``ltgrbfr. ¶Huàltf ablgrf tglaràl qug agpfsbt`r gn sâptbef aå`6
1) Xb hfnfhàr`efs P1,;;;.;; gl ul` huglt` ag `jfrrf gl n` qug i`l`rå`efs P1.7; pfr h`a` aå` qugp`sg gn ablgrf agpfsbt`af, ¶huàltf ablgrf j`mrå` nugif ag 0:; aå`s6
7) N` pfmn`hbøl ag pfnnfs ag ul` ir`lc` sg trbpnbh` h`a` `ðf. Xb agspuâs ag 7 ðfs j`y 1784 lbe`ngs,¶huàltfs j`må` gl gn prbegr `ðf6
4) Xb gn nulgs ustga i`lø P77.;; y gl nfs aå`s sbiubgltgs i`lø gn afmng agn `ltgrbfr.
`. ¶Huàltf i`lø gn sàm`af6
m. ¶Huàltf i`lø gl tft`n ag nulgs ` sàm`af6
00;BIG] ∝ [fnfhjbh
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]gsuegl
0. Wl` suhgsbøl e`tgeàtbh` gs ul hflcultf fragl`af ag lóegrfs. H`a` ulf ag gstfs lóegrfssg aglfebl` "târeblf", "gngegltf" f "ebgemrf", y `n lóegrf tft`n ag gngegltfs qug hfepflgn` suhgsbøl sg ng nn`e` „nflibtua―, gst` pugag sgr dblbt` f bldblbt`.
0.0 Wl` suhgsbøl f prfirgsbøl `rbteâtbh` gs ul` sgrbg ag lóegrfs t`ngs qug h`a` ulf ag gnnfs,gxhgptu`laf gn prbegr târeblf, sg fmtbglg sue`laf f rgst`laf ul` h`ltba`a dbc` `n târeblf`ltgrbfr, ` gst` h`ltba`a hflst`ltg sg ng hflfhg hfef abdgrglhb` y sg rgprgsglt` hfl n` n` ngtr` a .
08 =1 87 14 79 4: ...
+00 +00 +00 +00 +00
N` døreun` agn târeblf iglgr`n p`r` ul` suhgsbøl `rbteâtbh` lfs pgrebtg `vgrbiu`r hu`n-qubgr târeblf ag ul` suhgsbøl sb hflfhgefs gn prbegrf ag gnnfs y su abdgrglhb`.
`l 5 `0 + (l ‘ 0)a
[`r` d`hbnbt`r gn prfhgsf ag sue`r nfs gngegltfs ag ul` suhgsbøl `rbteâtbh`, gepng`efs n`døreun` sbiubgltg3
s 5 (`0 + `l )l
=
0.= Wl` suhgsbøl igfeâtrbh` gs ul` suhgsbøl gl n` qug h`a` târeblf sg fmtbglg euntbpnbh`laf `n `ltgrbfr pfr ul` h`ltba`a dbc` aglfebl`a` r`zøl (r ).
: =1 9= =04 41: 0211 ...
Ù8 Ù8 Ù8 Ù8 Ù8
N` døreun` agn târeblf iglgr`n p`r` ul` suhgsbøl igfeâtrbh` lfs pgrebtg `vgrbiu`r hu`n-
qubgr târeblf ag ul` suhgsbøl sb hflfhgefs gn prbegrf ag gnnfs y n` r`zøl.`l 5 `0 • r (l ‘ 0)
[`r` d`hbnbt`r gn prfhgsf ag sue`r nfs gngegltfs ag ul` suhgsbøl igfeâtrbh`, gepng`efs n`døreun` sbiubgltg3
s 5 (`l • r ) ‘ `0
r ‘ 0
Xb agsg` s`mgr ul pfhf eàs sfmrg n` suhgsbøl Dbmfl`hhb, ng blvbt`efs ` qug vbsbtg n`s pàibl`s wgmsbiubgltgs3
• ¶Puâ gs n` suhgsbøl ag Dbmfl`hhb6 iff.in/xnuDA[• N` sghuglhb` ag Dbmfl`hhb3 iff.in/cTfNxt
[`r` hflfhgr eàs sfmrg n`s prfpbga`ags ag n`s suhgsbflgs `rbteâtbh`s y igfeâtrbh`s, pugag vbsbt`r3• [rfirgsbflgs `rbteâtbh`s3 iff.in/mXz1L;• [rfirgsbflgs igfeâtrbh`s3 iff.in/;tIZHf
000E`tgeàtbh` ∝ Xge`l` 9
Blvgstbiug gl n` rga...
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@htbvba`a 0. Ageugstrg nf prglabaf
@. H`nhung n` abdgrglhb` gl n`s suhgsbflgs sbiubgltgs. Nugif, gshrbm` sb sfl hrghbgltgs, hflst`ltgs faghrghbgltgs.
0) 877, 100, 149, 7=8, 792...
=) =7, 0;8, 0:0, =72, 889...
8) 22, 98, 19, =0, ‘7, ‘80...
1) 49, 49, 49, 49, 49...
7) ‘0=, ‘18, ‘91, ‘0;7, ‘084...
4) 22, 0;;, 0;0, 0;=, 0;8...
M. H`nhung n` r`zøl ag n`s suhgsbflgs sbiubgltgs.
0) =, 0=, 9=, 18=, =72=...
=) 00, ==, 11, ::, 094...
8) =777, 700, 0;=.=, =;.11...
1) :, ‘::, 24:, ‘0;41:...
7) 0;9:, 782, =42.7, 081.97...
4) 8, ‘=0, 019, ‘0;=2, 9=;8...
@htbvba`a =. [r`htbqug nf `prglabaf
@. H`nhung nf qug sg ng sfnbhbt` ` hfltblu`hbøl.
0) `49 ag n` suhgsbøl3 99, 008, 012, 0:7... `49 5
=) `81 ag n` suhgsbøl3 02, 4, ‘9, ‘=;, ‘88... `81 5
8) `=;= ag n` suhgsbøl3 7, 00, 09, =8, =2... `=;= 5
1) `024 ag ul` suhgsbøl hfl a 5 01 y `0 5 =8 `024 5
7) Xue`tfrb` j`st` gn gngegltf `=;7, `0 5 87 y a 5 9 s 5
4) Xue`tfrb` j`st` gn gngegltf `08=, `0 5 01 y a 5 0= s 5
@utfhfltrfn
00=BIG] ∝ [fnfhjbh
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M. H`nhung nf qug sg ng sfnbhbt` hfltblu`hbøl.
0) `0= ag n` suhgsbøl3 7, 07, 17, 087... `0= 5
=) `2 ag n` suhgsbøl3 8, ‘0=, 1:, ‘02=... `2 5
8) `01 ag n` suhgsbøl3 4, 0=, =1, 1:... `01 5
1) N` r`zøl ag ul` suhgsbøl hfl `0 5 = y `7 5 080== r 5
7) N` r`zøl ag ul` suhgsbøl hfl `0 5 222 y `1 5 89 r 5
4) Xue`tfrb` j`st` gn gngegltf `07, `0 5 0 y r 5 8 s 5
H. ]gsugnv` nfs prfmnge`s sbiubgltgs, rghugrag nf qug pr`htbhø sfmrg n`s prfirgsbflgs `rbteâtbh`s yigfeâtrbh`s.
0) Wl gqubpf ag ruimy 0 pn`lg` aupnbh`r su pultgf hfl h`a` s`qug. Xb gl su prbegr s`qug tbglgl 8pultfs, ¶huàltfs pultfs tglarå`l nugif ag 7 s`qugs6
=) Wl àtfef ag pnutflbf pbgrag =: qu`ros= h`a` egs. Xb `n h`mf ag 84 egsgs ng quga`l =719qu`ros, ¶huàltfs qu`ros tglå` gl gn prbegr egs6
8) Gl ul gabdbhbf, gn prbegr pbsf sg glhugltr` ` 2.1; egtrfs ag `ntur`, y n` abst`lhb` gltrg h`a` pbsfgs ag =.2; egtrfs3
`. Fmtgli` n` døreun` iglgr`n qug lfs blabqug n` `ntur` ag ul agtgrebl`af pbsf.
m. ¶@ quâ `ntur` gst`rà gn 0=.³ pbsf6
1) Gl gn prbegr egs, gn `nqubngr ag ul tgrrglf hugst` P=,8;;.;;? sb pfr h`a` egs qug p`s` ng a`l ulagshugltf ag P087.;; gl gn `nqubngr3
`. ¶Huàltf agmgrà p`i`r gl gn 2.³ egs6
m. ¶Huàltf sgrà gn ablgrf qug j`mrà i`st`af gl tft`n nugif ag 9 egsgs6
7) N` pfmn`hbøl ag ul p`ås `ueglt` r`zøl ag 0.;= `lu`negltg, sb htu`negltg tbglg 4 ebnnflgs agj`mbt`ltgs3
`. ¶Huàltfs j`mbt`ltgs tglarà nugif ag 2 `ðfs6
m. ¶Hu`ltfs j`mbt`ltgs tglarà nugif ag =7 `ðfs6
4) Xb agpfsbt`efs P=,8;;.;; gl ul` huglt` gl n` qug sg euntbpnbh` ` r`zøl ag 0.;7 eglsu`n, ¶huàl-
tf ablgrf tglargefs nugif ag 84 egsgs6
9) Wl` pgrsfl` j` i`l`af gl h`a` egs 08
ag nf qug i`lø gn egs `ltgrbfr.
`. Xb gn prbegr egs i`lø P4,;;;.;;, ¶huàltf i`l`rà gl gn tgrhgr egs6
m. ¶Huàltf j`mrà i`l`af gl tft`n gl gsfs 8 egsgs6
0 ruimy3 agpfrtg ag hflt`htf frbibl`rbf ag Blin`tgrr`. Xg cugi` hfl ul m`nøl fv`n`af.= qu`ros3 p`rtåhun`s dåsbh`s dfre`afr`s ag nfs àtfefs. Xfl nfs hflstbtuygltgs dula`eglt`ngs ag n` e`tgrb`.
008E`tgeàtbh` ∝ Xge`l` 9
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@. ]gsugnv` n`s ghu`hbflgs sbiubgltgs. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.
0) ‘8 ‘ t 5 4 t 5
=) 2 + t 5 08 t 5
8) 1 ‘ t 5 =8 t 5
1) 9 ‘ t 5 ‘== t 5
7) ‘2 + t 5 =0 t 5
4) 00 ‘ t 5 00 t 5
9) =8 + t 5 ‘0 t 5
:) 0; + t 5 02 t 5
2) 09 + t 5 ‘= t 5
0;) =0 ‘ t 5 = t 5
00) 7 + t 5 ‘7 t 5
0=) ‘08 ‘ t 5 ‘0 t 5
08) t + 0= 5 =1 t 5
01) ‘ t ‘ 0 5 ; t 5
07) t + =9 5 8 t 5
04) t + 02 5 ‘4 t 5
09) t ‘ 0; 5 ‘0= t 5
0:) ‘ t + 00 5 04 t 5
02) ‘ t ‘ 08 5 0: t 5
=;) ‘ t + 9 5 ‘=2 t 5
=0) t + =7 5 ‘=7 t 5
==) t + 08 5 08 t 5
=8) t + 02 5 ‘8 t 5
=1) ‘ t ‘ 09 5 ‘=2 t 5
M. ]gsugnv` n`s ghu`hbflgs sbiubgltgs. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.
0) ‘08o 5 47 o 5
=) ‘:o 5 ‘9= o 5
8) 2o 5 :0 o 5
1) ‘ o 5 4; o 5
7) ‘=o 5 ‘4; o 5
4) 0=o 5 ‘4; o 5
9) 00o 5 ‘22 o 5
:) 7o 5 17 o 5
2) 8o 5 ‘0= o 5
0;) 1o 5 =; o 5
00) 0;o 5 ‘7; o 5
0=) ‘ o 5 ‘0 o 5
08) ‘:o + = 5 44 o 5
01) =o ‘ 9 5 89 o 5
07) 00o + 02 5 24 o 5
04) 0=o + 7 5 :2 o 5
09) 8o + : 5 =4 o 5
0:) ‘0;o ‘ = 5 1: o 5
02) 0;o ‘ 0 5 22 o 5
=;) ==o + 08 5 92 o 5
=0) ‘09o + 01 5 22 o 5
==) 9o ‘ 04 5 88 o 5
=8) ‘08o ‘ 09 5 87 o 5
=1) ‘=o ‘ 00 5 10 o 5
001BIG] ∝ [fnfhjbh
@ibnba`a ag hànhunf eglt`n
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[`r` tr`m`c`r hfl sghuglhb`s lueârbh`s f p`r` euhj`s ftr`s àrg`s ag n`s e`tgeàtbh`s, lghgsbt`efsul` ir`l h`p`hba`a ag `lànbsbs? hfl gn gcgrhbhbf sbiubgltg, pflargefs ` prugm` gst` j`mbnba`a.
Gl n`s sgrbgs sbiubgltgs, gnbc` y rfagg, ag gltrg n`s fphbflgs, gn lóegrf qug sbiug n` sghuglhb`.
0) ; = 1 4 : 0; 0= ...
`. 08 m. = h. 01 a. 0;
=) 4 2 0= 07 0: =0 =1 ...
`. =7 m. =: h. =4 a. =9
8) 2 : 9 4 7 1 8 ...
`. 0 m. = h. 1 a. ;
1) 02 09 07 08 00 2 9 ...
`. : m. 7 h. 4 a. 0;
7) 89 81 80 =: =7 == 02 ...
`. 07 m. 09 h. =0 a. 04
4) 0: =; =0 =8 =1 =4 =9 ...
`. 8; m. =2 h. =: a. =7
9) 02 =0 =0 =8 =7 =7... =9
`. =1 m. == h. =8 a. =4
:) 42 :2 22 002 0=2... 0820;2
`. 42 m. 92 h. 22 a. ::
2) 82 12 7111 42... 9172
`. 47 m. 92 h. 71 a. 41
0;) 00 11 ::== 9;1... 01;:87=
`. 22 m. 0== h. 071 a. 094
007E`tgeàtbh` ∝ Xge`l` 9
]`zfl`ebgltf nøibhf
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:0
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
]gp`sf3 sge`l`s 0 ` 9
Gst` sge`l` nfir`rà3 ]gp`s`r nfs hfltglbafs ag n` sge`l` 0 n` 9.
]gsfnvgr nfs gcgrhbhbfs ag rgp`sf p`r` gv nu`rsg gl n` prugm`
p`rhb`n.
[rgp`r`rsg mbgl p`r` n` prugm` p`rhb`n.
009E`tgeàtbh` ∝ Xge`l` :
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Pugrba` y qugrbaf gstuab`ltg3
Xg `prfxbe` n` prbegr` prugm` p`rhb`n y agmg prgp`r`rsg `aghu`a`egltg, rgp`s`laf nfs hfltg-
lbafs ag n`s sge`l`s 0 ` n` 9.
[`r` `prfvghj`r gstg rgp`sf, ng rghfegla`efs3
• J`i` ul pn`l ag nf qug gstuab`rà h`a` aå` y tr`tg ag huepnbrnf. Agabqug eàs tbgepf `
nfs tge`s qug ng rgsuntgl abdåhbngs.
• Musqug ul nui`r tr`lqubnf, bnuebl`af y sbnglhbfsf p`r` gstuab`r.
• Ng` nfs rgsóeglgs ag h`a` sge`l` y gshrbm` n`s bag`s eàs bepfrt`ltgs gl su hu`agrlf.
• Gshuhjg n` hn`sg r`ab`n. Xus e`gstrfs nfhutfrgs ng `hfep`ð`ràl gl gstg rgp`sf y ng yu-
a`ràl ` rgsfnvgr `niulfs gcgrhbhbfs.
• Hfeprugmg qug j`y` rg`nbz`af mbgl nfs `utfhfltrfngs. Xb tbglg aua`s, vugnv` ` nggr n`s
sge`l`s, `jå glhfltr`rà gxpnbh`hbflgs y gcgepnfs.
¶Høef sgrà n` prugm` ag gv`nu`hbøl6
N` prugm` p`rhb`n gv`nó` nfs ebsefs hfltglbafs y ag n` ebse` dfre` gl qug nfs j` tr`m`c`af
sge`l` ` sge`l`.
Gl n` prugm` glhfltr`rà3
• Wl` sgrbg ag `ibnba`a eglt`n p`r` egabr su agstrgz` y r`pbagz ag hànhunf gl ul tbgepf
nåebtg ag trgs eblutfs.
• Abdgrgltgs gcgrhbhbfs qug gv`nó`l nf prglabaf gl n`s sbgtg sge`l`s. Gstfs gcgrhbhbfs sgràl
sgegc`ltgs ` nfs qug ustga rgsfnvbø gl n`s `htbvba`ags agn `utfhfltrfn. Xg ng pgabrà3
rgspflagr prgiult`s,
e`rh`r hfl ul` gqubs "Z" gn hu`arf ag n` fphbøl hfrrght`,
rgsfnvgr fpgr`hbflgs y
rgsfnvgr prfmnge`s.
• Hu`laf rgsugnv` gcgrhbhbfs y prfmnge`s, agmg agc`r hflst`lhb` ag su prfhgabebgltf.
• Euy bepfrt`ltg3 h`a` sgrbg hfltbglg blstruhhbflgs gx`ht`s ag nf qug agmg rg`nbz`r gl
h`a` `p`rt`af, `så hfef n` v`nfr`hbøl `sbil`a`.
Xb ustga sg prgp`r` hfl tbgepf y agabh`hbøl, gn rgsunt`af sgrà s`tbsd`htfrbf.
Ng rghfra`efs qug agmg prgp`r`rsg p`r` n` prugm` p`rhb`n y p`r` n`
prugm` agn EBLGAWH pr`htbh`laf hfl nfs gcgrhbhbfs qug sg glhugltr`l
`n l`n ag su nbmrf gl gn Q`nngr e`tgeàtbhf.
00:BIG] ∝ [fnfhjbh
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Nøibh` B
0. N` nøibh` prfpfsbhbfl`n gstuab` glulhb`afs y gv`nó` su v`nfr ag vgra`a, gs aghbr, agtgrebl` sb sfl
vgra`agrfs f d`nsfs.
=. N`s prfpfsbhbflgs sfl glulhb`afs qug pugagl sgr `l`nbz`afs y `så agtgrebl`r su v`nfr nøibhf. N`s
prfpfsbhbflgs sbepngs sfl `qugnn`s qug lf pugagl sgr abvbaba`s sbl `dght`r su sgltbaf. Gst`s sfl
rgprgsglt`a`s hfl ul` ngtr` eblóshun`.
r 3 N` h`dgål` tbglg ul gdghtf gstbeun`ltg.
t 3 Gn `iu` gs lghgs`rb` p`r` n` vba` agn sgr jue`lf.
Wl glulhb`af lf gs ul` prfpfsbhbøl sb lf sg pugag agtgrebl`r su v`nfr ag vgra`a. Gstf suhgag
hfeólegltg hfl glulhb`afs bepgr`tbvfs, ag prgiult` f qugnnfs qug prgsglt`l ul cubhbf ag v`nfr
Gn dutmfn gs egcfr qug gn m`nflhgstf.
¶Puâ j`rgefs jfy6
=.0 N` lgi`hbøl gs ul` fpgr`hbøl qug hflsbstg gl blvgrtbr gn v`nfr nøibhf ag ul` prfpfsbhbøl? sg
blabh` `ltgpflbglaf gn sbilf "~" ` n` prfpfsbhbøl.
a 3 ]bh`raf @rcfl` gs iu`tge`ntghf. T Gs vgra`agrf. Gn h`lt`ltg l`hbø gl Cfhftgl`lif,
X`h`tgpâqugz.
~a 3 ]bh`raf @rcfl` lf gs iu`tge`ntghf. DN` lgi`hbøl ag n` prfpfsbhbøl a h`emb` ag v`nfr
ag vgra`a, `jfr` gs d`ns`, pfrqug @rcfl` så gs
iu`tge`ntghf.
8. Nfs hu`ltbdbh`afrgs sfl såemfnfs qug sg utbnbz`l p`r` gst`mnghgr huàltfs gngegltfs ag ul hflcultf
agtgrebl`af huepngl hfl hbgrt` prfpbga`a.
8.0 Gn hu`ltbdbh`afr ulbvgrs`n (∋) sg gepng` p`r` `dbre`r qug tfafs nfs gngegltfs ag ul hflcultf
huepngl hfl ul` hflabhbøl a`a`. Xg ngg "p`r` tfaf…".
∋ x ∊ @, = x ∊ M3 [`r` tfa` x qug pgrtglghg `n hflcultf @, sg huepng qug = x pgrtglghg `n
hflcultf M.
8.= Gn hu`ltbdbh`afr gxbstglhb`n (∈) sg utbnbz` p`r` blabh`r qug n eglfs ul gngegltf ag ul hflcultf
huepng hfl ul` hflabhbøl a`a`. Xg ngg "gxbstg `n eglfs ul…".
∈ x ∊ H ∩ x ∊ M3 Gxbstg `n eglfs ul` x qug pgrtglghg ` H y pgrtglghg t`embâl ` M.
002E`tgeàtbh` ∝ Xge`l` :
Gn eulaf ag n` e`tgeàtbh`
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Gcgrhbhbf 0@. Ng` n`s prfpfsbhbflgs sbiubgltgs y agtgreblg su v`nfr nøibhf. Nugif, pbglsg y rggshrbm` su lgi`-
hbøl, rghugrag gn sbilf "~". Iuågsg pfr gn gcgepnf.
;) r3 Pubhjâ gs ul agp`rt`egltf ag Iu`tge`n`.
~r3 Pubhjâ lf gs ul agp`rt`egltf ag Iu`tge`n`.
0) w3 Hum` gs ul p`ås qug pgrtglghg ` Gurfp`.
=) j3 Gn `iu` agn e`r gs aunhg.
8) i3 Gn babfe` fdbhb`n ag Iu`tge`n` gs gn `ngeàl.
1) s3 8 gs abvbsfr ag 2.
7) p3 Gn vgrag gs ul hfnfr sghula`rbf.
4) q3 Eâxbhf sg glhugltr` `n sur ag Iu`tge`n`.
M. Ng` n`s gxprgsbflgs sbiubgltgs. Nugif, tr`lsdøregn`s ` ngliu`cg hfeól. Ng a`efs ul gcgepnf.
;) ∈ x ∊ Q ∩ x ∊ I
Gxbstg `n eglfs ul` x qug pgrtglghg ` Q y pgrtglghg t`embâl ` I.
0) ∋ x ∊ G, : x ∊ J
=) ∈ x ∊ H ∩ x ∊ M
8) ∋ x ∊ @, x + 8 ∊ M
1) ∈ x ∊ ] ∩ x ∊ Y
7) ∈ x ∊ O ∩ x ∊ H
T
D
0=;BIG] ∝ [fnfhjbh
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Nøibh` BB
0. N`s prfpfsbhbflgs hfepugst`s sfl qugnn`s dfre`a`s pfr afs f eàs prfpfsbhbflgs sbepngs qug sg
ulgl ` tr`vâs ag hflghtbvfs nøibhfs.
Gn c`iu`r gs ul e`eådgrf `egrbh`lf y gstà gl pgnbirf ag gxtblhbøl.
Gl gn gcgepnf `ltgrbfr sg ulgl afs prfpfsbhbflgs sbepngs ` tr`vâs agn lgxf "y"3 Gn c`iu`r gs ul
e`eådgrf `egrbh`lf? gn c`iu`r gstà gl pgnbirf ag gxtblhbøl.
=. Nfs hflghtbvfs nøibhfs sfl p`n`mr`s y såemfnfs qug sbrvgl hfef lgxf p`r` ulbr afs f eàs prfpfsb-
hbflgs sbepngs. Nfs eàs hfeulgs sfl n` hflculhbøl, n` absyulhbøl, gn hflabhbfl`n y gn mbhflabhbfl`n.
=.0 N` hflculhbøl (∩) hfrrgspflag ` n` "y".
Gcgepnf Xg gshrbmg Xg ngg
I`mrbgn` gstà gstuab`laf y Nfrglzf gstà tr`m`c`laf. p ∩ q p y q
=.= N` absyulhbøl (∪) hfrrgspflag n` "f" y sg utbnbz` p`r` a`r sgltbaf ag gnghhbøl gltrg afs f eàs
gngegltfs.
Gcgepnf Xg gshrbmg Xg ngg
@l` pugag gstuab`r e`tgeàtbh` f gstuab`r dåsbh`. p ∪ q p f q
=.8 Gn hflabhbfl`n ( ) f bepnbh`hbøl tfe` dfre` gl n`s p`n`mr`s "Xb… gltflhgs". Gstg hflghtbvf
blabh` qug n` prbegr` prfpfsbhbøl gs rgqubsbtf p`r` qug sg huepn` n` sgiula`.
Gcgepnf Xg gshrbmg Xg ngg
Xb rgspgt`efs n`s ngygs, gltflhgs sgrgefs muglfs
hbua`a`lfs. p q Xb p gltflhgs q
=.1 Gn mbhflabhbfl`n ( ) sg dfre` hfl n`s p`n`mr`s "sb y sfnf sb" g blabh` ul` rgn`hbøl ag agpgl -
aglhb` eutu`. Gs aghbr, n` prbegr` sg huepng sfnf sb sg huepng n` sgiula` y vbhgvgrs`.
Gcgepnf Xg gshrbmg Xg ngg
E`ð`l` sgrà nulgs sb y sfnf sb jfy gs afeblif. p q p sb y sfnf sb q
8. [`r` agtgrebl`r gn v`nfr ag vgra`a ag ul` prfpfsbhbøl hfepugst` sg gepng`l n`s rgin`s ag nfshflghtbvfs nøibhfs. Xg prgsglt`l gl n`s t`mn`s ag vgra`a sbiubgltgs3
p q p ∩ q
v v v
v d d
d v d
d d d
Hflculhbøl p q p ∪ qv v v
v d v
d v v
d d d
Absyulhbøl p q p q
v v v
v d d
d v v
d d v
Hflabhbfl`n p q p q
v v v
v d d
d v d
d d v
Mbhflabhbfl`n
0=0E`tgeàtbh` ∝ Xge`l` :
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Lóegrfs rg`ngs
0. Gn hflcultf ag nfs lóegrfs rg`ngs sg rgprgsglt` hfl n` ngtr` ] y sg agdblg sbemønbh`egltg hfef
] 5 P ∣ B. Gstfs lóegrfs sg pugagl gxprgs`r pfr ul gltgrf f ul aghbe`n, sg` pgrbøabhf f lf.
]
B
P
S
L
...‘1, ‘8, ‘=, ‘0 ;, 0, =, 8, 1 ...0.=0=0=...
;.7
78
81
= ϋ
Nfs lóegrfs rg`ngs (] ) glinfm`l ` nfs brr`hbfl`ngs (B), l`tur`ngs (L), gltgrfs (S) y r`hbfl`ngs (P).
0.0 Nfs lóegrfs rg`ngs sg h`r`htgrbz`l pfr sgr ul hflcultf dblbtf, aglsf, hfepngtf y hfltbluf.
0.= Gn v`nfr `msfnutf ag ul lóegrf rg`n gs n` abst`lhb` qug j`y gltrg gn hgrf u frbigl y gstg
lóegrf. Xu rgsunt`af sbgeprg gs pfsbtbvf. Gstg v`nfr sg bagltbh` gshrbmbâlafnf gltrg m`rr`s |`|.
=. N`s fpgr`hbflgs hfembl`a`s sfl v`rb`s fpgr`hbflgs qug sg rgólgl gl ul` sfn`.
• Xb n`s fpgr`hbflgs lf blhnuygl sbilfs ag irup`hbøl, sg rgsugnvgl rgspgt`laf n` cgr`rquå` ag n`s
fpgr`hbflgs, prbegrf n`s euntbpnbh`hbflgs y abvbsbflgs y nugif n`s sue`s y rgst`s.
• Xb n`s fpgr`hbflgs tbglgl sbilfs ag `irup`hbøl, gstfs sg gnbebl`l ag `agltrf j`hb` `dugr`.[rbegrf p`râltgsbs, nugif hfrhjgtgs y pfr óntbef nn`vgs.
8. Gn hflcultf ag nfs lóegrfs rg`ngs huepng hfl abdgrgltgs prfpbga`ags.
1. N` pftglhb`hbøl ag lóegrfs rg`ngs gs n` dfre` ràpba` ag gshrbmbr ul lóegrf qug sg euntbpnbh`
v`rb`s vghgs pfr så ebsef.
7. N` r`abh`hbøl gs n` fpgr`hbøl blvgrs` n` pftglhb`hbøl, hflsbstg gl j`nn`r n` r`åz ag ul lóegrf a`af.
Gcgrhbhbf 8@. ]gsugnv` n`s fpgr`hbflgs hfembl`a`s sbiubgltgs.
0) 07 + 10 Ù = ‘ 24 + 0;; 5 7) :7 + _(8 Ù 01) ù (8 Ù 9)U 5
=) 7 Ù =8 + (‘=:) ‘ 09 5 4) {_(8 + :) ‘ 84 Ù =U ‘ :} ‘ = Ù : 5
8) 0;0 ‘ =7 + 84 ù 4 + = 5 9) (: + 4) ù 9 + 2 Ù = 5
1) 09 + =7 ‘ _8 Ù (= + 70)U 5 :) _0=7 ù (7 Ù 7)U ‘ =7 + 0 5
0=8E`tgeàtbh` ∝ Xge`l` :
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M. Bagltbdbqug y gshrbm` n` prfpbga`a agn hflcultf ag lóegrfs rg`ngs qug sg gcgepnbdbh` gl h`a`
blhbsf.
0) ;.97 • 0 5 ;.97
=) ( 8
4 ) • ( 4
8 ) 5 0
8) : + ; 5 :
1) 0: • 7 5 7 • 0:
7) 7(8 • 8) 5 (7 • 8)8
4) 1(7 + 1) 5 (1 • 7) + (1 • 1)
9) 87 + 9 5 1=
:) ( 9
4 ) + ( ‘ 9
4 ) 5 ;
H. @pnbqug n`s rgin`s ag n` pftglhb`hbøl p`r` rgsfnvgr nfs gcgrhbhbfs sbiubgltgs.0) 022; 5
=) 7= • 78 5
8) ( 8
= ) ‘8
5
1) (‘8)8 • (‘8)= 5
7) 7 ‘8 5
4) (8=)8 5
9) (= • 8)8 5
:) ( ‘ 0
= )7
5
2) 47
44 5
0;) (‘1) ‘= 5
00) (‘9); 5
0=) ( 4
8 )8
5
08) ( 7
= ) ‘=
5
01) (=8)8 5
A. @pnbqug n`s rgin`s ag n` r`abh`hbøl p`r` rgsfnvgr nfs gcgrhbhbfs sbiubgltgs.
0) 0: • = 5
=)41
1 5
8)
8 =98 : 5
1) 8 9=2 5
7)1 2= 5
4)2 0=78 5
9)8
1;24 5
:) 8= • = 5
2)1 04
1 =74
5
0;) 4=7 5
00)8 0=8 5
0=)0=7
7 5
08)1 4: 5
01) = • 12 5
0=1BIG] ∝ [fnfhjbh
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Lóegrfs hfepngcfs
0. Gn hflcultf ag nfs lóegrfs hfepngcfs gs ul` gxtglsbøl ag nfs lóegrfs rg`ngs y sbrvg p`r` a`rng
sfnuhbøl ` n`s r`åhgs p`rgs ag lóegrfs lgi`tbvfs.
[`r` gnnf, nfs e`tgeàtbhfs bltrfaucgrfl ul lugvf såemfnf aglft`af pfr b , nn`e`af ulba`a be`ibl`-
rb` y qug huepng hfl n` hflabhbøl sbiubgltg3
‘0 5 b y qug b = 5 ‘0.
Tg`efs ul gcgepnf ràpbaf, rgsfnv`efs3
• Euntbpnbh`efs gn r`abh`laf pfr ‘0.
• [fr agdblbhbøl, sustbtubefs ( ‘0) pfr b = .
• @pnbh`efs n` prfpbga`a agn prfauhtf ag ul` r`åz y rgsfnvgefs.
0.0 N` pftglhb`hbøl ag b sbiug ul p`trøl qug sg rgpbtg ` p`rtbr ag b7. Xus pftglhb`s màsbh`s sfl3
b
0
5 b , b=
5 ‘0, b8
5 ‘ b , b1
5 0, b7
5 b .
0.= Nfs rgsunt`afs ag n`s r`åhgs p`rgs ag lóegrfs lgi`tbvfs rghbmgl gn lfemrg ag lóegrfs hfe-pngcfs purfs (mb) y ` nfs qug sg hfembl`l hfl nfs lóegrfs rg`ngs sg ngs nn`e` dfre` mblfeb`nag ul lóegrf hfepngcf (` + mb).
=. [`r` sue`r y rgst`r lóegrfs hfepngcfs gs lghgs`rbf fpgr`r n`s p`rtgs rg`ngs, nugif sue`r f rgst`r
n`s p`rtgs be`ibl`rb`s y gn rgsunt`af agmg gshrbmbrsg gl dfre` mblfeb`n.
8. [`r` euntbpnbh`r lóegrfs hfepngcfs sg agmg `pnbh`r n` prfpbga`a abstrbmutbv` ag n` euntbpnbh`hbøl,
sustbtubr b= pfr ‘0, rgauhbr târeblfs sgegc`ltgs y gshrbmbr gn rgsunt`af gl dfre` mblfeb`n.
1. [`r` abvbabr lóegrfs hfepngcfs gs lghgs`rbf r`hbfl`nbz`r gn aglfebl`afr egab`ltg gn hflcui`af,
rg`nbz`r n`s fpgr`hbflgs hfrrgspflabgltgs, sustbtubr b= pfr ‘0 y gshrbmbr gn rgsunt`af ag n` dfre`
mblfeb`n.
Gcgrhbhbf 1@. @pnbqug sus hflfhbebgltfs sfmrg r`åhgs p`rgs ag lóegrfs lgi`tbvfs y rgsugnv` nfs h`sfs sbiubgltgs.
0) ‘2 5
=) ‘41 5
8) ‘0;; 5
1) ‘1;; 5
7) ‘:0 5
4) ‘12 5
9) ‘0 5
:) ‘= 5
2) ‘84 5
0;) ‘=7 5
00) ‘011 5
0=) ‘0=0 5
‘04
04 (‘0)
04b=
04 • b= 5 1b
0=7E`tgeàtbh` ∝ Xge`l` :
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M. ]g`nbhg n`s fpgr`hbflgs blabh`a`s hfl lóegrfs hfepngcfs y gxprgsg su rgspugst` gl dfre`
mblfeb`n.
0) (9 ‘ 0=b) + (8 + b) =) (0; + 00b) + (09 ‘ 7b)
8) (0;b) ‘ (: + 9b) 1) 4b(00 + 0;b)
7) ‘0=b(9 + 1b) 4) (== + 7b)(0: ‘ 1b)
9) ( ‘ 0
8 ‘ 2b)( 0
00 + =b) :) ( 8
1 + 8b)( ‘
=
7 ‘ b)
2) =(7 + 8b) 0;) ‘8(= ‘ b)
0=4BIG] ∝ [fnfhjbh
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Xue` y rgst` ag e`trbhgs
0. Wl` e`trbz gs ul hflcultf ag lóegrfs f gxprgsbflgs abspugstfs gl dfre` rght`liun`r, dfre`laf
dbn`s y hfnuel`s. Gn fragl f abeglsbøl ag ul` e`trbz sg agtgrebl` pfr gn lóegrf ag dbn`s y hfnue-
l`s qug n` hfepflgl.
`00 `0= ... `0c ... `0l
`=0 `== ... `=c ... `=l
... ... ... ... ... ...`b0 `b= ... `bc ... `bl
... ... ... ... ... ...`e0 `e= ... `ec ... `el
@ 5
Wl` e`trbz e Ù l tbglg e dbn`s y l hfnuel`s. Xb tbglg gn ebsef lóegrf ag dbn`s qug ag hfnuel`s,
aghbefs qug gs ul` e`trbz hu`ar`a`. @ h`a` lóegrf f gxprgsbøl qug dfre` n` e`trbz ng aglfebl`-
efs gngegltf.
M 5
8 ‘=
; 0
7 :
8 n`s
= hfnuel`s
E`trbz M3 8 Ù =
=. J`y `niulfs tbpfs ag e`trbhgs hfl h`r`htgråstbh`s p`rtbhun`rgs3
=.0 Wl` e`trbz tr`lspugst` (@t) gs n` qug sg dfre` `n h`emb`r fragl`a`egltg n`s dbn`s pfr n`s
hfnuel`s ag n` e`trbz a`a`.
=.= Wl` e`trbz gs sbeâtrbh` sb gs hu`ar`a` y gs biu`n ` su tr`lspugst`.
=.8 Wl` e`trbz bagltba`a gs ul` e`trbz gl huy` ab`ifl`n prblhbp`n tfafs sus gngegltfs sfl biu`-
ngs ` 0 y nfs ageàs gngegltfs sfl ; (hgrf). Xg lfemr` hfl n` ngtr` B.=.1 Wl` e`trbz lf sbliun`r f blvgrtbmng gs `qugnn` qug tbglg e`trbz blvgrs`.
=.7 Wl` e`trbz sbliun`r lf tbglg blvgrs`.
8. N`s fpgr`hbflgs qug sg pugagl j`hgr hfl e`trbhgs sfl n` `abhbøl, n` rgst` y n` euntbpnbh`hbøl3
8.0 [`r` sue`r f rgst`r e`trbhgs gs lghgs`rbf qug tgli`l n` ebse` abeglsbøl. [`r` fpgr`r, sg
agmgl sue`r nfs gngegltfs ag n`s e`trbhgs qug fhup`l n` ebse` pfsbhbøl. Xg pugagl
fpgr`r afs f eàs e`trbhgs.
8.= N` sue` ag e`trbhgs tbglg n`s prfpbga`ags sbiubgltgs3
• Gs bltgrl` pfrqug n` e`trbz rgsunt`af sgrà ag n`s ebse`s abeglsbflgs qug n`s e`trbhgs qug
sg fpgr`rfl.
• Gs `sfhb`tbv pfrqug gn rgsunt`af gs gn ebsef sbl bepfrt`r høef `irupgefs nfs târeblfs.
• Qbglg gngegltf lgutrf, hflfhba` hfef e`trbz hgrf (F).
• Qbglg gngegltf fpugstf, gn hu`n gs n` e`trbz hfl nfs ebsefs gngegltfs qug n` e`trbz frbib-
l`n, pgrf hfl nfs sbilfs hfltr`rbfs rgspghtf ` n` e`trbz frbibl`n.
• Gs hfleut`tbv pfrqug gn rgsunt`af pgre`lghg blv`rb`mng sbl bepfrt`r gn fragl gl qug
fpgr`efs nfs sue`lafs.
0=9E`tgeàtbh` ∝ Xge`l` :
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Gcgrhbhbf 7@. [bglsg y dfreg nfs tbpfs ag e`trbhgs qug sg ng pbagl.
0) Wl` e`trbz ag 1 Ù 8 =) N` e`trbz bagltba`a
8) Wl` e`trbz sbeâtrbh` 1) Wl` e`trbz hfl su tr`lspugst`
M. Xbglaf n`s e`trbhgs3
@ 5 2 ‘7 4 2
1 9 ; ‘9 M 5 ‘8 4 7 ‘1
1 : ‘2 ‘7 H 5 4 7 1 ;
; 7 ‘2 9
Z 5; ‘0 :
‘7 1 4
0 2 ‘9
R 5 ‘: 4 ‘1
0 9 1
; ; :
S 51 7 2
0 ; ‘0
: 2 0
]gsugnv` n`s sbiubgltgs sue`s y rgst`s ag e`trbhgs. [ugag tr`m`c`r gl jfc`s `p`rtg sb gs lghgs`rbf.
0) @ + M =) Z ‘ S
8) M ‘ H + @ 1) R ‘ Z
7) H ‘ M 4) Z + R + S
0=:BIG] ∝ [fnfhjbh
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Euntbpnbh`hbøl ag e`trbhgs
0. N` euntbpnbh`hbøl f prfauhtf ag e`trbhgs gs ul` fpgr`hbøl qug pugag gdghtu`rsg gltrg afs
e`trbhgs f mbgl gltrg ul` e`trbz y ul lóegrf rg`n f gsh`n`r. [`r` rgsfnvgr gstg prfhgabebgltf
agmgefs iub`rlfs pfr agtgrebl`a`s rgin`s.
0.0 [`r` euntbpnbh`r ul` e`trbz pfr ul lóegrf rg`n sg agmg euntbpnbh`r gsg lóegrf pfr h`a` ulfag nfs gngegltfs ag n` e`trbz. Xg pugag euntbpnbh`r ul` e`trbz ag hu`nqubgr fragl.
o @ 5 o
` m h
a g d
i j b
5
`o mo ho
ao go do
io jo bo
0.= [`r` euntbpnbh`r afs e`trbhgs gltrg så , lfs `pfy`efs gl gn prfhgabebgltf sbiubgltg3
• Xg hfeprugm` qug gn lóegrf ag hfnuel`s ag n` prbegr` e`trbz (@) hfblhba` hfl gn lóegrf
ag dbn`s ag n` sgiula` (M). Ag nf hfltr`rbf, lf sg pugagl fpgr`r.
• Xg euntbpnbh` n` prbegr` dbn` ag @ pfr n` prbegr` hfnuel` ag M. [rbegr gngegltf pfr prbeg-
rf, sgiulaf pfr sgiulaf, gth. y sue`efs nfs prfauhtfs. Hfl gstf hflsgiubefs gn prbegr
gngegltf ag n` e`trbz rgsunt`af.
• Xgiubefs euntbpnbh`laf n` prbegr` dbn` ag @ pfr n`s sbiubgltgs hfnuel`s ag M.
• ]gpgtbefs gn prfhgabebgltf hfl n`s sbiubgltgs dbn`s ag @ y h`a` hfnuel` ag M.
• Nugif ag tgrebl`r hfl tfa`s n`s dbn`s ag @, nngi`efs `n rgsunt`af dbl`n.
H 5 @ • M 5
` m h
a g d
i j b
•
r s t
u v w
x y z
5
`r + mu + hx `s + mv + hy `t + mw + hz
ar + gu + dx as + gv + dy at + gw + dz
ir + ju + bx is + jv +by it + jw + bz
Gcgrhbhbf 4@. ]gsugnv` n`s euntbpnbh`hbflgs sbiubgltgs. [ugag utbnbz`r su hu`agrlf p`r` rg`nbz`r gn prfhgabebgltf.
0) : : 0; ‘00
‘2 ; 0
9 1 ‘8
=) ‘8 ; ‘2 ‘1
‘2 7 0=
= ; ‘0
0=2E`tgeàtbh` ∝ Xge`l` :
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M. ]gsugnv` n`s euntbpnbh`hbflgs sbiubgltgs. [ugag utbnbz`r su hu`agrlf p`r` rg`nbz`r gn prfhgabebgltf.
0)= ‘2
•
‘7 9
7 4
=) ‘0 7 1 •
: 4
7 ‘7
0; ‘0
8) ; ‘:
; 0
•
2 4
0 ‘0
1)9 ; :
‘8 7 =
•2 ‘7
4 8
‘9 ;
7)= 1 ‘=
; ‘0 0
•: ‘0
7 ;
; ‘2
4)2 7 ‘8
0 ‘0 :
• ‘9 = 8
0 ; ‘=
= 9 ‘2
9) 0 8 ‘:
; 2 ‘0
2 0; 00
•
: 8 =
7 1 ‘4
0 0 =
:) ‘: 4 7
1 0 ‘8
1 ‘4 ;
•
4 : 9
‘2 ; =
00 0; 0
2) ‘0 ‘8 48 : 0
; : 2
• ‘2 8 =
: 1 4
8 ‘00 =
0;) 0= ‘: 29 9 :
0 8 08
• 00 : 80 0 0
8 ; ‘:
08;BIG] ∝ [fnfhjbh
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Xuhgsbflgs e`tgeàtbh`s
0. Wl` suhgsbøl e`tgeàtbh` gs ul hflcultf fragl`af ag lóegrfs. H`a` ulf ag gstfs lóegrfs
sg aglfebl` "târeblf", "gngegltf" f "ebgemrf", y `n lóegrf tft`n ag gngegltfs qug hfepflg n`
suhgsbøl sg ng nn`e` "nflibtua", gst` pugag sgr dblbt` f bldblbt`.
=. Wl` suhgsbøl f prfirgsbøl rbteâtbh` gs ul` sgrbg ag lóegrfs t`ngs qug h`a` ulf ag gnnfs, gxhgp-tu`laf gn prbegr târeblf, sg fmtbglg sue`laf f rgst`laf ul` h`ltba`a dbc` `n târeblf `ltgrbfr, `
gst` h`ltba`a hflst`ltg sg ng hflfhg hfef abdgrglhb` y sg rgprgsglt` hfl n` n` ngtr` a .
08 =1 87 14 79 4: ...
+00 +00 +00 +00 +00
`. N` døreun` agn târeblf iglgr`n p`r` ul` suhgsbøl `rbteâtbh` lfs pgrebtg `vgrbiu`r hu`n-qubgr târeblf ag ul` suhgsbøl sb hflfhgefs gn prbegrf ag gnnfs y su abdgrglhb`.
`l 5 `0 + (l ‘ 0)a
[`r` d`hbnbt`r gn prfhgsf ag sue`r nfs gngegltfs ag ul` suhgsbøl `rbteâtbh` gepng`efs n`døreun` sbiubgltg3
s 5 (`0 + `l )l
=
8. Wl` suhgsbøl igfeâtrbh` gs ul` suhgsbøl gl n` qug h`a` târeblf sg fmtbglg euntbpnbh`laf `n
`ltgrbfr pfr ul` h`ltba`a dbc` aglfebl`a` r`zøl (r ).
: =1 9= =04 41: 0211 ...
Ù8 Ù8 Ù8 Ù8 Ù8
`. N` døreun` agn târeblf iglgr`n p`r` ul` suhgsbøl igfeâtrbh` lfs pgrebtg `vgrbiu`r hu`n-qubgr târeblf ag ul` suhgsbøl sb hflfhgefs gn prbegrf ag gnnfs y n` r`zøl.
`l 5 `0 • r (l ‘ 0)
[`r` d`hbnbt`r gn prfhgsf ag sue`r nfs gngegltfs ag ul` suhgsbøl igfeâtrbh` gepng`efs n`
døreun` sbiubgltg3
s 5 (`l • r ) ‘ `0
r ‘ 0
N`s àrg`s gl n`s qug pfagefs `pnbh`r lugstrfs hflfhbebgltfs sfmrg suhgsbflgs e`tgeàtbh` sfl euy
v`rb`a`s. [fr gcgepnf3
Gl n` m`lh`, nfs blvgrsbflbst`s y n`s pgrsfl`s qug tbglgl huglt`s m`lh`rb`s, pugagl h`nhun`r n`s i`l`l-
hb`s pftglhb`ngs sfmrg ul lgifhbf gepng`laf n`s prfirgsbflgs `rbteâtbh`s.
Gl n` bliglbgrå`, nfs lóhngfs ag ul e`tgrb`n r`ab`htbvf sg agsbltgir`l gl dfre` ulbdfreg gl ul agtgreb-
l`af pgråfaf ag tbgepf, gstg hànhunf pugag sgr agshrbtf pfr ul` prfirgsbøl igfeâtrbh`.
[fr t`ltf, pfagefs rgsfnvgr euhjfs prfmnge`s hfl t`l sfnf pfagr bagltbh`r hfrrght`egltg nfs a`tfs
ag n` prfirgsbøl y `pnbh`r n`s døreun`s qug ustga y` afebl`.
080E`tgeàtbh` ∝ Xge`l` :
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Gcgrhbhbf 9@. H`nhung nf qug sg sfnbhbt` gl h`a` ulf ag nfs blhbsfs sbiubgltgs. [ugag utbnbz`r jfc`s `p`rtg.
]ghugrag tr`m`c`r gl fragl.
0) `18 ag n` suhgsbøl3 :, 0, ‘4, ‘08, ‘=;… `18 5
=) `74 ag n` suhgsbøl3 0, =.7, 1, 7.7, 9… `74 5
8) `81 ag n` suhgsbøl3 :0, :4, 20, 24, 0;0… `81 5
1) a ag ul` suhgsbøl hfl `0 5 08 y `1= 5 8;; a 5
7) a ag ul` suhgsbøl hfl `0 5 179 y `40 5 ‘=48 a 5
4) `0 ag ul` suhgsbøl hfl a 5 ‘: y `:: 5 ‘48: `0 5
9) `0 ag ul` suhgsbøl hfl a 5 =.7 y `40 5 =81.7 `0 5
:) Xue`tfrb` j`st` gn gngegltf `44, `0 5 19 y a 5 2
`44 5 s 5
M. H`nhung nf qug sg sfnbhbt` gl h`a` ulf ag nfs blhbsfs sbiubgltgs. [ugag utbnbz`r jfc`s `p`rtg.
]ghugrag tr`m`c`r gl fragl.
0) `0= ag n` suhgsbøl3 4, 0=, =1, 1:… `0= 5
=) `00 ag n` suhgsbøl3 1, ‘04, 41, ‘=74… `00 5
8) `07 ag n` suhgsbøl3 :, 0=, 0:, =9, 1;.7… `07 5
1) r ag ul` suhgsbøl hfl `0 5 ‘8 y `9 5 ‘14:97 r 5
7) r ag ul` suhgsbøl hfl `0 5 9 y `00 5 108818 r 5
4) `0 ag ul` suhgsbøl hfl r 5 7.7 y `1 5 0224.7 `0 5
9) `0 ag ul` suhgsbøl hfl r 5 =.7 y `7 5 82;.4=7 `0 5
:) Xue`tfrb` j`st` gn gngegltf `0;, `0 5 = y r 5 1
`0; 5 s 5
H. ]gsugnv` nfs prfmnge`s sbiubgltgs. [ugag j`hgrnf gl jfc`s `p`rtg.
0) Gn `nqubngr ag ul` eftfhbhngt` hugst` P8;.;; n` prbegr` jfr` y P:.;; eàs h`a` lugv jfr`.
`. ¶Huàn gs gn prghbf tft`n ag `nqubngr ag 4 jfr`s6
m. Glhugltrg ul` døreun` qug lfs blabqug gn prghbf ag nqubngr p`r` niul` jfr` gl gspghådbhf.
=) Wl` eàqubl` hfstø blbhb`negltg P1:,;;;.;;. Nugif ag ulfs `ðfs sg vglabø ` n` ebt`a ag su
prghbf. [`s`afs ulfs `ðfs, vfnvbø ` vglagrsg pfr n` ebt`a, y `så suhgsbv`egltg.
`. ¶Huàltf ng hfstø n` eàqubl` `n sgxtf prfpbgt`rbf6
m. Xb gn tft`n ag prfpbgt`rbfs j` sbaf fhjf, ¶huàn gs n` sue` tft`n p`i`a` pfr gs` eàqubl`6
08=BIG] ∝ [fnfhjbh
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@. ]gsugnv` n`s fpgr`hbflgs hfembl`a`s sbiubgltgs. ]ghugrag sbgeprg n` cgr`rquå` ag n`s fpgr`hbflgs.
0) ‘2 + 7 Ù 0 + 8 5
=) 7 Ù 4 ù (‘=) + 4 5
8) 4 Ù 7 ‘ =; ù 7 5
1) 1 Ù 4 ù = + 0 5
7) 8 ù 8 + 1 ‘ 4 5
4) ‘8 ‘ 4 ù = ‘ 9 5
9) 2 Ù : ‘ 9 Ù 4 5
:) 8 + ; ‘ 7 ‘ 0 5
2) 9 Ù : ‘ 1 ‘ 0 5
0;) ‘= + 0; ù 7 5
00) = ‘ 8 + : ù (‘:) 5
0=) 1 ‘ 0 Ù ; ‘ 8 5
08) ‘: ‘ 1 + 4 ù = 5
01) ‘0 Ù 4 + = ‘ 8 5
07) 2 Ù 9 + 2 ù 8 5
04) ; ù 0 + : Ù 2 5
M. ]gsugnv` n`s fpgr`hbflgs hfembl`a`s sbiubgltgs. ]ghugrag n` cgr`rquå` ag nfs sbilfs ag irup`hbøl.
0) _(2 + :) ‘ 1U Ù 8 5
=) (=; ù 1) ‘ 4 Ù = 5
8) 0 + 8 Ù (0 + 1) 5
1) 0; ù 7 ‘ (7 ‘ 1) 5
7) _(4 ‘ 9) Ù 7U + 1 5
4) (8 Ù 0) + = ‘ 0 5
9) ; ‘ _(1 + 1) ‘ 4U 5
:) = Ù 0 + (8 ‘ =) 5
2) (= Ù 8) ‘ (= + 8) 5
0;) _(2 ‘ 4) + 9U Ù 1 5
00) (7 ‘ 4) + 1 ‘ = 5
0=) _7 + (: ù 1)U ‘ 1 5
08) 9 ‘ _2 ù (4 ‘ 8)U 5
01) = Ù = ‘ (0 + :) 5
07) 7 Ù 1 ‘ (; + 0) 5
04) (9 + 8) Ù (0; ù 7) 5
H. ]gsugnv` n`s ghu`hbflgs sbiubgltgs. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.
0) ‘8 + e 5 0; e 5
=) : ‘ e 5 ‘0: e 5
8) 2 + e 5 ; e 5
1) 0 ‘ e 5 04 e 5
7) : + e 5 : e 5
4) ‘2 ‘ e 5 =0 e 5
9) 2 + e 5 02 e 5
:) ‘9 ‘ e 5 ‘09 e 5
2) ‘ e + 0: 5 01 e 5
0;) e ‘ 9 5 ‘=8 e 5
00) e + 00 5 == e 5
0=) ‘ e ‘ 4 5 =9 e 5
08) e + : 5 0= e 5
01) e + ; 5 ‘2 e 5
07) ‘ e + : 5 ‘= e 5
04) e ‘ : 5 08 e 5
088E`tgeàtbh` ∝ Xge`l` :
@ibnba`a ag hànhunf eglt`n
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E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfir`af gl
prfhgsflf
nfir`af
A g s p u â s a g
g s t u
a b ` r . . . ]gp`sf nfs hfltglbafs ag n` sge`l` 0 ` n` 9.
]gsugnvf nfs gcgrhbhbfs ag rgp`sf p`r` gv`nu`reg gl n` prugm` p`rhb`n.
Eg sbgltf mbgl prgp`r`af p`r` n` prugm` p`rhb`n.
Frbglt`hbflgs sfmrg n` prugm` p`rhb`n
¡Nngiø gn efegltf ag n` prugm`!
R` gstà nbstf p`r` su prbegr` prugm` ag E`tgeàtbh`. Ng prgsglt`efs n`s óntbe`s rghfegla`hbflgs
qug pugagl `yua`rng ` n` jfr` agn gx`egl.
Lf sg "`t`squg" gl lbliól gcgrhbhbf. Gepbghg pfr n`s prgiult`s qug sgp` egcfr y ng quga`rà eàs
tbgepf p`r` pgls`r gl n`s qug tgli` aua`s.
@n dbl`nbz`r su gx`egl, rgng` tfa`s sus rgspugst`s y vg` sb `nif sg ng p`sø pfr `ntf.
[rgsgltg su prugm` nbepb` y fragl`a`.
¡Àlbef! Gn rgsunt`af ag su gx`egl sgrà gn prfauhtf ag su gsdugrzf.
Irupf3 [fnfhjbh [rugm`3 p`rhb`n @ ‘ =;04E`tgrb`3 E`tgeàtbh` [rbegr sgegstrg
Lfemrg3
H`rlâ3
Hårhunf ag gstuabf L³3 Dghj`3
[ultgf3
B XG]BG 0 pultf h`a` rgspugst` hfrrght`. Qft`n 7 pultfs.
BLXQ]WHHBFLGX3 E`rqug hfl ul` "Z" gn hu`arf ag n` fphbøl hfrrght`.
0) ¶Huàn ag n`s gxprgsbflgs sbiubgltgs gs ul`prfpfsbhbøl6
Jgrefs` lfhjg ¶Puâ t`n tg j` baf6 9 gs ul lóegrf prbef Gn c`iu`r gs ul `lbe`n mgnnf.
Ng` `tglt`egltg n`s
blstruhhbflgs ltgs
ag hfltgst`r. Xb tbglg
aua`, hflsuntg ` sufrbglt`afr(`).
@n rghbmbr n` prugm`,y `ltgs ag gepgz`r
` rgsfnvgrn`, gshrbm`
su lfemrg, lóegrf
ag h`rlâ, lóegrf ag
hårhunf ag gstuabf y
dghj`.
081BIG] ∝ [fnfhjbh
]gvbsg su `prglabz`cg
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20
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
[ftglhb`hbøl ag gxprgsbflgs `nigmr`bh`s
¶Puâ glhfltr`rà gst` sge`l`6
]gp`sf ag n` pftglhb`hbøl ag lóegrfs rg`ngs
]gin`s ag n` pftglhb`hbøl y sbepnbh`hbøl ag gxprgsbflgs `nigmr`bh`s
[ftglhb`hbøl ag lóegrfs rg`ngs y ag gxprgsbflgs `nigmr`bh`s
Àrg`s ag iur`s igfeâtrbh`s
Gst` sge`l` nfir`rà3
@pnbh`r n`s rgin`s ag pftglhb`hbøl ` gxprgsbflgs `nigmr`bh`s.
Xbepnbh`r gxprgsbflgs `nigmr`bh`s. [r`htbh`r n` ibnba`a ag hànhunf hfl pftglhb`s ag lóegrfs rg`ngs
y gxprgsbflgs `nigmr`bh`s.
Wtbnbz`r n`s gxprgsbflgs `nigmr`bh`s p`r` h`nhun`r àrg`s ag iur`s igfeâtrbh`s.
087E`tgeàtbh` ∝ Xge`l` 2
8/20/2019 Libro Polochic Matemática 1er. Sem
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¡[`r` hfeglz`r!]gp`sf ag n` pftglhb`hbøl ag lóegrfs rg`ngs
@prglabefs gl n` sge`l` 8 sfmrg gn hflcultf ag nfs lóegrfs rg`ngs y n`s rgin`s
ag n` pftglhb`hbøl, rghfragefs `niul`s ag gnn`s.
Wl` pftglhb` sg rgprgsglt` ag dfre` sbemønbh` pfr3
`l 5 ` • ` • ` . . . • `
Aflag3
• ` rgprgsglt` n` m`sg y gs gn lóegrf qug sg
euntbpnbh` pfr så ebsef.
• l gs gn gxpflgltg g blabh` gn lóegrf ag
vghgs qug n` m`sg sg euntbpnbh` pfr så ebse`.
[fr gcgepnf3 81 5 8 • 8 • 8 • 8 5 :0
(‘7)8 5 (‘7)(‘7)(‘7) 5 ‘0=7
[rfauhtf ag pftglhb`s ag biu`n m`sg
Hu`laf sg euntbpnbh`l pftglhb`s ag biu`n m`sg, sg hfpb` n` m`sg y sg sue`l nfs
gxpflgltgs.
[fr gcgepnf3 4= • 41 5 4= + 1 5 44
(‘;.7)1(‘;.7)8 5 (‘;.7)1 + 8 5 (‘;.7)9
Hfhbgltg ag pftglhb`s ag biu`n m`sg
[`r` abvbabr pftglhb`s qug tbglgl biu`n m`sg, sg hfpb` n` m`sg y sg rgst`l nfs
gxpflgltgs.
[fr gcgepnf30;4
0;1 5 0;4 ‘ 1 5 0;=
(‘;.7)7
(‘;.7)= 5 (‘;.7)7 ‘ = 5 (‘;.7)8
¡@ tr`m`c`r!
@pnbqug n`s rgin`s ag n` pftglhb`hbøl qug jgefs rgp`s`af y gxprgsg h`a` fpgr`hbøl hfef ul` sfn`pftglhb`. Qbglg ul gcgepnf.
;) 78 • 71 5 =) (‘:)4(‘:)1 5 1)(2)4
(2)8 5
0) 87 • 87 5 8) (‘=)8(‘=)7 5
`lm`sggxpflgltg
79
084BIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
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0. ]gin`s ag n` pftglhb`hbølN`s rgin`s ag n` pftglhb`hbøl p`r` gxprgsbflgs `nigmr`bh`s sfl n`s ebse`s
qug vbefs gl gn hflcultf ag nfs lóegrfs rg`ngs, n` abdgrglhb` gs qug `jfr` glnui`r ag lóegrfs sg utbnbz`l ngtr`s y gl `niulfs h`sfs n` hfembl`hbøl ag ngtr`sy lóegrfs. @ltgs ag `v`lz`r hfl n`s rgin`s ag n` pftglhb`hbøl, j`i`efs ul`hfep`r`hbøl gltrg n`s pftglhb`s hfl lóegrfs rg`ngs y gxprgsbflgs `nigmr`bh`s.
[ftglhb`hbøl ag lóegrfsrg`ngs
[ftglhb`hbøl aggxprgsbflgs nigmr`bh`s
18 5 1 • 1 • 1 x8 5 x • x • x
(‘4)1 5 (‘4)(‘4)(‘4)(‘4) (‘ e)1 5 (‘ e)(‘ e)(‘ e)(‘ e)
(07 )8
5 (07 )(07 )(07 ) ( y
x )8
5 ( y
x )( y
x )( y
x )
0.0 ]gin` agn gxpflgltg hgrf
N` rgin` agn gxpflgltg hgrf blabh` qug hu`nqubgr m`sg gngv`a` n gxpflgltg hgrfgs biu`n ` n` ulba`a (0), gxhgptf gl gn h`sf qug n` m`sg sg` hgrf.
x ; 5 0
Gcgepnf3
o ; 5 0
( xy); 5 0
(‘1wz ); 5 0
0.= ]gin` agn prfauhtf ag pftglhb`s hfl biu`nm`sg
[`r` euntbpnbh`r gxprgsbflgs `nigmr`bh`s qug tbglgl n` ebse` m`sg, sg hfpb` n`m`sg y sg sue`l nfs gxpflgltgs.
x e • x l 5 x e + l
Gcgepnf3`7 • `8 5 `7 + 8 5 `:
x9 • x: 5 x9 + : 5 x 07
(8 y)=(8 y)= 5 (8 y)= + = 5 (8 y)1
089E`tgeàtbh` ∝ Xge`l` 2
Gn eulaf ag n` e`tgeàtbh`
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0.8 ]gin` agn hfhbgltg ag pftglhb`s ag biu`nm`sg
[`r` abvbabr gxprgsbflgs `nigmr`bh`s qug tbglgl n` ebse` m`sg y abdgrgltg gxpf-lgltg, sg e`ltbglg n` m`sg y sg rgst`l nfs gxpflgltgs.
x e
x l 5 x e ‘ l
Gcgepnf3m:
m7 5 m: ‘7 5 m8
(‘7 z )4
(‘7 z )1 5 (‘7 z )4 ‘ 1 5 (‘7 z )= 5 =7 z =
(= )7
(= )9 5
0
(= )9 ‘ 7 5
0
(= )= 5
0
1`=
0.1 ]gin` ag ul` pftglhb` gngv`a` ` ftr`pftglhb`
Xb n` gxprgsbøl nigmr`bh` sg glhugltr` agltrf ag sbilfs ag irup`hbøl, sg hfpb`n` m`sg y sg euntbpnbh`l nfs gxpflgltgs.
( x e)l 5 x e • l
Gcgepnf3( 1)= 5 `1 • = 5 `:
(w8
y1
z =
)1
5 (w8
)1
( y1
)1
( z =
)1
5 w0=
y04
z :
(‘7`8m:h=)8 5 (‘7)8( 8)8(m:)8(h=)8 5 ‘0=7`2m=1h4
0.7 ]gin` ag n` pftglhb` ag ul` dr`hhbøl`nigmr`bh`
Xb n` gxprgsbøl `nigmr`bh` gs ul` dr`hhbøl, sg hfpb` n` m`sg y sg euntbpnbh` gngxpflgltg agn luegr`afr y agn aglfebl`afr pfr gn gxpflgltg gxtgrlf.
( x y )e 5
x e
ye
Gcgepnf3
( `x
my )4
5`4 x 4
m4 y4
( ‘8e=l1
=w8 z = )=
5(‘8)=(e=)=(l1)=
(=)=(w8)=( z =)= 5
2e1l:
1w4 z 1
( 7w: x= y7
8 p4t 8u9 )8 5
(7)8(w:)8( x=)8( y7)8
(8)8( p4)8(t 8)8(u9)8 5
0=7w=1 x 4 y07
=9 p0:t 2u=0
08:BIG] ∝ [fnfhjbh
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Gcgrhbhbf 0Ng` n` rgin` blabh`a` gl h`a` p`rt`af y pnåqugn` n`s gxprgsbflgs nigmr`bh`s ag dfre` abrght`. J`yul gcgepnf p`r` h`a` h`sf.
@. Gxpflgltg hgrf
;) 7(= y); 5 8) ‘7(= x); 5
0) (= z ); 5 1) ( ‘8 x7
y1 );
5
=) (‘7 p=q=r ); 5 7) :j(7 x0; y8); 5
M. [rfauhtf ag pftglhb`s ag biu`n m`sg
;) `9 • `= • `8 5 8) (el)8(el)1 5
0) x7 • x8 • x 5 1) (=wy)8(=wy) 5
=) e7 • e7 • e7 5 7) (rs)=(rs)7(rs) 5
H. Hfhbgltg ag pftglhb`s ag biu`n m`sg
;)e0;
e4 5 1)
(‘1 z )1
(‘1 z )1 5
0)m2
m8 5 7)
( pqr )4
( pqr )1 5
=)w07
w: 5 4)
(7el)4
(7el)1 5
8) y=1
y0= 5 9) (‘8 xyz )
7
(‘8 xyz )07 5
A. [ftglhb` ag ftr` pftglhb`
;) ( =m7h4)= 5 1) ( p7r 9)= 5
0) ( x:)= 5 7) (‘7`m:h=); 5
=) (el1)1 5 4) ‘1(e=j8o 1)= 5
8) (w= y4)8 5 9) ‘7`8(m7ha =)8 5
G. [ftglhb` ag ul` dr`hhbøl
;) ( 8`
=m )1
5 8) (7e8l=
=w1 z 8 )=
5
0) ( x y)4
5 1) ( 8j1o 1
7 p1q4 )=
5
=) (`m
ha )4
5 7) ( `1m9h8
w= x8 y= )1
5
7(0) 5 7
`0=
e1
`1m0;h0=
:0`1
04m 1
082E`tgeàtbh` ∝ Xge`l` 2
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0.4 ]gin` agn gxpflgltg lgi`tbvf
`. Wl` nbtgr`n hfl gxpflgltg lgi`tbvf sg hflvbgrtg gl ul` pftglhb` pfsbtbv`gshrbmbglaf gn blvgrsf ag n` m`sg hfl gn ebsef gxpflgltg.
x ‘ e 5 0
x
e
Gcgepnf3i ‘7 5
0
i 7
(1`m) ‘= 50
(1`m)= 5
0
04`=m=
: p ‘4 5 : •0
p4 5
:
p4
m. Wl` dr`hhbøl `nigmr`bh` hfl gxpflgltg lgi`tbvf sg hflvbgrtg gl ul` pf-tglhb` pfsbtbv` gshrbmbglaf gn blvgrsf ag n` m`sg hfl gn ebsef gxpflgltg.
( `
m ) ‘ e
5 ( m
` )e
Gcgepnf3
( x y ) ‘7
5 ( y x )7
5 y7
x7
(=m
8` ) ‘=
5 (8`
=m )=
52`=
1m=
(7w8 x=
1 y= z = ) ‘8
5 ( 1 y= z =
7w8 x= )8
541 y4 z 4
0=7w2 x4
Gcgrhbhbf =@pnbqug n` rgin` agn gxpflgltg lgi`tbvf p`r` rgsfnvgr nfs gcgrhbhbfs sbiubgltgs.
0) ( `m ) ‘7
5 =) ( 8 x
1 y ) ‘8
5
8)
(7m=
8h8
)
‘=
5 1)
(`=m8
h1a 7 )
‘8
5
7) (1e8l=
= p1q8 ) ‘8
5 4) ( ‘7w 1 x7
y8 z 9 ) ‘8
5
01;BIG] ∝ [fnfhjbh
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=. Xbepnbdbh`hbøl ag gxprgsbflgs`nigmr`bh`s
Xbepnbh`r ul` gxprgsbøl `nigmr`bh` dfre`a` hfl pftglhb`s ag lóegrfs rg`ngsgs tr`lsdfre`rn` ` ul` gxprgsbøl, gl n` qug h`a` lóegrf rg`n `p`rgzh` sfnf ul`
vgz y tfafs nfs gxpflgltgs sg`l pfsbtbvfs.Xbepnbqugefs
• @pnbh`efs n` rgin` ag ul` pftglhb` gngv`a`` ftr` pftglhb` gl gn d`htfr agrghjf.
• @pnbh`efs n` rgin` agn prfauhtf.
• Fmtglgefs gn rgsunt`af.
Xbepnbqugefs
• @pnbh`efs n` rgin` agn hfhbgltg ag pftglhb`sag biu`n m`sg.
• Hfef x: tbglg gn gxpflgltg e`yfr, sg quga`gl gn aglfebl`afr y sg ng rgst` gn gxpflgl-tg ag x4 y fmtglgefs gn rgsunt`af.
Xbepnbqugefs
• @pnbh`efs n` rgin` agn hfhbgltg ag pftglhb`sag biu`n m`sg gl n` dr`hhbøl.
• @pnbh`efs n` rgin` ag n` pftglhb` ag ul`pftglhb`.
• Fmtglgefs gn rgsunt`af.
Xbepnbqugefs
• @pnbh`efs n` rgin` agn prfauhtf ag pftgl-
hb`s ag biu`n m`sg.• Fmtglgefs gn rgsunt`af.
Gl `niulfs h`sfs, p`r` sbepnbh`r ul` gxprgsbøl `nigmr`bh` gs pfsbmng `pnbh`reàs ag ul` rgin` sbl cgr`rquå` niul`, gs aghbr, lf gxbstg ul fragl aglbaf p`r`fpgr`r.
(8`=h1a 7)(=`=ha 8)1 5
(8`=h1a 7)(04`:h1a 0=) 5
8 • 04`= + :h 1 + 1a 7 + 0= 5
1:`0;h:a 09
=7 x4 y2 z :
7 x:
y4
z =
5
7 y2 ‘ 4 z : ‘ =
x: ‘ 4 5
7 y8 z 4
x =
(1w: x1 y7
=w7 x= y8 )1
5
(=w:
‘ 7
x1
‘ =
y7
‘ 8
)1
5
(=w8 x= y=)1 5
04w0= x : y:
(0
= x1 y4 z :)(8
1 x= y8 z 7)
8
: x1 + = y4 + 8 z : + 7 5
8
: x 4 y2 z 08
010E`tgeàtbh` ∝ Xge`l` 2
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Gcgrhbhbf 8Xbepnbqug n`s gxprgsbflgs `nigmr`bh`s ag dfre` abrght`. Qbglg ul gcgepnf.
;) ( 04w7 x4 y2
1w= x1 y8 )8
5 0) (8`=m1h8)(=`1m7h=) 5
(1w8 x= y4)8 5
41w2 x 4 y0:
=) (=`1m8h=)(8`8mh7)8 5 8) (‘= x8 y1)(4 x= y4) 5
1) (7w8 x= y1)(8 x= y8)8 5 7)0: p:q4r 7
4 p7q=r 5
4)41`0;m:h0=
:`1m8h9 5 9) (07 x4 y2 z 8
7 x1 y7 z 8 )1
5
:) ( w4 x9 y1
w1 x8 y= )7 5 2) ( 7
1 x4 y1 z 4 )( =8 x: y= z ) 5
01=BIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
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]gsuegl
0. N`s rgin`s ag n` pftglhb`hbøl ag gxprgsbflgs `nigmr`bh`s sfl n`s ebse`s qug sg gepng`l gl gnhflcultf ag nfs lóegrfs rg`ngs. [`r` `pnbh`rn`s, agmgefs prgst`r `tglhbøl ` nfs gxpflgltgs qugn`s `hfep`ð`l.
0.0 ]gin` agn gxpflgltg hgrf
Blabh` qug hu`nqubgr m`sg gngv`a` n gxpflgltg hgrf gs biu`n n` ulba`a(0), gxhgptf gl gn h`sf ag qug n` m`sg sg` hgrf.
0.= ]gin` agn prfauhtf ag pftglhb`s hfl biu`n m`sg
[`r` euntbpnbh`r gxprgsbflgs `nigmr`bh`s qug tbglgl n` ebse` m`sg,hfpb`efs n` m`sg y sue`efs nfs gxpflgltgs.
0.8 ]gin` agn hfhbgltg ag pftglhb`s ag biu`n m`sg
[`r` abvbabr gxprgsbflgs nigmr`bh`s qug tbglgl n` ebse` m`sg y abdgrgltg
gxpflgltg, sg e`ltbglg n` m`sg y sg rgst`l nfs gxpflgltgs.
0.1 ]gin` ag ul` pftglhb` gngv`a` ` ftr` pftglhb`
Xb n` gxprgsbøl nigmr`bh` sg glhugltr` agltrf ag sbilfs ag irup`hbøl,sg hfpb` n` m`sg y sg euntbpnbh`l nfs gxpflgltgs.
0.7 ]gin` ag n` pftglhb` ag ul` dr`hhbøl `nigmr`bh`
Xb n` gxprgsbøl nigmr`bh` gs ul` dr`hhbøl, sg hfpb` n` m`sg y sg euntbpnbh`gn gxpflgltg agn luegr`afr y agn aglfebl`afr pfr gn gxpflgltggxtgrlf.
0.4 ]gin` agn gxpflgltg lgi`tbvf`. Wl` nbtgr`n hfl gxpflgltg lgi`tbvf, sg hflvbgrtg gl ul` pftglhb`
pfsbtbv` gshrbmbglaf gn blvgrsf ag n` m`sg hfl gn ebsef gxpflgltg.
m. Wl` dr`hhbøl `nigmr`bh` hfl gxpflgltg lgi`tbvf, sg hflvbgrtg glul` pftglhb` pfsbtbv` gshrbmbglaf gn blvgrsf ag n` m`sg hfl gn ebsefgxpflgltg.
=. Xbepnbdbh`hbøl ag gxprgsbflgs nigmr`bh`s
Xbepnbdbh`r ul` gxprgsbøl `nigmr`bh` dfre`a` hfl pftglhb`s ag lóegrfs rg`ngs gs tr`lsdfre`rn`` ul` gxprgsbøl, gl n` qug h`a` lóegrf rg`n `p`rgzh` sfnf ul` vgz y tfafs nfs gxpflgltgs sg`lpfsbtbvfs.
@låegsg ` vbsbt`r n` pàibl` wgm qug ng suigrbefs p`r` rgdfrz`r y `epnb`r sus hflfhbebgltfs sfmrgpftglhb`hbøl3 jttp3//iff.in/Zo1Bzrn
x ; 5 0
x e • x l 5 x e + l
x e
x l 5 x e ‘ l
( x e)l 5 x e • l
( x y )e 5
x e
ye
x ‘ e 5 0
x e
( `
m ) ‘ e
5 ( m
` )e
018E`tgeàtbh` ∝ Xge`l` 2
Blvgstbiug gl n` rga...
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@htbvba`a 0. Ageugstrg nf prglabaf
E`rqug hfl ul` "Z" gn hu`arf ag n` fphbøl qug prgsglt` n` rgspugst` hfrrght` ` h`a` prgiult`.
0) ¶Huàn gs gn gxpflgltg gl n` gxprgsbøl ‘1`=6
=) ¶Puâ rgsunt`af sg fmtbglg ag n` pftglhb` 0;;;6
8) ¶Huàn gs gn rgsunt`af ag (‘= y)
8
6
1) ¶Puâ rgsunt`af gxprgs` ul` hfrrght` `pnbh`hbøl ag n`s rgin`sag n` pftglhb`hbøl ag gxprgsbflgs `nigmr`bh`s6
7) ¶Huàn gs gn rgsunt`af sbepnbdbh`af ag
( x4
x7
)
=
6
4) ¶Puâ gxprgsbøl gs gqubv`ngltg ` n` sbiubgltg3 ` • ` • ` • ` • `6
9) ¶Puâ gxprgsbøl gs hfrrght`6
:) ¶Puâ gxprgsbøl huepng hfl n` rgin` agn gxpflgltg hgrf6
@utfhfltrfn
= `
1
‘1
;
0
0;
0;;
‘: y ‘= y8
‘4 y4
‘: y8
x ‘0 5 ‘ x
x8 ù x= 5 x7
x8 • x8 5 x4
(= x8)8 5 4 x4
x=
x0=
x0;
x ‘=
`
`7
7`
7`7
7 x; 5 x
7 x; 5 ;
7 x; 5 0
7 x; 5 7
(8 xy); 5 0
(8 xy); 5 ;
(8 xy); 5 8
(8 xy); 5 xy
011BIG] ∝ [fnfhjbh
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@htbvba`a =. [r`htbqug nf prglabaf
@. @pnbqug n` rgin` agn gxpflgltg hgrf p`r` sbepnbdbh`r n`s gxprgsbflgs `nigmr`bh`s. J`y ul gcgepnf.
;) ( m); 5
0) (7e); 5
=) ‘7(= x); 5
8) (04 x1 y1); 5
1) ( 8j1
7o );
5
7) (:e7
2l= );
5
4)
(
07m:
0;m4
)
;
5
9) ( ‘ m7
0=`m );
5
:) 8 x(9 x7 y1); 5
2) 0;(1`:m8); 5
0;) xy(7 x0; y8); 5
00) ‘=7( p=q8); 5
0=) 7el(0;;); 5
08) 0:j=(ha 0=); 5
M. @pnbqug n` rgin` agn prfauhtf ag pftglhb`s ag biu`n m`sg p`r` sbepnbdbh`r n`s gxprgsbflgs nigmr`bh`sJ`y ul gcgepnf.
;) `1 • `7 • `= 5
0) x4 • x1 5
=) z : • z = 5
8) w: • w9 5
1) m7
• m8
• m4
5 7) j4 • j1 • j= 5
4) m7 • m8 • m4 5
9) a : • a 7 • a 4 5
:) h0; • h0= • h ‘1 5
2) ( xy)7( xy): 5
0;) ( m)1( m) 5
00) ( pq):
( pq)2
5 0=) (1w)1(1w)= 5
08) ( st )8( st )9( st )1 5
H. @pnbqug n` rgin` agn hfhbgltg ag pftglhb`s ag biu`n m`sg p`r` sbepnbdbh`r n s gxprgsbflgs `nigmr`bh`sJ`y ul gcgepnf.
;)(0; p)4
(0; p)= 5
0)
`0=
`0; 5
=)h4
h8 5
8)m:
m: 5
1)t 2
t 0; 5
7)o 2
o : 5
4)
z 04
z 4 5
9) y07
y: 5
:) x0;
x0= 5
2)(el)7
(el)8 5
0;)(wxy)2
(wxy)2 5
00)
(1wx)07
(1wx)7 5
0=)(=7rs)4
(=7rs)01 5
08)(‘7 m)0=
(‘7 m)0; 5
01)(‘07w)01
(‘07w)0= 5
0
`00
(0; p)1
017E`tgeàtbh` ∝ Xge`l` 2
8/20/2019 Libro Polochic Matemática 1er. Sem
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A. @pnbqug n` rgin` ag ul` pftglhb` gngv`a` ` ftr` pftglhb` p`r` sbepnbdbh`r n`s gxprgsbflgs `nigmr`bh`s.J`y ul gcgepnf.
;) (‘7 =m1h8)8 5
0) (e8)8 5
=) (w:)= 5
8) ( 1m8)1 5
1) (=w8 x=)8 5
7) (‘7e1l:)= 5
4) (8`1m8h=)= 5
9) (=7 x4 y: z 4); 5
:) ( `1
m7
h=a 8 )8
5
2) (w1 x:
y= z 1 )9
5
0;) ( ‘8e1
7l8 )=
5
00) (8wx1
= y= )8
5
G. @pnbqug n` rgin` ag n` pftglhb` ag ul` dr`hhbøl y n` pftglhb` ag ftr` pftglhb` p`r` sbepnbdbh`r n`sgxprgsbflgs `nigmr`bh`s. J`y ul gcgepnf.
;) (8e
=l )8
5
0) ( `
m )7
5
=) ( `8
m8 )1
5
8) ( p7
q8 )=
5
1) ( j=
o 8 )4 5
7) ( `=m1
h7a 8 )8
5
4) ( w7 x4
y1 z : )=
5
9) (=`=
8m8 )8
5
:) ( 1 x8
7 y1 )8
5
2) ( ‘=w
‘8 z 7 )=
5
0;) ( 7 x= y=
8w8 z 8 )= 5
00) (0;e1l7
0; p1q1 )8
5
D. @pnbqug n` rgin` agn gxpflgltg lgi`tbvf p`r` sbepnbdbh`r n`s gxprgsbflgs. J`y ul gcgepnf.
;) (8 x
= y ) ‘1
5
0) x ‘: 5
=) ` ‘8 5
8) m ‘= 5
1) (7 x) ‘= 5
7) (‘8 y) ‘8 5
4) 1 x(8 y=) ‘= 5
9) 7`m(=h1) ‘8 5
:) 1 x(= y= z 8) ‘= 5
2) (8m
1`) ‘=
5
0;) ( 7 y=
8 x8 ) ‘=
5
00) (=m8
8h1 ) ‘8
5
‘0=7`4m0=h2
=9e8
:l8
( = y
8 x )1
504 y1
:0 x1
014BIG] ∝ [fnfhjbh
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@. Gngvg h`a` lóegrf ` n` pftglhb` blabh`a`. Qbglg ul gcgepnf.
;) ‘71 5
0) 18 5
=) 2= 5
8) 88 5
1) =; 5
7) := 5
4) 02 5
9) ‘1= 5
:) ‘2; 5
2) ‘9= 5
0;) ‘88 5
00) ‘4= 5
0=) ‘0: 5
08) ‘1; 5
01) (‘8)8 5
07) (‘9)= 5
04) (‘2)= 5
09) (‘=)7 5
0:) (‘7)8 5
02) (‘0); 5
=;) (‘1)8 5
M. Abvba` n`s gxprgsbflgs `nigmr`bh`s. Qbglg ul gcgepnf.
;)`:
`4 5
0)m7
m= 5
=)h:
h8 5
8) x2
x= 5
1) y:
y1 5
7) z 0=
z 0= 5
4)w4
w8 5
9) y7
y; 5
:) x1
x= 5
2)t 2
t 7 5
0;)h01
h2 5
00) z 0;
z 7 5
0=) p0;
p8 5
08)m0=
m1 5
01)`7
`8 5
07)h:
h4 5
04)o 2
o 1 5
09)q00
q: 5
0:) x:
x ‘= 5
02)j4
j ‘4 5
=;) y0;
y ‘4 5
H. Gngvg h`a` gxprgsbøl `nigmr`bh` n` pftglhb` blabh`a`. Qbglg ul gcgepnf.
;) (‘8 x8)1 5
0) (:m8)= 5
=) (= z 7)1 5
8) (7 y=)8 5
1) (‘4`1)= 5
7) (0;h=)8 5
4) (9a 7)= 5
9) (1 z 4)8 5
:) (8`4)1 5
2) (: y0;); 5
0;) (‘2 x1)= 5
00) (‘7h4)= 5
0=) (: y8)= 5
08) (4m1)= 5
01) (‘2a :); 5
07) (‘= x8)1 5
04) ( ‘2 z 8)= 5
09) (‘7w9)8 5
`=
‘4=7
:0 x0=
01:BIG] ∝ [fnfhjbh
@ibnba`a ag hànhunf eglt`n
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Glhugltrg ul` gxprgsbøl `nigmr`bh` p`r` h`nhun`r gn àrg` tft`n ag h`a` iur`.
Gn gcgepnf qug gstà ` hfltblu`hbøl prgsglt` nfs p`sfs ` sgiubr. Wstga pugag rgsfnvgr nfs ageàs
gcgrhbhbfs ag dfre` abrght`.;)
x
x
Gn àrg` ag ulf ag nfs hu`ar`afs gs3 @ 5 x • x 5 x=
[`r` glhfltr`r gn àrg` tft`n, agmgefs euntbpnbh`r gn àrg` agn hu`ar`af pfr 4.
Gl dfre` `nigmr`bh` gs3 @tft`n 5 4 x =
0) y
y
=)
x
x
x
x x
8) x
x
x
y y
y
012E`tgeàtbh` ∝ Xge`l` 2
]`zfl`ebgltf nøibhf
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0;0
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
]`abh`hbøl ag gxprgsbflgs `nigmr`bh`s
¶Puâ glhfltr`rà gst` sge`l`6
Biu`n lóegrf ag n`s y hfnuel`s
]gin`s ag nfs r`abh`ngs p`r` gxprgsbflgs `nigmr`bh`s Xue`, rgst`, euntbpnbh`hbøl y abvbsbøl ag r`abh`ngs hfl gxprgsbflgs `nigmr`bh`s
Gnbebl`hbøl ag d`htfrgs gl nfs r`abh`ngs ]`hbfl`nbz`hbøl ag gxprgsbflgs `nigmr`bh`s
[rfmnge`s ag àrg`, pgråegtrf y vfnuegl, `pnbh`laf n` r`abh`hbøl
Gst` sge`l` nfir`rà3
@pnbh`r n`s rgin`s ag nfs r`abh`ngs ` gxprgsbflgs `nigmr`bh`s.
]g`nbz`r fpgr`hbflgs `rbteâtbh`s hfl gxprgsbflgs qug hfltbglgl r`abh`ngs. Xbepnbh`r r`abh`ngs qug hfltbglgl gxprgsbflgs `nigmr`bh`s.
]`hbfl`nbz`r gxprgsbflgs `nigmr`bh`s.
@pnbh`r n` r`abh`hbøl ` n` sfnuhbøl ag prfmnge`s.
070E`tgeàtbh` ∝ Xge`l` 0;
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¡[`r` hfeglz`r!Biu`n lóegrf ag dbn`s y hfnuel`s
Be`iblg qug ustga tbglg ul lgifhbf ag `nqubngr ag sbnn`s y egs`s p`r` dbgst`s y
ul hnbgltg ng pbag qug, gl ul s`nøl ag dfre` hu`ar`a`, hfnfqug sbnn`s ` n` ebse`
abst`lhb`, ag t`n e`lgr` qug, j`y` n` ebse` h`ltba`a ag dbn`s y hfnuel`s.
Gstfs sfl nfs a`tfs qug s`mg3
Gn `nqubngr ag h`a` sbnn` hugst` P8.;;.
N` h`ltba`a eàxbe` qug gn hnbgltg qubgrg p`i`r pfr gn `nqubngr ag n`s sbnn`s gs
P18=.;;.
Hfl n` bldfre`hbøl prgsglt`a` hfeglhgefs ` rgsfnvgr gn prfmnge`.
[rbegrf, agmgefs hflfhgr gn lóegrf ag sbnn`s qug sg pugag utbnbz`r, p`r` gnnf
abvbabefs n` h`ltba`a ag ablgrf abspflbmng P18=.;; agltrf ag P8.;;, qug gs gn
prghbf agn `nqubngr ag h`a` sbnn`. P18=.;; ù P8.;; 5 011
@jfr` qug y` hflfhgefs gn lóegrf ag sbnn`s ` utbnbz`r, pfagefs `vgrbiu`r gn
lóegrf ag dbn`s y hfnuel`s, hfef gn gsp`hbf gs hu`ar`af h`mg n` ebse` h`l-
tba`a ag dbn`s y hfnuel`s. Wtbnbz`rgefs n` ngtr` x p`r` rgprgsglt`r n`s dbn`s y n`s
hfnuel`s qug gs gn a`tf agshflfhbaf.
Hfef s`mgefs qug gn àrg` qug fhup`ràl n`s 011 sbnn`s gs hu`ar`a`, gst` sg
fmtbglg `n euntbpnbh`r x • x 5 x=
[`r` fmtglgr gn v`nfr ag x, gnbebl`efs n` pftglhb` qug`hfep`ð` ` n` v`rb`mng, `pnbh`laf r`åz hu`ar`a` gl `emfs
ebgemrfs ag n` ghu`hbøl.
Gn v`nfr ag x gs 0=, pfr nf t`ltf sg agmgl hfnfh`r 0= dbn`s y 0= hfnuel`s hfl
0= sbnn`s gl h`a` ul`.
[`r` rgsfnvgr gstg pn`ltg`ebgltf dug lghgs`rbf j`hgr usf ag nfs r`abh`ngs, tge`
qug tr`t`rgefs gst` sge`l`.
¡@ tr`m`c`r!
Xupfli` qug ustga tbglg ul tgrrglf ag dfre` hu`ar`a` hfl ul àrg` ag 4=7 e=, agstbl`af p`r`sgemr`r h`dâ. ¶Huàlt`s pn`lt`s pugag sgemr`r, abstrbmuba`s gl n` ebse` h`ltba`a ag n`s y hfnuel`s,sb tfa`s gst`ràl ` 0 egtrf ag abst`lhb`6
x= 5 011
x= 5 011
x 5 0=
07=BIG] ∝ [fnfhjbh
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=. Hflvgrsbøl ag ul` r`åz ` ul`pftglhb` dr`hhbfl`rb`
Gst` hflvgrsbøl sg rg`nbz` gl `niulfs h`sfs p`r` d`hbnbt`r n`s fpgr`hbflgs hflr`abh`ngs, y` qug rgsunt` eàs dàhbn tr`m`c`r hfl gxpflgltgs.
Wl r`abh`n gs gqubv`ngltg ` ul` pftglhb` dr`hhbfl`rb` hu`laf gn aglfebl`afrag n` dr`hhbøl gs gn ålabhg agn r`abh`n y gn luegr`afr gs gn gxpflgltg agn r`ab-h`laf. [rgstg `tglhbøl `n rghu`arf y ` nfs gcgepnfs.
e l 5 `l/e
`8 5 `8/=8 x2 5 x2/8 5 x8
1 m7 5 m7/1
7 z 0; 5 z 0;/7 5 z =
1
w
9
5 w9/1
( x
y )1
5 ( x
y )1/=
5 ( x
y )=
[`r` hflvgrtbr ul gxpflgltg dr`hhbfl`rbf ` r`abh`n, sfnf j`hgefs n` fpgr`hbølblvgrs` ` nfs h`sfs `ltgrbfrgs. Tg`efs `niulfs gcgepnfs.
x=/7 57 x= e9/: 5
: e9
y8/1 51 y8 ( pq )
8/1
5 ( pq )8
1
x4/2 52 x4 ( w
z )=/7
5 ( w
z )=
7
Gcgrhbhbf =@. Hflvbgrt` nfs r`abh`ngs gl pftglhb`s dr`hhbfl`rb`s y sbepnbdbqug gn rgsunt`af aflag sg` pfsbmng.
Qbglg ul gcgepnf.
;)8 4 5
0)7 : 5
=) a :
5
8)8 (=; x)1 5
1)1 (:l)= 5
7)
(
p
q )4 5
4) ( jo )=
5
9): ( yz )1 5
:)
7
( x=
y8
) 5 M. Hflvbgrt` nfs gxpflgltgs dr`hhbfl`rbfs gl r`abh`ngs. Qbglg ul gcgepnf.
;) ( m)=/7 5
0) x8/1 5
=) w7/9 5
8) z =/8 5
1) (el)=/2 5
7) ( pq)8/1 5
4) ( jo )7/2
5
9) ( r
s )8/7
5
:) ( xyz )0/8 5
`4/8 5 `=
7 ( m)=
[`r` gvbt`r nfs v`nfrgs`msfnutfs, tfa`sn`s nbtgr`ngs ag nfsr`abh`lafs rgprgsglt`llóegrfs rg`ngspfsbtbvfs.
071BIG] ∝ [fnfhjbh
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8.= Euntbpnbh`hbøl ag r`abh`ngs
[`r` euntbpnbh`r r`abh`ngs, hflsbagrgefs nfs h`sfs sbiubgltgs3
`. ]`abh`ngs qug tbglgl gn ebsef ålabhg
Gl gstg h`sf agmgefs `pnbh`r n` rgin` agn prfauhtf ag r`åhgs. [rgstg `tglhbøl `n
rghu`arf y ` nfs p`sfs ag nfs gcgepnfs.
l • l
m 5 l` • m
Euntbpnbqugefs
• Gshrbmbefs nfs r`abh`lafs gl ul sfnf r`abh`n.
• Euntbpnbh`efs nfs r`abh`lafs y hflvgrtbefs gnr`abh`n gl ul gxpflgltg dr`hhbfl`rbf.
• Xbepnbdbh`efs gn rgsunt`af.
Euntbpnbqugefs
• Gshrbmbefs nfs r`abh`lafs gl ul sfnf r`abh`n.
• Euntbpnbh`efs nfs r`abh`lafs. Gl gstg h`sf, gnrgsunt`af lf sg pugag sbepnbdbh`r.
m. ]`abh`ngs qug tbglgl abdgrgltg ålabhg g biu`n r`abh`laf
Hu`laf nfs ålabhgs sfl abdgrgltgs, h`a` r`abh`n sg hflvbgrtg gl gxpflgltg dr`h-
hbfl`rbf y sg `pnbh` n` euntbpnbh`hbøl ag gxpflgltgs hfl m`sgs biu`ngs. Xb n`lugv` pftglhb` gs ul` dr`hhbøl, sg hflvbgrtg lugv`egltg gl r`abh`n, hfef sgeugstr` gl nfs gcgepnfs.
e •
l 5 `0/e • `0/l
Euntbpnbqugefs
• Hflvgrtbefs h`a` r`abh`n gl gxpflgltg dr`hhbfl`rbf.
• Hfpb`efs n` m`sg y sue`efs nfs gxpflgltgs.
• Hflvgrtbefs gn lugvf gxpflgltg dr`hhbfl`rbf gl r`abh`n.
Euntbpnbqugefs
• Hflvgrtbefs h`a` r`abh`n gl gxpflgltg dr`hhbfl`rbf.
• Hfpb`efs n` m`sg y sue`efs nfs gxpflgltgs.
• Hflvgrtbefs gn lugvf gxpflgltg dr`hhbfl`rbf gl r`abh`n.
18`7 •
1=9 8 5
18`7 • =9 8 5
1:0 : 5 8`:/1 5
8`=
:07 x= y8 •
:7 xy1 5
:07 x= y8 • 7 xy1 5
:97 x 8 y9
8`m • `m 5
( m)0/8 • ( m)0/= 5
( m)0/8 + 0/= 5 ( m)7/4 5
4
(`m)7
7( xy)8 • ( xy)7 5
( xy)8/7 • ( xy)7/= 5
( xy)8/7 + 7/= 5 ( xy)80/0; 5
0;( xy)80
074BIG] ∝ [fnfhjbh
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8.8 Abvbsbøl ag r`abh`ngs
[`r` abvbabr r`abh`ngs qug tbglgl gn ebsef ålabhg y abdgrgltg r`abh`laf, j`hgefsusf ag n` rgin` agn hfhbgltg ag nfs r`abh`ngs? sg abvbagl nfs r`abh`afs y ag sgrpfsbmng sg sbepnbh` gn rgsunt`af. N` abvbsbøl ag r`abh`ngs sbiug gst` rgin`3
l
l m
5 l `m
Abvba`efs
• Gxprgs`efs gn hfhbgltg hfef ul sfnf r`abh`n.
• Xbepnbdbh`efs gn r`abh`laf.
• Gshrbmbefs gn rgsunt`af.
Abvba`efs
• Gxprgs`efs gn hfhbgltg hfef ul sfnf r`abh`n.
• Xbepnbdbh`efs gn r`abh`laf.
• Gshrbmbefs gn rgsunt`af.
17`8
=7`=
5
1
=7
`8
`= 5
=7`8 ‘ = 5
=7`
4=; x7 y1
41 x= y8
5
4=; x7 y1
1 x= y8 5
47 x7 ‘ = y1 ‘ 8 5
47 x 8 y
Gcgrhbhbf 7[r`htbqug n` abvbsbøl ag r`abh`ngs hfl gn ebsef ålabhg y abdgrgltg r`abh`laf.
0)0:
2`4
42`=
5 =)4 x7 y1
4 x8 y=
5
8)9 =4e4l7
908e8l1
5 1): 04 p7q4r 1:1 p=q8r 1
5
07:BIG] ∝ [fnfhjbh
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8.1 Gnbebl`hbøl ag d`htfrgs gl nfs r`abh`ngs
Gl `niulfs h`sfs pfagefs sbepnbh`r ul r`abh`n gnbebl`af nfs d`htfrgs agnr`abh`laf j`st` qug n`s pftglhb`s ag n` p`rtg lueârbh` y ag n`s nbtgr`ngs sg`leglfrgs qug gn ålabhg agn r`abh`n. Nfs gcgepnfs sbiubgltgs ageugstr`l gst`fpgr`hbøl.
Xbepnbqugefs
• Hflvgrtbefs gn r`abh`laf gl d`htfrgs.
• @pnbh`efs n` rgin` agn prfauhtf y ag n` r`åz agul` pftglhb`.
• Gshrbmbefs gn rgsunt`af.
Xbepnbqugefs
• Hflvgrtbefs gn r`abh`laf gl d`htfrgs.
• @pnbh`efs n` rgin` agn prfauhtf y ag n` r`åz agul` pftglhb`.
• Gshrbmbefs gn rgsunt`af.
Xbepnbqugefs
• Gshrbmbefs nfs d`htfrgs hfef ul sfnf r`abh`n.
• Hflvgrtbefs gn r`abh`laf gl d`htfrgs.
• @pnbh`efs n` rgin` agn prfauhtf y ag n` r`åz agul` pftglhb`.
• Gshrbmbefs gn rgsunt`af.
0:`8 5
2 • = • `= • ` 5
(8 )=(= ) 5
(8 )= • (= ) 5
8` =`
841 x8 y7 z : 5
8
(1 xyz =)8 • ( y= z =) 58(1 xyz =)8 •
8( y= z =) 5
1 xyz =8 y= z =
Gcgrhbhbf 4Xbepnbqug nfs r`abh`ngs hfl nfs gcgrhbhbfs sbiubgltgs.
0) 1:`7 5 =) 804 x7 y1 z 4 5
8) =9e8l7 p9 5 1) 9 x1 y= • 9 x= y8 5
= x7 y8 • : x= y 5
04 x9 y1 5
(1 x8 y=)=( x) 5
(1 x8 y=)= • ( x) 5
1 x 8 y= x
072E`tgeàtbh` ∝ Xge`l` 0;
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1. ]`hbfl`nbz`hbøl ag aglfebl`afrgsWl` gxprgsbøl `nigmr`bh` gxprgs`a` gl su dfre` eàs sbepng f sbepnbh`a` lf agmg tglgr r`abh`ngs gl gn aglfebl`afr. [`r` gnnf, tr`m`c`rgefs gl n` r`hbfl`nb-z`hbøl, tge` qug y` vbefs gl nfs lóegrfs hfepngcfs y qug jfy `epnb`efs ` n`r`abh`hbøl. [rgstg `tglhbøl ` h`a` h`sf.
`. Hu`laf sg euntbpnbh` gn luegr`afr y gn aglfebl`afr pfr gn r`abh`n agn ag-lfebl`afr.
]`hbfl`nbhgefs
• Euntbpnbh`efs gn luegr`afr y gn aglf-ebl`afr pfr = x
• Xbepnbdbh`efs n`s gxprgsbflgs y fmtglg-efs gn rgsunt`af.
]`hbfl`nbhgefs
• Euntbpnbh`efs gn luegr`afr y gn aglf-ebl`afr pfr 8w
• Xbepnbdbh`efs n`s gxprgsbflgs y fmtglg-efs gn rgsunt`af.
m. Hu`laf gn aglfebl`afr tbglg ul hfgdbhbgltg, sg euntbpnbh` gn luegr`afr y gnaglfebl`afr pfr gn r`abh`n, sbl tfe`r gl huglt` gn hfgdbhbgltg.
]`hbfl`nbhgefs
• Euntbpnbh`efs gn luegr`afr y gn aglf-ebl`afr pfr 7 x
• Xbepnbdbh`efs n`s gxprgsbflgs y fmtglg-efs gn rgsunt`af.
]`hbfl`nbhgefs
• Euntbpnbh`efs gn luegr`afr y gn aglf-ebl`afr pfr =l
• Xbepnbdbh`efs n`s gxprgsbflgs y fmtglg-efs gn rgsunt`af.
1
= x 5
1
= x ( = x
= x ) 5 1 = x
( = x )= 5
1 = x
( = x )= 5
1 = x
= x 5
= = x
x
7 y
8w 5
7 y
8w ( 8w
8w ) 5 7 y 8w
( 8w)= 5
7 y 8w
( 8w)= 5
7 y 8w
8w
:e
1 =l 5
:e
1 =l ( =l
=l ) 5 :e • =l
1( =l)= 5
04el
1( =l)= 5
1 el
1(=l) 5
el
=l
9
8 7 x 5
9
8 7 x ( 7 x
7 x ) 5 9 • 7 x
8( 7 x)= 5
87 x
8( 7 x)= 5
87 x
8(7 x) 5
87 x
07 x
04;BIG] ∝ [fnfhjbh
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h. Hu`laf gn aglfebl`afr gs ul mblfebf sg euntbpnbh` gn luegr`afr y gn ag-lfebl`afr pfr gn hflcui`af agn aglfebl`afr.
]`hbfl`nbhgefs
• Euntbpnbh`efs gn luegr`afr y gn aglf-ebl`afr pfr gn hflcui`af.
• Xbepnbdbh`efs n`s gxprgsbflgs.
• Gshrbmbefs gn rgsunt`af.
]`hbfl`nbhgefs
• Euntbpnbh`efs gn luegr`afr y gn aglf-ebl`afr pfr gn hflcui`af.
• Xbepnbdbh`efs n`s gxprgsbflgs.
• Gshrbmbefs gn rgsunt`af.
7
8 x + = x
7
8 x + = x ( 8 x ‘ = x
8 x ‘ = x ) 5
7( 8 x ‘ = x )
( 8 x )= ‘ ( = x )= 5
7( 8 x ‘ = x )
( 8 x )= ‘ ( = x )= 5
7( 8 x ‘ = x )
8 x ‘ = x 5
7( 8 x ‘ = x )
x
8 x
x ‘ 4
8 x
x ‘ 4 ( x + 4
x + 4 ) 5
8 x( x + 4 )
( x )= ‘ ( 4 )= 5
8 x( x + 4 )
( x )= ‘ ( 4 )= 5
8 x ( x + 4 )
x ‘ 4
Gcgrhbhbf 9]`hbfl`nbhg n`s gxprgsbflgs `nigmr`bh`s y sbepnbqug gn rgsunt`af aflag sg` pfsbmng.
0)7
= 8 x 5 =)
2 y
y + 8 5
040E`tgeàtbh` ∝ Xge`l` 0;
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]gsuegl
0. N`s rgin`s ag n` r`abh`hbøl sfl n`s ebse`s qug utbnbz`efs hfl nfs lóegrfs rg`ngs. Ag dfre`sbemønbh` sg rgprgsglt`l `så3
l
l
5
l
l
5 `
l
` • m 5
l
•l
m
l
`
m 5
l
l m
l e` 5 l • e `
=. [`r` hflvgrtbr ul r`abh`n gl gxpflgltg dr`hhbfl`rbf, hfpb`-efs gn r`abh`laf y abvbabefs su pftglhb` gltrg gn ålabhgagn r`abh`n.
8. Afs f eàs r`abh`ngs sfl sgegc`ltgs sb gn r`abh`laf y gnålabhg sfl biu`ngs.
8.0 N` sue` y rgst` ag r`abh`ngs sg pugag rg`nbz`r sb y sfnfsb sfl r`abh`ngs sgegc`ltgs. [`r` rg`nbz`r gst`s fpgr`-hbflgs, sg sue`l f rgst`l nfs hfgdbhbgltgs y sg hfpb` gnr`abh`n.
8.= Gl n` euntbpnbh`hbøl ag r`abh`ngs sg pugagl prgsglt`r afs h`sfs3 nfs qug tbglgl gn ebsefålabhg y nfs qug tbglgl abdgrgltg ålabhg.
`. Hfl gn ebsef ålabhg
m. Hfl abdgrgltg ålabhg g biu`n r`abh`laf
8.8 [`r` abvbabr r abh`ngs qug tbglgl gn ebsef ålabhg, sgabvbagl nfs r`abh`afs y ag sgr pfsbmng sg sbepnbdbh` gnrgsunt`af.
1. [`r` gnbebl`r d`htfrgs gl nfs r`abh`ngs, sg gxtr`gl nfs d`htfrgs ag n` p`rtg lueârbh` y ag n`s nbtgr`ngsj`st` qug n`s pftglhb`s sg`l eglfrgs qug gn ålabhg agn r`abh`n.
7. N` r`hbfl`nbz`hbøl pgrebtg gxprgs`r nfs r`abh`ngs gl ul` dfre` sbepnbdbh`a`, `n lf tglgr r`abh`ngsgl gn aglfebl`afr. Xg agmgl tfe`r gl huglt` nfs h`sfs sbiubgltgs3
`. Gn aglfebl`afr hflst` ag ul sfnf târeblf r`abh`n y lf tbglg hfgdbhbgltgs.
m. Gn aglfebl`afr gs ul r`abh`n y tbglg ul hfgdbhbgltg.h. Gn aglfebl`afr gstà dfre`af pfr afs r`abh`ngs qug dfre`l ul` sue` f abdgrglhb`, gs ulmblfebf.
[`r` pr`htbh`r n` r`abh`hbøl gl n` rga, blirgsg ` gst` abrghhbøl gnghtrølbh`3 jttp3//iff.in/CALY`z
e l 5 `l/e
== y y 0: x == y
`l x + m
l x ‘ h
l x 5 (` + m ‘ h)
l x
l
l m
5 l`
m
e •
l 5 `0/e • `0/l
l • l
m 5 l` • m
04=BIG] ∝ [fnfhjbh
Blvgstbiug gl n` rga...
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@htbvba`a 0. Ageugstrg nf prglabaf
E`rqug ul` "Z" gl gn hu`arf ag n` fphbøl qug hfepngt` h`a` glulhb`af.
0) N` gxprgsbøl gqubv`ngltg ` x7 gs…
=) Gn rgsunt`af qug eugstr` n` dfre` hfrrght` ag fpgr`r l e x gs…
8) N` fphbøl qug rgprgsglt` gn rgsunt`af sbepnbdbh`af ag8 x4 gs…
1) Wl r`abh`n sgegc`ltg ` ‘7 x = x7
gs…
7) N` fphbøl qug rgprgsglt` gn hflcui`af ag 7 x ‘ = x gs…
4) N` fphbøl qug rgprgsglt` gn rgsunt`af hfrrghtf ag n` fpgr`hbøl( x ‘ 4 )( x + 4 ) gs…
@utfhfltrfn
x7
x7/=
x=/7
7 x=
l + e
x
l ‘ e
x
l • e
x
l ù e
x
x
x=
x8/4
x=
= x7
7 x7
x = x8
‘7 x8 = x7
= x ‘ 7 x
7 x + = x
‘ 7 x ‘ = x
‘ 7 x + = x
x ‘ 4
x + 4
x ‘ 4
x + 4
048E`tgeàtbh` ∝ Xge`l` 0;
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Gl gstg `p`rt`af pr`htbh`rà n` r`abh`hbøl ag lóegrfs rg`ngs. [`r` rgsfnvgr nfs gcgrhbhbfs, gs lghgs`rbfqug rghugrag n`s døreun`s p`r` gn àrg` ag hu`ar`afs, rghtàliunfs y gn hårhunf, `så t`embâl n` agnvfnuegl p`r` gn humf.
0) Gn àrg` ag ul tgrrglf ag dfre` hu`ar`a` gs 042 e=. ¶Huàltf ebag gn pgråegtrf agn tgrrglf6
=) Gn vfnuegl ag ul humf gs 0;;; he8. ¶Huàn gs gn àrg` ag ul` ag sus h`r`s6
8) Gn àrg` ag ul tgrrglf hu`ar`af gs 4=7 e=. ¶Huàn gs gn àrg` ag ftrf tgrrglf ag n` ebse` dfre`huyf n`af ebag gn trbpng agn qug tbglg gn àrg` ag 4=7 e=6
1) Xg qubgrg abstrbmubr ` 7=2 gstuab`ltgs gl ul s`nøl ag dfre` hu`ar a`. ¶Huàltfs gstuab`ltgsj`mrà gl h`a` dbn`6
7) N` ebt`a agn hu`ar`af ag n` abst`lhb` qug rghfrrg ul hbhnbst` gl 8; eblutfs gs 04= oe.
¶Huàltf rghfrrgrà gl = jfr`s6 4) Xg qubgrg hflstrubr ul t`mngrf hu`ar`af qug tgli` ul` supgrdbhbg ag ==7 he= y qug ` su vgz
hfltgli` 011 h`sbnn`s biu`ngs, sbl gsp`hbf gltrg ul` y ftr`. ¶Huàltf egabrà gn n`af ag h`a`h`sbnn`6
9) Gn àrg` ag ul hu`ar`af gs 1;24 he=. ¶Huàltf egabrà gn pgråegtrf ag ftrf hu`ar`af huyfn`af gs n` r`åz hómbh` ag nf qug ebag ul n`af agn prbegr hu`ar`af6
:) Gl ul` j`mbt`hbøl sg qubgrgl hfnfh`r 8 egs`s hu`ar`a`s ag = e= h`a` ul` y = egs`s,t`embâl hu`ar`a`s, ag : e= h`a` ul`. [ugst`s ul` ` hfltblu`hbøl ag n` ftr`, ¶quâ nflibtuafhup`ràl tfa`s n`s egs`s6
2) Wl tgrrglf hu`ar`af tbglg ul` supgrdbhbg ag 2;; e=. ¶Huàltf hfst`rà hgrh`rnf sb gn egtrf age`nn` tbglg ul v`nfr ag P97.;;6
0;) Gn augðf ag ul tgrrglf, huy`s abeglsbflgs sfl 8= e ag n`rif pfr : e ag `lhjf, qubgrgpgreut`rnf pfr ul tgrrglf hu`ar`af ag n` ebse` supgrdbhbg. ¶Huàltf agmg egabr h`a` n`afagn tgrrglf hu`ar`af6
00) Wl hfegrhb`ltg hfeprø hbgrtf lóegrf ag pn`ygr`s pfr P=74.;;. Xb gn lóegrf y gn prghbf agn`s pn`ygr`s sfl biu`ngs, ¶huàlt`s pn`ygr`s hfeprø6
0=) Wl` h`c` gl dfre` hómbh` tbglg ul vfnuegl ag 0=7 ;;; he8. Xb sg hfrt` pfr n` ebt`a, ¶huàngs
sgràl n`s abeglsbflgs ag n` h`c` rgsunt`ltg6
08) J`nng gn àrg` ag n` hfrfl` hbrhun`r hfeprglaba` gltrg n`shbrhuldgrglhb`s blshrbt` y hbrhulshrbt` gl ul hu`ar`af ag4.=7 he= ag àrg`.
8 . 7 1
049E`tgeàtbh` ∝ Xge`l` 0;
]`zfl`ebgltf nøibhf
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E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfir`af gl
prfhgsflf
nfir`af
A g s p u â s a g g s t u a b ` r . . . @pnbhf n`s rgin`s ag nfs r`abh`ngs ` gxprgsbflgs `nigmr`bh`s.
]g`nbzf fpgr`hbflgs `rbteâtbh`s hfl gxprgsbflgs qug hfltbglglr`abh`ngs.
Xbepnbhf r`abh`ngs qug hfltbglgl gxprgsbflgs `nigmr`bh`s.
]`hbfl`nbzf gxprgsbflgs `nigmr`bh`s.
@pnbhf n` r`abh`hbøl ` n` sfnuhbøl ag prfmnge`s.
Gn ebstgrbfsf lóegrf 4
[`r` hfeglz`r hflfzh`efs gn såemfnf ! Gl e`tgeàtbh` sg ng nn`e` d`htfrb`n ysg aglg hfef gn prfauhtf ag tfafs nfs lóegrfs gltgrfs pfsbtbvfs agsag gn
lóegrf blabh`af j`st` 0. Tg`efs gn d`htfrb`n ag 1 y 7.1! 5 1 Ù 8 Ù = Ù 0 5 =1
7! 5 7 Ù 1 Ù 8 Ù = Ù 0 5 0=;
Gstg såemfnf gs gn qug lfs sgrvbrà p`r` hfepngt`r n` prbegr` n` agn hu`arf ag`m`cf, aflag, hfl hu`nqubgr fpgr`hbøl e`tgeàtbh` y utbnbz`laf nfs såemfnfs
, +, 8 , ‘ , ù, Ù agmg a`r hfef rgsunt`af 4.
[rgstg `tglhbøl høef sg fmtbglg 4 gl n` prbegr` n`.
(0 + 0 + 0 )! 5 (8)! 5 8 Ù = Ù 0 5 4
@jfr` ng tfh` ` ustga hfepngt`r n`s ageàs.
(0 + 0 + 0)! 5 4
= = = 5 4
8 8 8 5 4
1 1 1 5 4
7 7 7 5 4
4 4 4 5 4
9 9 9 5 4: : : 5 4
2 2 2 5 4
04:BIG] ∝ [fnfhjbh
]gvbsg su `prglabz`cg
Ags`rrfnng lugv`s j`mbnba`ags
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000
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
Fpgr`hbflgs pfnblfeb`ngs
¶Puâ glhfltr`rà gst` sge`l`6
N` bepfrt`lhb` agn ànigmr`
[fnblfebfs Fpgr`hbflgs hfl pfnblfebfs
Fpgr`hbflgs hfl eflfebfs
H`nhun`r gn àrg` sfemrg`a` gl iur`s pn`l`s.
Gst` sge`l` nfir`rà3
Hflhgptu`nbz`r g bagltbh`r ul pfnblfebf. Hn`sbh`r pfnblfebfs ag `hugraf ` su lóegrf ag târeblfs.
Bagltbh`r y rgsfnvgr fpgr`hbflgs gltrg pfnblfebfs.
]gsfnvgr fpgr`hbflgs ag eflfebfs hfl `ibnba`a.
042E`tgeàtbh` ∝ Xge`l` 00
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¡[`r` hfeglz`r!N` bepfrt`lhb` agn ànigmr`
Gl `niól pultf ag lugstrfs gstuabfs, sg lfs pugag hruz`r pfr gn pgls`ebgltf
n` prgiult`3 ¶pfr quâ gs bepfrt`ltg `prglagr ànigmr`6
[ugs mbgl, gst` r`e` ag n` e`tgeàtbh` gs ul` pfagrfs` jgrr`ebglt` ag lànbsbs
y tbglg ir`l v`rbga`a ag `pnbh`hbflgs? gs utbnbz`a` pfr prfdgsbfl`ngs qug v`l
agsag nfs gnghtrbhbst`s, nfs `rqubtghtfs y nfs bldfreàtbhfs.
@ hfltblu`hbøl, vgrgefs `niulfs h`sfs gspghådbhfs gl nfs qug sg gepng` gn
ànigmr`3
• [`r` h`nhun`r n` nblg`hbøl ag `ltgl`s gnghtrfe`ilâtbh`s. Gn absgðf ag ul`
`ltgl` p`r` rgags ag bltgrlgt bl`nàemrbh` sg j`hg gepng`laf n` døreun`
ag Hjgvysjgv, ul e`tgeàtbhf rusf. Hfl m`sg gl nfs a`tfs h`nhun`afs, sg
gst`mnghgl n`s abeglsbflgs baølg`s p`r` e`ltglgr ul` sgð`n ulbdfreg y
hfl n` eàxbe` hfmgrtur` pfsbmng.
• Gl n`s hbglhb`s mbfnøibh`s, sg utbnbz` p`r` prfyght`r gn rbtef ag hrghbebgl-
tf ag ul` pfmn`hbøl ag m`htgrb`s f n` pfsbmbnba`a ag prfp`i`hbøl ag ul`
gldgrega`a.
• Gl n` egtgfrfnfiå`, gn ànigmr` `yua` p`r` n`s prgabhhbflgs agn hnbe`? gn
BLXBTWEGJ utbnbz` efagnfs e`tgeàtbhfs p`r` h`nhun`r n` h`ltba`a ag nnuvb`,
vbgltfs, gth. qug sg gspgr`l gl `niul` rgibøl.
• Gl ghflfeå , n` e`tgeàtbh` tbglg ul rfn dula`eglt n? jgrr`ebglt`s yhflhgptfs agn ànigmr` sg utbnbz`l p`r` prflfstbh`r nfs hfepfrt`ebgltfs
qug tglarà n` ghflfeå` gl gn duturf.
[fr gcgepnf, gn M`lhf ag Iu`tge`n`, n` blstbtuhbøl glh`ri`a` ag n` ghflf-
eå` l`hbfl`n, utbnbz` n`s prfyghhbflgs p`r` tfe`r n`s egaba`s lghgs`rb`s
y e`ltglgr `så n` gst`mbnba`a gl nfs prghbfs ag nfs prfauhtfs gl gn p`ås.
Hfef puabefs lft`r, gn ànigmr` pugag sgr `mstr`ht`, pgrf tbglg ir`l h`ltba`a
ag `pnbh`hbflgs gl n` vba` rg`n, pfr nf qug rgsunt` euy bepfrt`ltg su afeblbf.
¡@ tr`m`c`r!
[bglsg y gshrbm` niul` ftr` pnbh`hbøl p`r` n` qug gs lghgs`rb` gn ànigmr`. [ugag pfy`rsg gl bltgrlgtu ftrf rghursf ag blvgstbi`hbøl.
09;BIG] ∝ [fnfhjbh
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0. [fnblfebfsWl pfnblfebf gs ul` gxprgsbøl e`tgeàtbh` dfre`a` ` p`rtbr ag n` ulbøl ag
afs f eàs v`rb`mngs y hflst`ltgs, qug sg vblhun`l ` tr`vâs ag n`s fpgr`hbflgs`rbteâtbh`s ag sue`, rgst` y euntbpnbh`hbøl. Gl ftr`s p`n`mr`s, ul pfnblfebf gs n`ulbøl ag afs f eàs eflfebfs.
0.³ târeblf =.³ târeblf 8.³ târeblf
9 x= + 1 xy ∝ 8
0.0 Hn`sbdbh`hbøl ag nfs pfnblfebfs
Xgiól su lóegrf ag târeblfs, ul pfnblfebf pugag rghbmbr abstbltfs lfemrgs3
• Mblfebf3 gs gn pfnblfebf eàs rgauhbaf, hfl afs târeblfs.• Qrblfebf3 pfnblfebf hfl trgs târeblfs.
• Qgtr`lfebf3 pfnblfebf hfl hu`trf târeblfs.
Lfs rgdgrbefs ` n`s gxprgsbflgs hfl hblhf târeblfs gl `agn`ltg sbepngegltghfef pfnblfebfs.
Xgiól nfs târeblfs qug hfeprglag3
[fnblfebf hfepngtf3 gs `qugn qug tbglg tfafs nfs târeblfs, agsag gn târeblfag e`yfr ir`af j`st` gn târeblf blagpglabgltg.
1 y8 ∝ = y= + 7 y + 0;
Hfef lft`efs, gn pfnblfebf `ltgrbfr v` agsag gn târeblf hfl gxpflgltg 8j`st` gn târeblf blagpglabgltg, gs aghbr, gn qug y` lf tbglg v`rb`mng.
[fnblfebf blhfepngtf3 gs `qugn qug lf tbglg tfafs nfs târeblfs agsag gntâreblf ag e`yfr ir`af j`st` gn târeblf blagpglabgltg.
4 y8 ∝ 7 y + 8 5
[fagefs vgr qug gl n` gxprgsbøl `ltgrbfr j`hg d`nt` gn târeblf ag sgiulafir`af ( y =) p`r` qug sg` hfepngtf.
Xgiól su ir`af3
@n biu`n qug hfl nfs eflfebfs, nfs pfnblfebfs tbglgl "ir`af". Gn ir`af ag ulpfnblfebf nf agtgrebl` gn eflfebf f târeblf hfl gn e`yfr gxpflgltg. Tg`efs3
7w8 + 4w= ‘ 0;w + 4
E`yfr gxpflgltg
Gl gn gcgepnf `ltgrbfr, gn e`yfr gxpflgltg gs 8, pfr nf qug aghbefs qug gs ulpfnblfebf ag "tgrhgr ir`af" f "hómbhf"? sb su gxpflgltg e`yfr dugr` =, sgrå` ag"sgiulaf ir`af" y `så hfltbló` hfl tfafs nfs lóegrfs.
]ghfragefs qug ul
eflfebf gs ul`gxprgsbøl `nigmr`bh`,dfre`a` pfr ulhfghbgltg y ul`p`rtg nbtgr`n (v`rb`mng)qug sg euntbpnbh`l.
090E`tgeàtbh` ∝ Xge`l` 00
Gn eulaf ag n` e`tgeàtbh`
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Gcgrhbhbf 0@. Gshrbm` gl n` nålg` hfrrgspflabgltg sb sg tr`t` ag ul mblfebf, trblfebf f ag ul tgtr`lfebf.
0) ‘0;t + 7t 1 + t ‘ 0; 1) w8 + =v
=) 1 x8 + 7 xy= 7) ‘ d + d =
8) 7` ‘ 1`m1 + 1m8 4) 8a = + a 8 ‘ 7
=. Xue` y rgst` ag pfnblfebfs[`r` n` `abhbøl y n` sustr`hhbøl gltrg afs f eàs pfnblfebfs, sg sue`l f rgst`lnfs hfghbgltgs ag nfs târeblfs sgegc`ltgs, gs aghbr, ag nfs târeblfs qug tgl-i`l n` ebse` v`rb`mng hfl gn ebsef ir`af. T`efs, lf sg prgfhupg, qug gs euysglhbnnf.
]gsfnv`efs n` sbiubgltg sue` pfnblfeb`n3
• Fragl`efs nfs pfnblfebfs sb lf nf gstàl.
• Gnbebl`efs nfs p`râltgsbs tglbglaf hub-a`af hfl n` ngy ag sbilfs.
• @irup`efs nfs eflfebfs f târeblfssgegc`ltgs.
• Fpgr`efs nfs hfgdbhbgltgs ag nfs târeb-lfs sgegc`ltgs. @qugnnfs eflfebfs quglf tbglgl lbliól târeblf sgegc`ltg `nhu`n sue`rsg sfn`egltg sg hfpb`l.
N` rgst` sbiug ul prfhgsf euy sbebn`r, pfli`efs `tglhbøl.
]g`nbhgefs n` rgst` sbiubgltg3
• Fragl`efs nfs pfnblfebfs sb lfnf gstàl.
• Gnbebl`efs nfs p`râltgsbs tglbgl-af huba`af hfl n` ngy ag sbilfs.
• @irup`efs nfs eflfebfs ftâreblfs qug tbglgl n` ebse`v`rb`mng y gn ebsef ir`af.
• Fpgr`efs nfs hfgdbhbgltgs ag nfstâreblfs sgegc`ltgs. @qugnnfseflfebfs qug lf tbglgl lblióltâreblf sgegc`ltg `n hu`n fpg-r`rsg sfn`egltg sg hfpb`l.
(: x1 + 7 x8 ‘ 4) + (1 x ∝ 9 x8 + = x1)
(: x1 + 7 x8 ‘ 4) + (= x1 ∝ 9 x8 + 1 x) 5
: x1 + 7 x8 ‘ 4 + = x1 ∝ 9 x8 + 1 x 5
: x1 + = x1 + 7 x8 ∝ 9 x8 + 1 x ‘ 4 5
0; x 1 ∝ = x 8 + 1 x ‘ 4
(‘02 x8 + =0 x7 + 2) ‘ (01 x7 ‘ 2 x8 + : x= + 0)
(=0 x7 ‘ 02 x8 + 2) ‘ (01 x7 ∝ 2 x8 + : x= + 0) 5
=0 x7 ‘ 02 x8 + 2 ‘ 01 x7 + 2 x8 ‘ : x= ‘ 0 5
=0 x7 ‘ 01 x7 ‘ 02 x8 + 2 x8 ‘ : x= + 2 ‘ 0 5
9 x 7 ‘ 0; x 8 ‘ : x = + :
09=BIG] ∝ [fnfhjbh
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M. Gshrbm` gl n` nålg` hfrrgspflabgltg gn ir`af ag h`a` ulf ag nfs pfnblfebfs sbiubgltgs.
0) 8 x8 + = x= + x ‘ 2 4) 0= + y + x
=) y1 ‘ 0; y8 + y= + y 9) ‘== + 2r 8 + 1r = + r 1
8) ‘0;a 8 + a 1 ‘ 8a = :) 88 x8 ‘ x= ‘ 0; + 8 x
1) j= + j7 + 1 2) ‘=0 ‘ `= ‘ `
7) 4t 8 ‘ t = + 02 0;) r = + r
H. ]gsugnv` n`s sbiubgltgs sue`s y rgst`s ag pfnblfebfs. Xb lghgsbt` eàs gsp`hbf, pugag tr`m`c`r gljfc`s `p`rtg.
0) ( x8 + x=) + (00 x= ‘ x) 4) ‘(r + 0:r = + 08) + (‘:r = + 7r + 7)
=) (0; ‘ 0= x=) ‘ (7 x ‘ = x= + 7) 9) (18 x= y + 08 xy) ‘ (‘4 xy + 08 x= y + 08)
8) (4 y + 8 y8 ‘ 1) ‘ ( y + 09 ‘ 8 y8) :) (8=w8 ‘ 8w= ‘ w + =) + (0=w ‘ 8 + 1w8)
1) ( t 8 + t ) + (02t ‘ 00`t 8 + 0) 2) ‘(=8 i + 04 i = + =) + (‘: i = + 0)
7) (14`1 + =0`8) + ( = ‘ 01`1 + `8) 0;) ‘(:wy8 ‘ 1 y= + wy) + (=wy + 1w + 8wy8)
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8. Euntbpnbh`hbøl ag pfnblfebfs[`r` euntbpnbh`r pfnblfebfs sg utbnbz` n` prfpbga`a abstrbmutbv`? gn prfauhtf pug-ag sgr ag ul lóegrf pfr ul pfnblfebf, ag ul eflfebf pfr ul pfnblfebf ygltrg afs pfnblfebfs.
8.0 Euntbpnbh`hbøl ag ul lóegrf pfr ulpfnblfebf
Xg euntbpnbh` gn lóegrf pfr h`a` ulf ag nfs târeblfs ag n` gxprgsbøl pfnblfeb`n. Gnrgsunt`af gs ftrf pfnblfebf qug pfsgg gn ebsef ir`af agn pfnblfebf qug sgfpgrø y hfef hfghbgltgs sg tbglgl nfs prfauhtfs ag nfs hfghbgltgs agn pfnb-lfebf pfr gn lóegrf.
Euntbpnbqugefs3
• @pnbh`efs n` prfpbga`aabstrbmutbv` `n euntbpnbh`r(‘1) pfr h`a` ulf ag nfstâreblfs agn pfnblfebf.
• Fpgr`efs y fmtglgefs gnrgsunt`af dbl`n. Qgli`efssbgeprg gl egltg n` ngy agsbilfs.
8.= Euntbpnbh`hbøl ag ul eflfebf pfr ulpfnblfebf
Xg euntbpnbh` gn eflfebf pfr tfafs y h`a` ulf ag nfs târeblfs qug dfre`l gnpfnblfebf, gs aghbr, hfghbgltg pfr hfghbgltg y v`rb`mng pfr v`rb`mng. Gl gstgprfhgabebgltf sg gepng` n` prfpbga`a abstrbmutbv`.
[`r` gst` fpgr`hbøl agmgefs rghfra`r n`s ngygs ag nfs gxpflgltgs, sb ng surig`niul` aua`, pugag rgirgs`r y hflsunt`r n` sge`l` 2 ag su nbmrf.
Fpgrgefs3
• @pnbh`efs n` prfpbga`aabstrbmutbv` `n euntbpnbh`r(7wx8)
pfr h`a` ulf ag nfstâreblfs agn pfnblfebf.
Hfef aghå`efs `rrbm`, `ntr`m`c`r hfl n`s v`rb`mngsagmgefs `pnbh`r n`s ngygsag nfs gxpflgltgs.
• ]gsfnvgefs y fmtglgefsgn rgsunt`af dbl`n.
(‘1)(‘9 x1 + 04 x8 y ‘ x= + 08)
(‘1)(‘9 x1 + 04 x8 y ‘ x= + 08) 5
(‘1)(‘9 x1) + (‘1)(04 x8 y) + (‘1)(‘ x=) + ( ‘1)(08) 5
=: x 1 ‘ 41 x 8 y + 1 x = ‘ 7=
(7wx8)(1 x1 + : x8 ∝ = x ∝ 00)
(7wx8)(1 x1 + : x8 ∝ = x ∝ 00) 5
(7wx8)(1 x1) + (7wx8)(: x8) + (7wx8)(‘= x) + (7wx8)(∝00) 5
=;wx 9 + 1;wx 4 ‘ 0;wx 1 ‘ 77wx 8
091BIG] ∝ [fnfhjbh
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8.8 Euntbpnbh`hbøl gltrg pfnblfebfs
[`r` euntbpnbh`r afs pfnblfebfs gltrg så, sg euntbpnbh` h`a` târeblf agn prbegrpfnblfebf pfr tfafs y h`a` ulf ag nfs târeblfs agn sgiulaf pfnblfebf.
]gsfnv`efs3
• Ag lugvf, hfef gl nfs h`sfs `ltgrbf-rgs, `pnbh`efs n` prfpbga`a abstrbmu-tbv`.
• Euntbpnbh`efs gn prbegr târeblf agnprbegr pfnblfebf (8 x8) pfr h`a` ulfag nfs târeblfs agn sgiulaf pfnblf-ebf.
• Euntbpnbh`efs gn sgiulaf târeblfagn prbegr pfnblfebf (1 x) pfr h`a`ulf ag nfs târeblfs agn sgiulafpfnblfebf.
• Fragl`efs nfs eflfebfs y sue`-efs nfs qug tgli`l gn ebsef ir`af.
• Xg fmtbglg gn rgsunt`af dbl`n, ftrfpfnblfebf huyf ir`af gs n` sue` agnfs ir`afs ag nfs pfnblfebfs qug sgeuntbpnbh`l.
Gcgrhbhbf =Gl su hu`agrlf, rgsugnv` n`s euntbpnbh`hbflgs pfnblfeb`ngs sbiubgltgs. Nugif, gshrbm` su rgspugst` gl
n` nålg` hfrrgspflabgltg.
0) 0;(7 x8 + 08 x ‘ 8) 5
=) (8`m= + 7`=)(8`8m= ‘ 1m) 5
8) (7w8 x)(9w= + 8 x= ‘ 7) 5
1) (‘:)(2 xy= + y8 + x8) 5
7) (2 i =j8 + 9ij= ‘ :j ‘ 8)(1 i =j= + 0= ij ‘ 07) 5
4) (01m=h8)(9 m8 + 00`h= ‘ :) 5
9) 09(8l8e ‘ :l=e= + 4le8 + :) 5
:) (0=w= xy ‘ 1w= x= y= + 2wx8 y8)(7w= x + : xy8 ‘ 9wxy) 5
2) (2 x8 y)(8 xy= ‘ 7 x= y + 9 x) 5
0;) (‘1w= z + 4wz)(:wy= ‘ 7 x8 z =) 5
(8 x8 + 1 x)(8 x1 ∝ 1 x= ∝ 2)
(8 x8 + 1 x)(8 x1 ∝ 1 x= ∝ 2) 5
(8 x8 + 1 x)(8 x1 ∝ 1 x= ∝ 2) 5
(8 x8)(8 x1) + (8 x8)(∝1 x=) + (8 x8)(∝2) 5
2 x 9 ‘ 0= x 7 ‘ =9 x 8
(8 x8 + 1 x)(8 x1 ∝ 1 x= ∝ 2) 5
(1 x)(8 x1) + (1 x)(∝1 x=) + (1 x)(∝2) 5
0= x 7 ‘ 04 x 8 ‘ 84 x
2 x 9 ‘ 0= x 7 + 0= x 7 ‘ 04 x 8 ‘ =9 x 8 ‘ 84 x 5
2 x 9 ‘ 18 x 8 ‘ 84 x
2 x 9 ‘ 18 x 8 ‘ 84 x
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Gcgrhbhbf 8Gl su hu`agrlf, rgsugnv` n`s abvbsbflgs pfnblfeb`ngs sbiubgltgs. Nugif, gshrbm` su rgspugst` gl n` nålg`hfrrgspflabgltg.
0) (4 x1 y ‘ 2 x8 y= + 0= x= y8 ‘ 4 xy1) ù (8 xy) 5
=) (1 x8 + 4 x= ‘ : x ‘ =) ù (= x) 5
8) (0=`8m= ‘ =;`=m + 04`m ‘ 1) ù (∝ 1`=m) 5
1) (=;wx=
+ 07w=
x ‘ =7w) ù (7wx) 5
7) (:0`8m= ‘ =9 =m8 + 71`m1 + m) ù (2`=m8) 5
4) (2 x= y= + 8 xy8 ‘ 0=) ù (8 xy=) 5
9) (1;m1h8 ‘ ==m=h= + 0:mh) ù (∝ =m=h8) 5
:) (0= xy8 + 0: x= y= ‘ 1= x8 y + 4 x1 ‘ 0=) ù (4 x= y8) 5
2) (1:`1m1 ‘ 8=`8m= + :`=m) ù (:`8m8) 5
1. Abvbsbøl ag ul pfnblfebf gltrg uleflfebf
[`r` abvbabr ul pfnblfebf gltrg ul eflfebf, sg abvbag h`a` ulf ag nfs târeblfsagn pfnblfebf pfr gn eflfebf, tglbglaf gl huglt` n` ngy ag nfs sbilfs.
]gsfnv`efs3
• Abvbabefs h`a` ulf ag nfs târeblfsagn pfnblfebf gltrg gn eflfebf.
• Fpgr`efs, abvbabglaf hfgdbhbgltgsgltrg hfgdbhbgltgs y v`rb`mngs gltrgv`rb`mngs.
• Fmtglgefs gn rgsunt`af dbl`n.
Tg`efs ul óntbef gcgepnf, fpgrgefs3• Abvbabefs h`a` ulf ag nfs târeblfs agn
pfnblfebf gltrg gn eflfebf. Qglgefshuba`af hfl n` ngy ag sbilfs.
• Fpgr`efs, abvbabglaf hfgdbhbgltgsgltrg hfgdbhbgltgs y v`rb`mngs gltrgv`rb`mngs.
• Fragl`efs nfs târeblfs y fmtglgefsgn rgsunt`af dbl`n.
(8 x8
y=
+ 7 x=
y ‘ 4 xy=
+ 1 xy) ù (∝1 x=
y) 5
8 x8 y=
∝1 x= y +
7 x= y
∝1 x= y ‘
4 xy=
∝1 x= y +
1 xy
∝1 x= y 5
∝8 xy
1 ∝
7
1 +
8 y
= x ∝
0 x
5
∝ 8 xy
1 +
8 y
= x ‘
0 x
‘ 7
1
(=; x8 ∝ 07 x= + 1; x + 7) ù (7 x=) 5
=; x8
7 x= ∝07 x=
7 x= +1; x7 x= +
77 x= 5
=; x8
7 x= ∝07 x=
7 x= +1; x7 x= +
77 x= 5
1 x ‘ 8 +:
x + 0
x =
094BIG] ∝ [fnfhjbh
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]gsuegl
0. Wl pfnblfebf gs ul` gxprgsbøl e`tgeàtbh` dfre`a` ` p`rtbr ag n` ulbøl ag afs f eàs v`rb`mngsy hflst`ltgs, qug sg vblhun`l ` tr`vâs ag n`s fpgr`hbflgs `rbteâtbh`s ag sue`, rgst` y euntbpnbh`-hbøl. Gl ftr`s p`n`mr`s, ul pfnblfebf gs n` ulbøl ag afs f eàs eflfebfs.
0.0 Nfs pfnblfebfs sg pugagl hn`sbdbh`r sgiól su lóegrf ag târeblfs, sgiól nfs târeblfs qughfeprglag y sgiól gn ir`af ag sus v`rb`mngs.
=. [`r` n` `abhbøl y n` sustr`hhbøl gltrg afs f eàs pfnblfebfs, sg sue`l f rgst`l nfs hfgdbhbgltgsag nfs târeblfs sgegc`ltgs, gs aghbr, ag nfs târeblfs qug tgli`l n` ebse` v`rb`mng hfl gn ebsefir`af.
8. [`r` euntbpnbh`r pfnblfebfs sg utbnbz` n` prfpbga`a abstrbmutbv`? gn prfauhtf ag pfnblfebfs pugagprgsglt`r abdgrgltgs h`sfs3
8.0 Euntbpnbh`hbøl ag ul lóegrf pfr ul pfnblfebf3 sg euntbpnbh` gn lóegrf pfr h`a` ulf agnfs târeblfs ag n` gxprgsbøl pfnblfeb`n.
8.= Euntbpnbh`hbøl ag ul eflfebf pfr ul pfnblfebf3 sg euntbpnbh` gn eflfebf pfr tfafsy h`a` ulf ag nfs târeblfs qug dfre`l gn pfnblfebf, gs aghbr, hfgdbhbgltg pfr hfgdbhbgltg yv`rb`mng pfr v`rb`mng.
8.8 Euntbpnbh`hbøl gltrg pfnblfebfs3 sg euntbpnbh` h`a` târeblf agn prbegr pfnblfebf pfr tf-afs y h`a` ulf ag nfs târeblfs agn sgiulaf pfnblfebf.
1. [`r` abvbabr ul pfnblfebf pfr ul eflfebf, sg abvbag h`a` ulf ag nfs târeblfs agn pfnblfebfgltrg gn eflfebf.
Ng `lbe`efs ` fmsgrv`r nfs vbagfs sbiubgltgs sfmrg n`s fpgr`hbflgs hfl pfnblfebfs3
• Euntbpnbh`hbøl ag pfnblfebfs3 iff.in/tB=RoS
• Abvbsbøl ag pfnblfebfs3 iff.in/bvdOEo
[ugag glhfltr`r eàs gcgrhbhbfs p`r` pr`htbh`r gl nfs sbtbfs wgm sbiubgltgs3
• Xue`s y rgst`s3 iff.in/S@wD07
• Euntbpnbh`hbøl3 iff.in/t19YOQ
099E`tgeàtbh` ∝ Xge`l` 00
Blvgstbiug gl n` rga...
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@htbvba`a 0. Ageugstrg nf prglabaf
@. Fmsgrvg g blabqug sb nfs pfnblfebfs sfl hfepngtfs f blhfepngtfs. Iuågsg pfr gn gcgepnf.
;) ‘ 1 + 0=`8 + 0:`= ‘ 9 + 09
0) = x8 ‘ x= + 1 x ‘ 02
=) m8 + 4m= ‘ 1
8) ‘00r 7 + 02r 1 + =9r = ‘ =:
1) 0; y7 ‘ 1 y1 ‘ : y8 + 02 y= + y ‘ =0
M. Fmsgrvg g blabqug gn ir`af ag nfs pfnblfebfs sbiubgltgs. Iuågsg pfr gn gcgepnf.
;) 8 x8 + =0 x1 ‘ 2 x + 09
0) ‘09`7 + 9 8 ‘ 0:`= + =:
=) =7e= ‘ 07e8 + 0=e= + 17
8) ‘01 x= + 0: x + 01 x ‘ ==
1) 7 y + 8 x ‘ 4 y + =7
@htbvba`a =. [r`htbqug nf `prglabaf
@. ]gsugnv` n`s sue`s y n`s rgst`s ag pfnblfebfs sbiubgltgs. [ugag j`hgrnf gl su hu`agrlf f gl jfc`s
`p`rtg.
0) (‘4m x= + : x1 ‘ 0= ‘ = x8 +0
=mx) + (‘0; x1 ‘ 7 + 4mx + 2 x8 + m x=)
=) (:`1 ‘ 01`4 ‘ 04`8 ‘ =` + 07) ∝ (∝8`4 ‘ =0`= + 01` ∝ =8)
8) (∝== x8 + =8 x= ∝ 07 x7 + =0 x1 + =7 x ∝ 09) ∝ (9 x8 ∝ =0 x1 + 09 x= ∝ 0: x7)
1) (=: x= z ∝ 10 x1 + 01 xz ∝ 2) ∝ (∝09 x1 ∝ 8 x= z ‘ : ∝ =9 xz )
7) (17 x9 + 09 x7 ∝ =7 x + 40 x8 ∝ 0=) + (=2 x8 ∝ 8: x7 + 7 x9 ∝ 0: x8)
4) (∝=:el= + =7l ‘ e=l + =:) ∝ (84l ∝ 10e=l + = ∝ :el= + 9e)
9) (7:el8 + 9e=l ‘ 10el ‘ 8e) ∝ (8el8 + =7el + 1 ‘ 4e=l)
:) (8= y8 ∝ 0= y1 + 81 y ‘ == + 8 y=) + (=1 y1 ∝ 1 y= ‘ 81 ∝ 09 y8 + 1 y)
2) (∝:=w= z = + 89w8 + 4w8 z ‘ 7wz =) + (4 z + =8w8 z ∝ 94 ‘ 8w= z = ‘ =w8)
0;) (∝08r = s + =1 s= ‘ 7rs8 + 1) ∝ (∝48r = s + 1=r + :0 s= ‘ :rs8)
@utfhfltrfn
hfepngtf
hu`rtf ir`af
09:BIG] ∝ [fnfhjbh
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M. ]gsugnv` n`s euntbpnbh`hbflgs pfnblfeb`ngs sbiubgltgs. [ugag j`hgrnf gl su hu`agrlf f gl jfc`s `p`rtg
0) (∝0=)(7 xy= + : x= y ‘ 0= x + : y + 0=)
=) (4`m8)(7`=m + 4`m8 ‘ 2 + 1`=)
8) (=o =v + 8ov= ∝ o )(00 + =ov ‘ v8)
1) (∝: y= z =)(9 yz 8 + : y8 z = ‘ 0; y)
7) (:)(‘1`m8 ‘ 4` + 2`=m)
4) (8 x ∝ 8 y=)( ‘7 y8 + 4 x= y ‘ :)
9) (∝2 xy= + 4 x= y8 ‘ 7 x8 y)(2 + 1 x= y ‘ x8 y=)
:) (0;e8l=)(7el= + 0;e=l8 ‘ 2)
2) (∝2)(‘ u8 + 4u= ‘ u)
0;) (0;`=m1 ‘ 7`8m + 4m)( = + =`m=)
00) (∝:m=h1)(0= + 1h8 ‘ m1)
0=) (∝02e= + 2e ∝ 9)( 2e8 ‘ 00e= + :e ∝ 00)
H. ]gsugnv` n`s sbiubgltgs abvbsbflgs ag pfnblfebfs gltrg eflfebfs. [ugag j`hgrnf gl su hu`agrlf fgl jfc`s `p`rtg.
0) ( x= ‘ xz =) ù ( x)
=) (4`8m8 ‘ 2`= x1) ù (∝8`=)
8) (∝7 x8 + 0; xy= + 1`=m8) ù (∝= x)
1) (04 x7 ‘ : x1 + 9 x8 ∝ 0=) ù (1 x1)
7) (∝12`= + 8`=m ‘ =0`m=) ù (9`m=)
4) (0;e8l1 + e1l8 ‘ 4e=l=) ù (∝=el)
9) ( m8 ‘ 00`1 + 1`8 ∝ 7) ù ( 8)
:) (∝41 + 04 x1 ‘ = + 1 x8) ù (∝: x)
2) (84wz 8 + =w= z = ∝ 2w) ù (8w= z 1)
0;) (0:o 8e1 + 8e8 ‘ 2o 1) ù (2o 1e8)
00) (∝: s + 4r = s=) ù (∝:rs=)
0=) (0;;m=h + 7;mh= ‘ 7h1) ù (0;m8h8)
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@. ]gsugnv` n`s sbiubgltgs sue`s y rgst`s ag eflfebfs. Gshrbm` su rgspugst` gl n` nålg`hfrrgspflabgltg.
0) 1 x
=
+ 9 x
=
5
=) 0; y1 + 0; y1 5
8) = x8 ‘ 2 x8 5
1) ‘0; x= ‘ 00 x= 5
7) 02 x + 8 x 5
4) 0=t 1 ‘ 0=t 1 5
9) ‘00 y8 + 07 y8 5
:) 7 y= + 8 y= 5
2) 2w8 ‘ 8w8 5
0;) ‘1r + 1r 5
00) ‘==w
4
‘ 09w
4
5
0=) 00e8 + =e8 5
08) 8 d 1 + 7 d 1 5
01) ‘ 1j7 + 8j7 5
07) i + 7 i 5
04) ‘ c + = c 5
09) 8 y8 + 1 y8 5
0:) 0= x1 + 9 x1 5
02) ‘8t 8 ‘ 2t 8 5
=;) w ‘ =w 5
M. ]gsugnv` n`s sbiubgltgs euntbpnbh`hbflgs ag eflfebfs. Gshrbm` su rgspugst` gl n` nålg`hfrrgspflabgltg.
0) 8 x1 • 8 x7 5
=) 7 xy • (‘8 x8) 5
8) (‘= y1) • 8 y7 5
1) (‘8 y8) • (‘1 xy=) 5
7) 1 xy • ‘9 x1 5
4) 7w8
• 4 y8
5
9) 1`8 • (‘7`1m) 5
:) (‘0;m=) • 0;m= 5
2) (‘8u8) • 2u= 5
0;) (‘1t 1) • 9t 8 y 5
00) 7y • 4 y 5
0=) 0= xy • 1 x8 5
08) 00r = • (‘9rt ) 5
01) 2w= • 2w7 5
07) (‘00 d 8) • 7 d = c 5
04) 1u1
• 1u=
5
09) (‘0= xy8 z ) • 0; x8 y= 5
0:) t = • =t 8 5
02) 9r 1 • 9r = p8 5
=;) = xyz 8 • (‘8 x8 y=) 5
0:;BIG] ∝ [fnfhjbh
@ibnba`a ag hànhunf eglt`n
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E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.
nfir`af glprfhgsf
lfnfir`af
A g s p u â s a g
g s t u a b ` r . . .
Hflhgptu`nbzf g bagltbhf ul pfnblfebf.
Hn`sbhf pfnblfebfs ag `hugraf ` su lóegrf ag târeblfs.
Bagltbhf y rgsugnvf fpgr`hbflgs gltrg pfnblfebfs.
]gsugnvf fpgr`hbflgs ag eflfebfs hfl `ibnba`a.
@jfr` lfs gltrgtglargefs `n ebsef tbgepf qug `prglagefs. Qr`m`c`rgefshfl `niulfs `hgrtbcfs ag `rbteâtbh` hfl p`nbnnfs. [`r` gst` `htbvba`a lghgsbt`efs08 p`nbnnfs ag abgltgs, f `niól e`tgrb`n sbebn`r qug tgli` n` ebse` nflibtua.
Gstg tbpf ag `hgrtbcfs tbglg ulfs =;; `ðfs ag jbstfrb` y sfl bepfrt`ltgs pfrqug
eftbv`l `n gstuab`ltg ` j`hgr usf ag sus `prglabz`cgs ag `rbteâtbh` màsbh` p`r`nngi`r ` ul` sfnuhbøl s`tbsd`htfrb` agn gcgrhbhbf pn`ltg`af, `n ebsef tbgepf qugags`rrfnn` n` nøibh` y pr`htbh` sus hflfhbebgltfs ag n` luegr`hbøl rfe`l`.
]gsugnv` nf qug sg ng pbag3
0) Dfreg hfl p`nbnnfs n` dbiur` agn rghu`arf. Nugif, h`embg ag nui`r 0 ag nfs 0=p`nbnnfs y j`i` qug qugag dfre`a` ul` biu`na`a vgra`agr`, pfrqug 4 ‘ 1 lfgs biu`n 2. Gl gn gsp`hbf gl mn`lhf abmucg høef quga`rå` n` dbiur` hfrrght`.
Tn ‘ nT 5 nZ
=) Dfreg hfl p`nbnnfs n` dbiur` agn rghu`arf. Nugif, h`embg ag nui`r 0 ag nfs 08p`nbnnfs y j`i` qug qugag ul` biu`na`a vgra`agr`, pfrqug = + : lf gs biu`n` 7. Gl gn gsp`hbf gl mn`lhf abmucg høef quga`rå` n` dbiur` hfrrght`.
T 5 nn + Tnnn
0:=BIG] ∝ [fnfhjbh
]gvbsg su `prglabz`cg
Ags`rrfnng lugv`s j`mbnba`ags
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0=0
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
[rfauhtfs lft`mngs
¶Puâ glhfltr`rà gst` sge`l`6
@n-Cw`rbzeb, gn hrg`afr agn ànigmr`
Gn hu`ar`af y gn humf ag n` sue` y ag n` abdgrglhb` ag ul mblfebf
Qâreblfs agn hu`ar`af y agn humf ag ul mblfebf
Àrg` `nigmr`bh` ag hu`ar`afs
Gst` sge`l` nfir`rà3
Hflfhgr n` jbstfrb` ag n-Cw`rbzeb y su `pfrtg ` n` e`tgeàtbh`.
]gsfnvgr gn hu`ar`af y gn humf ag n` sue` f abdgrglhb` ag ul mblfebf.
[r`htbh`r gn hànhunf eglt`n n hfepngt`r prfauhtfs lft`mngs.
H`nhun`r gl dfre` `nigmr`bh` gn àrg` ag iur`s igfeâtrbh`s.
0:8E`tgeàtbh` ∝ Xge`l` 0=
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¡[`r` hfeglz`r!@n-Cw`rbzeb, gn hrg`afr agn ànigmr`
¡@ tr`m`c`r!
¶X`mg gn lfemrg ag ftr`s pgrsfl`s qug jbhbgrfl `pfrtgs bepfrt`ltgs ` n` e`tgeàtbh`6 Gshråm`nfs gln`s nålg`s.
Gs euy prfm`mng qug ustga rghflfzh` f j`y` vbstf gn rgtr`tf ag gstg e`tgeà-
tbhf, gs n` rgprgsglt`hbøl be`ibl`rb` ag `n-Cw`rbzeb, ` qubgl sg ng hflsbagr` gn
dula`afr agn ànigmr`.
[fhf sg s`mg ag n` vba` ag gstg `strølfef, igøir`df y e`tgeàtbhf eusuneàl,
`dbhbfl`af ` n`s hbglhb`s gx`ht`s? qug vbvbø `prfxbe`a`egltg gltrg nfs `ðfs
9:; ‘ :7; y sg hrgg qug rgsbabø gl Br`o.
Gl su Nbmrf ag n` rgauhhbøl, `n-Cw`rbzeb gshrbmbø n`s prbegr`s rgin`s agn hànhunf
`nigmr`bhf, n` tr`lspfsbhbøl ag târeblfs gl ul` ghu`hbøl, prgvbf h`embf ag
sbilf, n` `lun`hbøl ag târeblfs baâltbhfs gl `emfs ebgemrfs, n`s ghu`hbflgs
ag sgiulaf ir`af y ftrfs tge`s rgn`hbfl`afs hfl gn ànigmr`.
Hfl su Obt`m `n-y`mrw`-n-euq`m`n` f Nbmrf agn ànigmr`, `n-Cw`rbzeb blbhbø n` nbtg-r`tur` e`tgeàtbh` gl n` qug ageugstr` n` sfnuhbøl ag ghu`hbflgs ag prbegr y
sgiulaf ir`af, qug lf abdbgrg ag nfs prfhgabebgltfs gepng`afs jfy aå`.
Gl n` ulbvgrsba`a Fxdfra, Blin`tgrr`, j`y ul` hfpb` qug rgprfauhg gn frbibl`n agn
ànigmr` ag `n-Cw`rbzeb qug prfpfrhbfl` n` bldfre`hbøl eàs `ltbiu` `hgrh` agn
usf qug j`hå`l nfs àr`mgs agn sbstge` ag luegr`hbøl ag m`sg abgz, jfy us`af
gl tfaf gn eulaf.
(9:; ‘ :7;)
0:1BIG] ∝ [fnfhjbh
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0. [rfauhtfs lft`mngsNfs prfauhtfs lft`mngs sfl h`sfs gspghb`ngs ag n` pftglhb`hbøl y ag n` euntb-
pnbh`hbøl ag pfnblfebfs qug huepngl hfl rgin`s c`s, y sg pugagl rgsfnvgr pfrsbepng blspghhbøl, gs aghbr, rgsfnvgr eglt`negltg n`s fpgr`hbflgs, sbl lghgsba`aag gshrbmbr tfaf gn prfhgabebgltf. Gst` sge`l` prgsglt`efs gcgepnfs gl nfsqug rg`nbz`efs n`s fpgr`hbflgs qug blabh` h`a` døreun`. Nugif, ustga, hfl n`pràhtbh`, agmgrà gshrbmbr sfnf gn rgsunt`af.
0.0 Hu`ar`af ag n` sue` ag ul mblfebf (` + m)=
Hfl nfs lóegrfs rg`ngs `prglabefs qug gngv`r ul` h`ltba`a `n hu`ar`af gs nfebsef qug euntbpnbh`r n` m`sg pfr så ebse` afs vghgs. N` ebse` rgin` `pnbh` sbags`rrfnn`efs gn hu`ar`af ag ul mblfebf.
( + m)= 5 ( + m)( + m)
5 `( + m) + m( + m)
5 `= + `m + `m + m=
5 `= + =`m + m=
Gl nui`r ag rg`nbz`r tfa`s n`s fpgr`hbflgs qug efstr`efs, ul h`eblf eàs hfrtfp`r` fmtglgr gn ebsef rgsunt`af gs ` tr`vâs ag n` døreun` iglgr`n agn hu`ar`afag ul mblfebf qug blabh`3
( + m)= 5 `= + =`m + m=
Xg ngg3 Gn hu`ar`af ag n` sue` ag ul mblfebf gs biu`n `n hu`ar`af agn prbegr
târeblf, eàs gn afmng prfauhtf agn prbegrf pfr gn sgiulaf, eàs gn hu`ar`af agn
sgiulaf.
Gcgepnf3
( x + 7)= 5 ( x)= + =( x • 7) + (7)= 5 x = + 0; x + =7
( y + :)= 5 ( y)= + =( y • :) + (:)= 5 y= + 04 y + 41
(=`8 + 1m=)= 5 (=`8)= + =(=`8 • 1m=) + (1m=)= 5 1`4 + 04`8m= + 04m1
Gcgrhbhbf 0Ags`rrfnng gn hu`ar`af ag n` sue` ag ul mblfebf. Iuågsg pfr gn gcgepnf.
;) ( x= + m8)= 5
0) ( + 8)= 5
=) (=h= + 1a =)= 5
( x=)= + =( x= • m8) + (m8)= 5 x1 + = x=m8 + m4
0:7E`tgeàtbh` ∝ Xge`l` 0=
Gn eulaf ag n` e`tgeàtbh`
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0.= Hu`ar`af ag n` abdgrglhb` ag ulmblfebf (` ‘ m)=
Gn h`sf ag n` rgst` ag ul mblfebf `n hu`ar`af gs euy p`rghbaf `n `ltgrbfr, hfln` abdgrglhb` qug nfs sbilfs sg `ntgrl`l gltrg pfsbtbvf y lgi`tbvf. Tg`efs n`euntbpnbh`hbøl y nugif su døreun`.
( ‘ m)= 5 ( ‘ m)( ‘ m)
5 `( ‘ m) ‘ m( ‘ m)
5 `= ‘ `m ‘ `m + m=
5 `= ‘ =`m + m=
N` døreun` iglgr`n agn hu`ar`af ag n` abdgrglhb` ag ul mblfebf gs3
( ‘ m)= 5 `= ‘ =`m + m=
Xg ngg3 Gn hu`ar`af ag n` abdgrglhb` ag ul mblfebf gs biu`n n hu`ar`af agn prbegr
târeblf, eglfs gn afmng prfauhtf agn prbegrf pfr gn sgiulaf, eàs gn hu`ar`afagn sgiulaf.
Gcgepnf3 ( ‘ 8)= 5 ( )= ‘ =( • 8) + (8)= 5 `= ‘ 4` + 2
(1 x ‘ 7)= 5 (1 x)= ‘ =(1 x • 7) + (7)= 5 04 x = ‘ 1; x + =7
(7 y1 ‘ = z 8)= 5 (7 y1)= ‘ =(7 y1 • = z 8) + (= z 8)= 5 =7 y: ‘ =; y1 z 8 + 1 z 4
0.8 [rfauhtf ag n` sue` pfr n`abdgrglhb` (` + m)(` ‘ m)
Gl gstg prfauhtf agmg fmsgrv`r qug nfs târeblfs ag nfs mblfebfs tbglgl n`sebse`s nbtgr`ngs, pgrf gstàl ulba`s pfr sbilfs abdgrgltgs.
( + m)( ‘ m) 5 `( ‘ m) + m( ‘ m)
5 `= ‘ `m + `m ‘ m=
5 `= ‘ ; ‘ m=
N` døreun` iglgr`n agn prfauhtf ag n` sue` pfr n` abdgrglhb` ag afs mblfebfs gs3
( + m)( ‘ m) 5 `= ‘ m=
Xg ngg3 Gn prfauhtf ag n` sue` pfr n` abdgrglhb` ag afs mblfebfs gs biu`n ` n`
abdgrglhb` ag sus hu`ar`afs.
Gcgepnf3
( + 7)( ‘ 7) 5 ( )= ‘ (7)= 5 `= ‘ =7
(4 x + :)(4 x ‘ :) 5 (4 x)= ‘ (:)= 5 84 x = ‘ 41
(8 x + 1 y)(8 x ‘ 1 y) 5 (8 x)= ‘ (1 y)= 5 2 x = ‘ 04 y=
@tglhbøl3 `n ags`rrfnn`rn` abdgrglhb` ag ul
mblfebf ` hu`nqubgrpftglhb`, nfs sbilfssbgeprg v`l `ntgrlfs+, ‘, +, ‘, gth.
0:4BIG] ∝ [fnfhjbh
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0.1 [rfauhtf ag afs mblfebfs hfl ul târeblfhfeól y sbilfs biu`ngs
Gl gstg prfauhtf sg pugagl prgsglt`r afs h`sfs3 qug nfs sbilfs sg`l pfsbtbvfsf lgi`tbvfs. [rgstg `tglhbøl ` h`a` ulf.
`. (` + m)(` + h)
[`r` ags`rrfnn`r gn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól y sbilfpfsbtbvf, sg prfhgag `så3
( + m)( + h) 5 `( + h) + m( + h)
5 `= + `h + `m + mh
5 `= + `(m + h) + mh
N` døreun` iglgr`n agn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól ysbilfs pfsbtbvfs gs3
( + m)( + h) 5 `= + `(m + h) + mh
Xg ngg3 Gn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól y sbilfs biu`ngs gs
biu`n `n hu`ar`af agn târeblf hfeól, eàs gn târeblf hfeól pfr n` sue` ag nfs
târeblfs lf hfeulgs, eàs gn prfauhtf ag nfs târeblfs lf hfeulgs.
Gcgepnf3 ( + 1)(` + =) 5 ( )= + `(1 + =) + (1 • =) 5 `= + 4` + :
(1 y= + 4)(1 y= + 8) 5 (1 y=)= + 1 y=(4 + 8) + (4 • 8) 5 04 y1 + 84 y= + 0:
m. (` ‘ m)(` ‘ h)
[`r` ags`rrfnn`r gn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól y sbilflgi`tbvf, sg prfhgag `så3
( ‘ m)( ‘ h) 5 `( ‘ h) ‘ m( ‘ h)
5 `= ‘ `h ‘ `m + mh
5 `= + `(‘ m ‘ h) + mh
N` døreun` iglgr`n agn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól ysbilfs lgi`tbvfs gs3
( ‘ m)( ‘ h) 5 `=
+ `(‘ m ‘ h) + mh
Xg ngg3 Gn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól y sbilfs biu`ngs gs
biu`n `n hu`ar`af agn târeblf hfeól, eàs gn târeblf hfeól pfr n` sue` ag nfs
târeblfs lf hfeulgs, eàs gn prfauhtf ag nfs târeblfs lf hfeulgs.
Gcgepnf3
( ‘ 2)( ‘ 8) 5 ( )= + `(‘2 ‘ 8) + (‘2)(‘8) 5 `= ‘ 0=` + =9
(8m ‘ 1)(8m ‘ 4) 5 (8m)= + 8m(‘1 ‘ 4) + (‘1)(‘4) 5 2m= ‘ 8;m + =1
0:9E`tgeàtbh` ∝ Xge`l` 0=
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Gcgrhbhbf =Ags`rrfnng nfs prfauhtfs lft`mngs blabh`afs gl h`a` `p`rt`af. J`y ul gcgepnf p`r` h`a` h`sf.
@. Hu`ar`af ag n` abdgrglhb` ag ul mblfebf.
;) ( x8 ‘ 4)= 5
0) ( x ‘ y)= 5
=) (w ‘ z )= 5
8) (8 x ‘ 1)= 5
1) (4 y ‘ 7)= 5
7) ( 0
=w ‘ = z )
=
5
4) ( x ‘ 0
7 z )
=
5
9) (8 x1 ‘ 1 y8)= 5
:) (= x= ‘ 8 y=)= 5
2) (: x7 ‘ 1 y8)= 5
0;) (1 x1 ‘ 7 y2)= 5
00) (8e=l8 ‘ : pq=)= 5
0=) ( 0
8w ‘ 2 z )
=
5
08) ( 1
8 x= ‘
7
= y8)
=
5
M. [rfauhtf ag n` sue` pfr n` abdgrglhb`.
;) (= x + 8 y)(= x ‘ 8 y) 5
0) ( p + q)( p ‘ q) 5
=) ( y + 1)( y ‘ 1) 5
8) (w + 4)(w ‘ 4) 5
1) (8 x + y)(8 x ‘ y) 5
7) ( =8
` + m)( =8
` ‘ m) 5
4) ( :
7e= + 8)( :
7e= ‘ 8) 5
9) (7 z + :)(7 z ‘ :) 5
:) (1 y= + z )(1 y= ‘ z ) 5
2) (= x= + 8)(= x= ‘ 8) 5
0;) (8 x + 1 y)(8 x ‘ 1 y) 5
00) (7 x + 0; y)(7 x ‘ 0; y) 5
0=) ( 0= x + 7 y)( 0
= x ‘ 7 y) 5
08) ( 7
1e= + 8l)( 7
1e= ‘ 8l) 5
H. [rfauhtf ag afs mblfebfs hfl ul târeblf hfeól.
;) ( + 7)( + 8) 5
0) ( x + :)( x + 2) 5
=) ( y + 8)( y + 1) 5
8) ( z + 9)( z + =) 5
1) ( + 1)( + 1) 5
7) (m + 4)(m + 4) 5
4) (o = + :)(o = + =) 5
9) (8w8 + 4)(8w8 + 7) 5
:) ( ‘ 4)(` ‘ 1) 5
2) (=h ‘ 8)(=h ‘ =) 5
0;) (7 x ‘ :)(7 x ‘ 1) 5
00) (8 y ‘ :)(8 y ‘ 9) 5
0=) (= x ‘ 7)(= x ‘ 8) 5
08) (0; x ‘ 8)(0; x ‘ 8) 5
01) (2 z = ‘ 4)(2 z = ‘ 0;) 5
07) (8 x= y ‘ =)(8 x= y ‘ 1) 5
x4 ‘ 0= x8 + 84
1 x= ‘ 2 y=
`= + :` + 07
0::BIG] ∝ [fnfhjbh
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0.7 Humf ag n` sue` ag ul mblfebf (` + m)8
[`r` gngv`r ul mblfebf `n humf, nf agshfepflgefs gl afs d`htfrgs y ags`rrf-nn`efs n`s fpgr`hbflgs hfrrgspflabgltgs.
( + m)8 5 ( + m)=( + m)
5 (=
+ =`m + m=
)( + m)5 `=( + m) + =`m( + m) + m=( + m)
5 `8 + `=m + =`=m + =`m= + `m= + m8
5 `8 + 8`=m + 8`m= + m8
Wl h`eblf eàs hfrtf p`r` fmtglgr gn ebsef rgsunt`af gs ` tr`vâs ag n` døreun`iglgr`n agn humf ag ul mblfebf.
( + m)8 5 `8 + 8`=m + 8`m= + m8
Xg ngg3 Gn humf ag n` sue` ag ul mblfebf gs biu`n n humf agn prbegr târeblf, eàs
gn trbpng agn prbegrf `n hu`ar`af pfr gn sgiulaf, eàs gn trbpng agn prbegrf pfr gnsgiulaf `n hu`ar`af, eàs gn humf agn sgiulaf.
Gcgepnf3
(e + l)8 5 (e)8 + 8(e)=(l) + 8(e)(l)= + (l)8
5 e8 + 8(e=)l + 8e(l=) + l8
5 e8 + 8e=l + 8el= + l8
(1 x + y)8 5 (1 x)8 + 8(1 x)=( y) + 8(1 x)( y)= + ( y)8
5 41 x8 + 8(04 x=)( y) + 8(1 x)( y=) + y8
5 41 x 8
+ 1: x =
y + 0= xy=
+ y8
(8 p= + =t 8)8 5 (8 p=)8 + 8(8 p=)=(=t 8) + 8(8 p=)(=t 8)= + (=t 8)8
5 =9 p4 + 8(2 p1)(=t 8) + 8(8 p=)(1t 4) + :t 2
5 =9 p4 + 71 p1t 8 + 84 p=t 4 + :t 2
Gcgrhbhbf 8Gshrbm` nfs târeblfs qug hfepngt`l gn ags`rrfnnf agn mblfebf `n humf.
0) (1w + z )8 5 ( )8 + 8(1w)=( ) + 8( )( z )= + ( )8
5 + 8( )( z ) + 8(1w)( ) +
5 41w8 + 1:w= z + 0=wz = + z 8
=) (8 x= + = y=)8 5 ( )8 + 8( )=(= y=) + 8(8 x=)( )= + ( )8
5 =9 x4 + 8(2 x1)( ) + 8( )(1 y1) + : y4
5 + 71 x 1 y= + + : y4
0:2E`tgeàtbh` ∝ Xge`l` 0=
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0.4 Humf ag n` abdgrglhb` ag ul mblfebf (` ‘ m)8
Gl gstg h`sf, n` abdgrglhb` hfl n` sue` gs qug nfs sbilfs agn rgsunt`af sg ntgrl`l+, ‘, +, ‘ gl h`a` ulf ag nfs târeblfs.
( ‘ m)8 5 ( ‘ m)=( ‘ m)
5 (=
‘ =`m + m=
)( ‘ m)5 `=( ‘ m) ‘ =`m( ‘ m) + m=( ‘ m)
5 `8 ‘ `=m ‘ =`=m + =`m= + `m= ‘ m8
5 `8 ‘ 8`=m + 8`m= ‘ m8
N` døreun` iglgr`n ag n` abdgrglhb` ag ul mblfebf `n humf gs3
( ‘ m)8 5 `8 ‘ 8`=m + 8`m= ‘ m8
Xg ngg3 Gn humf ag n` abdgrglhb` ag ul mblfebf gs biu`n `n humf agn prbegr târeblf,
eglfs gn trbpng agn prbegrf `n hu`ar`af pfr gn sgiulaf, eàs gn trbpng agn prbegrf
pfr gn sgiulaf `n hu`ar`af, eglfs gn humf agn sgiulaf.
Gcgepnf3
(h ‘ a )8 5 (h)8 ‘ 8(h)=(a ) + 8(h)(a )= ‘ (a )8
5 h8 ‘ 8(h=)a + 8h(a =) ‘ a 8
5 h8 ‘ 8h=a + 8ha = ‘ a 8
( x ‘ = y)8 5 ( x)8 ‘ 8( x)=(= y) + 8( x)(= y)= ‘ (= y)8
5 x8 ‘ 8( x=)(= y) + 8( x)(1 y=) ‘ : y8
5 x 8 ‘ 4 x = y + 0= xy= ‘ : y8
(1w ‘ 7)8 5 (1w)8 ‘ 8(1w)=(7) + 8(1w)(7)= + (7)8
5 41w8 ‘ 8(04w=)(7) + 8(1w)(=7) ‘ 0=7
5 41w8 ‘ =1;w= + 8;;w ‘ 0=7
Gcgrhbhbf 1Gshrbm` nfs târeblfs qug hfepngt`l gn ags`rrfnnf agn mblfebf `n humf.
0) (=h ‘ a )8 5 (=h)8 ‘ 8( )=( ) + 8(=h)( )= ‘ (a )8
5 ‘ 8(1h=)( ) + 8( )(a =) ‘
5 :h8 ‘ 0=h=a + 4ha = ‘ a 8
=) (1 y8 ‘ = z )8 5 ( )8 ‘ 8( )=(= z ) + 8( )(= z )= ‘ ( )8
5 41 y2 ‘ 8( )( ) + 8( )(1 z =) ‘
5 41 y2 ‘ 24 + 1: ‘ : z 8
02;BIG] ∝ [fnfhjbh
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]gsuegl
Nfs prfauhtfs lft`mngs sfl h`sfs gspghb`ngs ag n` pftglhb`hbøl y euntbpnbh`hbøl ag pfnblfebfs qughuepngl hfl rgin`s c`s y qug sg pugagl rgsfnvgr pfr sbepng blspghhbøl, gs aghbr, rgsfnvgr eglt`n-egltg n`s fpgr`hbflgs, sbl lghgsba`a ag gshrbmbr tfaf gn prfhgabebgltf.
Nfs prfauhtfs lft`mngs qug gstuab`efs gst` sge`l` sg rgsuegl gl n` t`mn` sbiubgltg3
[`r` rgdfrz`r sus hflfhbebgltfs sfmrg prfauhtfs lft`mngs, vbsbtg gst` abrghhbøl3 jttp3//iff.in/IbCH`9
Lfemrg [rfauhtf lft`mng Døreun` iglgr`n
Hu`ar`af ag n` sue` ag ul mblfebf ( + m)= 5 `= + =`m + m=
Hu`ar`af ag n` abdgrglhb` ag ul mblfebf ( ‘ m)= 5 `= ‘ =`m + m=
[rfauhtf ag n` sue` pfr n` abdgrglhb` ( + m)( ‘ m) 5 `= ‘ m=
[rfauhtf ag afs mblfebfs hfl ul târeblf
hfeól y sbilfs biu`ngs
( + m)( + h) 5
( ‘ m)( ‘ h) 5
`= + `(m + h) + mh
`= + `(‘ m ‘ h) + mh
Humf ag n` sue` ag ul mblfebf ( + m)8 5 `8 + 8`=m + 8`m= + m8
Humf ag n` abdgrglhb` ag ul mblfebf ( ‘ m)8 5 `8 ‘ 8`=m + 8`m= ‘ m8
020E`tgeàtbh` ∝ Xge`l` 0=
Blvgstbiug gl n` rga...
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@htbvba`a 0. Ageugstrg nf prglabaf
E`rqug ul` "Z" gl gn hu`arf ag n` fphbøl qug rgspflag hfrrght`egltg h`a` prgiult`.
0) ¶Puâ gxprgsbøl gs gqubv`ngltg ( + m)=6
=) ¶Huàn gs gn prbegr târeblf agn rgsunt`af ag (8 x + =)(8 x ‘ =)6
8) ¶Huàn gs gn prbegr târeblf agn rgsunt`af ag (1 x + =)=6
1) ¶Huàn gs ul gcgepnf agn prfauhtf ag n` sue` pfr n` abdgrglhb`6
7) ¶Huàn gs gn sgiulaf târeblf `n fpgr`r (= y + 8)(= y + 7)6
4) ¶Huàn gs gn rgsunt`af ag ( x + y)( x ‘ y)6
9) ¶Huàltfs târeblfs rgsunt`l ag gngv`r ( x + y)
8
6
:) ¶Huàn gs gn óntbef târeblf `n gngv`r (= x= + 8 y8)86
@utfhfltrfn
( + m)( ‘ m) ( + m)( + m)
( ‘ m)( ‘ m)
( ‘ m)( + m)
4 x
2 x
8 x=
2 x=
1 : x
04 x
04 x=
( + 8)(` + 1)
( + 7)( ‘ 7)
( ‘ 4)( ‘ =)
( ‘ 8)( ‘ 8)
: y 07
1 y=
04 y
x + y
x ‘ y
x= + y=
x= ‘ y=
= 8
1
4
: x4
: x7
=9 y2
=9 y4
02=BIG] ∝ [fnfhjbh
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@htbvba`a =. [r`htbqug nf prglabaf
@. Fpgrg gn prfauhtf ag n` sue` pfr n` abdgrglhb` pfr sbepng blspghhbøl y gshrbm` gn rgsunt`af gl n`nålg`. Qbglg ul gcgepnf.
;) (7` + 1)(7` ‘ 1) 5
0) (1m + 4)(1m ‘ 4) 5
=) (=h + 2)(=h ‘ 2) 5
8) (4 x + 1)(4 x ‘ 1) 5
1) ( z + 0;)( z ‘ 0;) 5
7) (7 y ‘ :)(7 y + :) 5
4) (9w ‘ 2)(9w + 2) 5
9) (0; x + =)(0; x ‘ =) 5
:) ( 0
=
x ‘ 1)( 0
=
x + 1) 5
2) ( 7
8 y + 2)( 7
8 y ‘ 2) 5
0;) ( 8
7` + 7)( 8
7` ‘ 7) 5
00) ( :
8a +
4
9 )( :
8a ‘
4
9 ) 5
0=) (8m + = )(8m ‘ = ) 5
08) (=`= + 4m)(=`= ‘ 4m) 5
01) (7j8 ‘ 8o )(7j8 + 8o ) 5
07) (4e ‘ 8l)(4e + 8l) 5
M. Fpgrg gn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól y sbilfs biu`ngs pfr sbepng blspghhbøl.
;) ( + 7)( + :) 5
0) ( x ‘ 4)( x ‘ 9) 5
=) (h ‘ :)(h ‘ =) 5
8) ( z ‘ 4)( z ‘ :) 5
1) (a ‘ 1)(a ‘ 7) 5
7) (m ‘ 4)(m ‘ :) 5
4) (w + :)(w + 2) 5
9) ( y + 0;)( y + 7) 5
:) ( z ‘ 1)( z ‘ 4) 5
2) ( y ‘ 8)( y ‘ 7) 5
0;) ( x + =)( x + 9) 5
00) (w + :)(w + 9) 5
0=) (8 y ‘ 7)(8 y ‘ 8) 5
08) (7 p + 7)(7 p + 7) 5
H. Ags`rrfnng pfr sbepng blspghhbøl gn humf ag nfs mblfebfs sbiubgltgs.
;) ( + 1)8 5
0) (m + =)8 5
=) (h + 1)8 5
8) (a + 0)8 5
1) (= x + 8)8 5
7) (8m + 8)8 5
4) (7e + =)8 5
9) (= z ‘ 8)8 5
:) (1w ‘ 0)8 5
2) (1 x ‘ =)8 5
0;) (7 y= ‘ 0)8 5
00) (= z = ‘ 1)8 5
0=) ( =
8m ‘ 7)
8
5
08) ( 7
4o = ‘ 8)
8
5
=7`= ‘ 04
`= + 08` + 1;
`8 + 0=`= + 1:` + 41
028E`tgeàtbh` ∝ Xge`l` 0=
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[r`htbqug nfs prfauhtfs lft`mngs hfl n`s àrg`s ag h`a` iur`. Iuågsg pfr gn gcgepnf.
;)
x + 8
x ‘ 8
@ 5 ( x + 8)( x ‘ 8)@ 5 x= ‘ 2
0)
x + 1
x ‘ 1
=)
1 x + 7
1 x + 8
8)
x + 0;
x + 7
1)
x= + 4
x= ‘ 1
7)= x + :
= x ‘ :
027E`tgeàtbh` ∝ Xge`l` 0=
]`zfl`ebgltf nøibhf
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E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfir`af glprfhgsf lfnfir`af
A g s p u â s a g
g s t u a b ` r . . .
Hflfzhf n` jbstfrb` ag `n-Cw`rbzeb y su `pfrtg ` n` e`tgeàtbh`.
]gsugnvf gn hu`ar`af y gn humf ag n` sue` f abdgrglhb` ag ul mblfebf.
[r`htbhf gn hànhunf eglt`n `n hfepngt`r prfauhtfs lft`mngs.
H`nhunf gl dfre` `nigmr`bh` gn àrg` ag iur`s igfeâtrbh`s.
Nfs prfauhtfs lft`mngs y n`s iur`s igfeâtrbh`s
[fagefs rgn`hbfl`r nfs prfauhtfs lft`mngs hfl gn hànhunf ag àrg`s ag iur`sigfeâtrbh`s.
[fr gcgepnf, gn hu`ar`af ag n` agrghj` sg dfreø hfl n`s iur`s sbiubgltgs3
• Wl hu`ar`af ag n`af `
• Wl hu`ar`af ag n`af m
• Afs rghtàliunfs ag n`rif ` y `lhjf m
Xb qugrgefs glhfltr`r gn àrg` tft`n, sue`efs gn àrg` ag h`a` iur`.
`= + `m + `m + m= 5 `= + =`m + m=
[grf sb j`hgefs usf ag nfs prfauhtfs lft`mngs, pfagefs h`nhun`rn` ag dfre`abrght`, `så3 ( + m)= 5 `= + =`m + m= y nngi`efs `n ebsef rgsunt`af.
@jfr` ng hfrrgspflag ` ustga glhfltr`r gn àrg` agnrghtàliunf. ]g`nbhg nf qug sg pbag gl h`a` blhbsf.
`. Fmtgli` gn àrg` ag h`a` sghhbøl y sóegn`s p`r` fmtglgr gn àrg` tft`n.
+ + + 5m. @jfr` utbnbhg ul prfauhtf lft`mng p`r` glhfltr`r gn àrg` tft`n.
h. ¶Puâ prfauhtf lft`mng utbnbzø6
x x
x
x
m m
`
`
`
`
`
`
m m
m
m
`
`
`
`m m
m
m
024BIG] ∝ [fnfhjbh
]gvbsg su `prglabz`cg
Ags`rrfnng lugv`s j`mbnba`ags
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080
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
D`htfrbz`hbøl B¶Puâ glhfltr`rà gst` sge`l`6
D`htfrgs prbefs
D`htfrbz`hbøl ag pfnblfebfs
Euntbpnbh`hbøl ag eflfebfs
[rfmnge` ag nøibh`
Gst` sge`l` nfir`rà3
]ghflfhgr n`s h`r`htgråstbh`s ag ul pfnblfebf d`htfrbz`mng.
[r`htbh`r n` d`htfrbz`hbøl ag ul pfnblfebf pfr d`htfr hfeól.
[r`htbh`r n` d`htfrbz`hbøl ag ul pfnblfebf pfr `irup`hbøl ag târeblfs.
[r`htbh`r n` d`htfrbz`hbøl ag ul trblfebf hu`ar`af pgrdghtf.
[r`htbh`r n` d`htfrbz`hbøl ag ul` abdgrglhb` ag hu`ar`afs.
[r`htbh`r n` euntbpnbh`hbøl ag eflfebfs.
029E`tgeàtbh` ∝ Xge`l` 08
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¡[`r` hfeglz`r!
D`htfrgs prbefs
@ nf n`rif ag n`s sge`l`s `ltgrbfrgs jgefs tr`m`c`af agshfepflbglaf nfs
lóegrfs gl sus d`htfrgs prbefs. Gstf nf j`hgefs hu`laf sg agsg` sbepnbdbh`r
`niul` gxprgsbøl f p`r` e`lbpun`r `niul` fpgr`hbøl ` lugstr` hflvglbglhb`.
@jfr`, lfs tfe`rgefs ul tbgepf p`r` hflfhgr gl prfdulaba`a gstg tge`, pfr-
qug sgrà gsglhb`n p`r` tr`m`c`r hfl nf qug `prglagrgefs gst` sge`l`. Tg`efs
quâ sfl nfs d`htfrgs prbefs.
[rbegrf, agmgefs tglgr hn`rf n` agdblbhbøl ag ul lóegrf prbef3 gstg gs ul
lóegrf l`tur`n qug pugag agshfepflgrsg sfn`egltg gl afs d`htfrgs abstbltfs3
ân ebsef y gn 0. Gstfs lóegrfs sfl hfltr`rbfs ` nfs hfepugstfs, gs aghbr,
`qugnnfs qug tbglgl `n eglfs ul abvbsfr l`tur`n abstbltf agn ulf y ag så ebsefs.
Nfs lóegrfs prbefs eglfrgs ` 8; sfl3
0, =, 8, 7, 9, 00, 08, 09, 02, =8...
Wl lóegrf hfepugstf sg pugag "agsinfs`r" f "agshfepflgr" gl sus d`htfrgs
prbefs, gs aghbr, gl `qugnnfs lóegrfs prbefs abvbsfrgs gx`htfs ag gsg lóegrf
gltgrf? gn prfhgsf p`r` glhfltr`r gstfs abvbsfrgs sg aglfebl` d`htfrbz`hbøl ag
gltgrfs. Gstg prfhgabebgltf ng rgsunt`rà d`ebnb`r, vg`efs ul gcgepnf3
07; =
97 8
=7 77 7
0
Ag e`lgr` sbebn`r qug gl n` rbteâtbh`, gl gn ànigmr` sg utbnbz` n` d`htfrbz`hbøl p`r`
e`lbpun`r y agshfepflgr `niól pfnblfebf gl eflfebfs f gl ftrf pfnblfebf
eàs sbepnbdbh`af, hfl gn fmcgtbvf ag j`hgr eàs dàhbn lugstrf tr`m`cf.
¡@ tr`m`c`r!
H`nhung nfs d`htfrgs prbefs ag nfs lóegrfs sbiubgltgs.
2; 89: =:4
02:BIG] ∝ [fnfhjbh
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0. D`htfrbz`hbøl ag pfnblfebfsGspghåh`egltg gl ànigmr`, hu`laf j`mn`efs ag d`htfrbz`r ul` gxprgsbøl lfs
rgdgrbefs n prfhgsf ag sbepnbh`rn` j`nn`laf afs f eàs d`htfrgs huyf prfauhtfgs biu`n ` n` gxprgsbøl prfpugst`. Iglgr`negltg sg gepng` gstg prfhgabebgltf
p`r` tr`m`c`r hfl pfnblfebfs.
J`y abdgrgltgs eâtfafs f "h`sfs" p`r` d`htfrbz`r, agpglabglaf ag n` gxprgsbøl
qug agsggefs sbepnbh`r? gst` sge`l`, lfsftrfs gstuab`rgefs nfs prbegrfs
hu`trf h`sfs ag ul tft`n ag sbgtg qug vgrgefs. Muglf, ¡e`lfs ` n` fmr`!
0.0 D`htfr hfeól eflfebf
Gstg eâtfaf hflsbstg gl gxtr`gr ag ul pfnblfebf, ul eflfebf hfef d`htfr
hfeól ` h`a` ulf ag nfs târeblfs agn pfnblfebf gl hugstbøl. Nugif, sg agc`blabh`a` n` euntbpnbh`hbøl gltrg gn d`htfr hfeól gxtr`åaf y nfs târeblfs qug
quga`l ag n` gxprgsbøl frbibl`n. [fli`efs `tglhbøl `n gcgepnf qug sbiug.
D`htfrbhgefs gn pfnblfebf3
• [rbegrf, h`nhun`efs gn Eàxbef Hfeól Abvb-
sfr (EHA) ag nfs hfgdbhbgltgs agn pfnblfebf.
Gl gstg h`sf, fmtglgefs qug gn EHA ag 01,
=: y 74 gs 01. Hfl gstf glhfltr`efs gn hfgdb-
hbgltg agn d`htfr hfeól. Xb nfs hfgdbhbgltgs lf
tuvbgr`l EHA, lf sg prgfhupg, sglhbnn`egltg
hfltbló` hfl gn prfhgabebgltf.
• Abvbabefs h`a` ulf ag nfs târeblfs agn pfnb-
lfebf gltrg su EHA, hfl gstf nf "gxtr`gefs",
pfr nf qug agc`efs n` euntbpnbh`hbøl blabh`a`.
• @jfr`, bagltbdbh`efs quâ v`rb`mngs f nbtgr`ngs
pfsggl gl hfeól tfafs nfs eflfebfs. Gl gstg
h`sf, n` v`rb`mng x gs n` qug sg rgpbtg f gs "hf-
eól" gl tfafs nfs târeblfs. Gngibefs n` x qug
tgli` gn eglfr gxpflgltg, gl gstg h`sf sgrå`
x =, y n` "gxtr`gefs", abvbabglaf h`a` târeblf
gltrg n` ebse`.
• @n gxtr`gr n` x, hfepngt`efs gn d`htfr hfeól,
qug sgrå` 01 x=. Fragl`efs gn pfnblfebf y hfl
gstf quga`rå` n` gxprgsbøl d`htfrbz`a`3
[fagefs hfeprfm`r qug ul` d`htfrbz`hbøl gs hfrrght` sb n fpgr`r n` euntbpnbh`-
hbøl blabh`a`, fmtglgefs gn pfnblfebf frbibl`n. Hfeprfmgefs gn gcgepnf qug
`h`m`efs ag vgr3 01 x =( y = + 1 x = ‘ = x ) 5 01 x = y= ‘ =: x 8 + 74 x 1
01 x= y= ‘ =: x8 + 74 x1
01 =: 74 =
9 01 =: 9
0 = 1
= • 9 5 01
01(0 x= y= ‘ = x8 + 1 x1)
01( x= y= ‘ = x8 + 1 x1) 5
x= y=
x= ‘
= x8
x= +
1 x1
x= 5
01 x=( y= ‘ = x + 1 x=) 5
01 x =( y = + 1 x = ‘ = x )
022E`tgeàtbh` ∝ Xge`l` 08
Gn eulaf ag n` e`tgeàtbh`
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0.= D`htfr hfeól pfr `irup`hbøl ag târeblfs
Gepng`efs gn h`sf ag d`htfr hfeól pfr irup`hbøl ag târeblfs, sb nfs eflfebfs
qug hfepflgl gn pfnblfebf pugagl rgulbrsg gl irupfs hfl ul d`htfr hfeól
abdgrgltg gl h`a` irupf. Nugif, gepng`efs n` ebse` tâhlbh` qug gl gn d`htfr
hfeól. Gltglagrgefs hfl ul gcgepnf3
D`htfrbhgefs n` gxprgsbøl sbiubgltg3
• Lft`efs qug pfagefs `irup`r
‘4 xy y 1ey, pfrqug tbglgl gl hfeól
y. Ag biu`n dfre`, pfagefs `irup`r
=ep= y ‘8 xp= pfrqug tbglgl gl
hfeól p=.
• @jfr`, utbnbz`laf gn prfhgabebgltf
qug `prglabefs gl n` sghhbøl 0.0,
prfhgagefs ` gxtr`gr gn d`htfr
hfeól ag h`a` ulf ag nfs "irupfs"ag târeblfs.
• @n fragl`r n`s gxprgsbflgs agltrf
ag nfs p`râltgsbs, lfs a`efs huglt`
qug sfl biu`ngs, nf qug qubgrg aghbr,
qug gstfs sfl ftrf d`htfr hfeól.
• [fagefs gxtr`gr gn lugvf d`htfr
hfeól qug lfs qugaø y rgulbr nfs
eflfebfs qug fmtuvbefs dugr` ag
nfs p`râltgsbs. Hfl gstf fmtglarå`-efs lugstr` gxprgsbøl d`htfrbz`a`.
¶Tg qug gs dàhbn6 Xfnf tglgefs qug br pfhf ` pfhf, sbiubglaf gn prfhgabebgltf.
Tg`efs ftrf gcgepnf, d`htfrbhgefs3
• @irup`efs 87 8 z y =;`8 y, pfrqug tbglgl
gl hfeól `8. @irup`efs ‘0=wy y ‘=0wz
pfrqug tbglgl gl hfeól w. Agmgefs
tglgr huba`af hfl nfs sbilfs.
• @jfr`, gxtr`gefs gn d`htfr hfeól ag
h`a` ulf ag nfs irupfs ag târeblfs. Gl gstg h`sf, gxtr`gefs ( ‘8w) agn
sgiulaf irupf p`r` qug n` gxprgsbøl
sg` biu`n ` n` agn prbegr irupf
• ]gulbefs nfs eflfebfs qug quga`-
rfl dugr` agn p`râltgsbs. Fmtglgefs
`så lugstr` gxprgsbøl d`htfrbz`a`.
87`8 z ‘ =0wz ‘ 0=wy + =;`8 y 5
(87`8 z + =;`8 y) + (‘=0w z ‘ 0=w y) 5
7`8(9 z + 1 y) + (‘8w)(9 z + 1 y) 5
7`8(9 z + 1 y) ‘ 8w(9 z + 1 y) 5
(7`8 ‘ 8w)(9 z + 1 y)
‘4 xy + =ep= + 1ey ‘ 8 xp= 5
(‘4 x y + 1e y) + (=e p= ‘ 8 x p=) 5
= y(‘8 x + =e) + p=(=e ‘ 8 x) 5
= y(=e ‘ 8 x) + p=(=e ‘ 8 x) 5
(= y + p=)(=e ‘ 8 x )
=;0E`tgeàtbh` ∝ Xge`l` 08
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Gcgrhbhbf =
D`htfrbhg nfs pfnblfebfs sbiubgltgs. [ugag tr`m`c`r gl jfc`s `p`rtg sb gs lghgs`rbf.
0) 4`8 ‘ 0;`x ‘ 7mx= + 8`=mx =) 0=my ‘ 4m + 4 x= y ‘ 8 x=
8) 9`=my + :mx= y= + 1m= x= + 01`= y8 1) ‘0=`z + 04`xy= + 8m= z ‘ 1m= xy=
7) ‘2mx= + 2`= x= ‘ 8my= + 8`= y= 4) 7ml ‘ 0;me + 0:e8 x ‘ 2e=lx ‘ =l1 y + 1el8 y
9) =e8 yz ‘ 4e=l8 y ‘ 07ml8 + 7mez :) ‘0: y8 z + 00`8 x= ‘ 2 xy= z + ==`8 xy
2) 2e=l8 + :`=ez ‘ 8`l8 z ‘ =1`e8 0;) ‘2mxy + 0: x + 0;m8r ‘ 7m1ry
00) 07rx8 + =1 sz + =; x= z + 0:rsx ‘ 8; s1 y ‘ =7 s8 x= y 0=) : xy8 ‘ 8wz = ‘ 0; yz + 7 z 8 + 4wy ‘ 1 xy= z =
=;=BIG] ∝ [fnfhjbh
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Gcgrhbhbf 8
D`htfrbhg nfs pfnblfebfs sbiubgltgs. @niul`s gxprgsbflgs pugagl gst`r agsfragl`a`s, `så qug tglarà
qug fragl`rn`s `ltgs ag fpgr`r. [ugag tr`m`c`r gl jfc`s `p`rtg sb gs lghgs`rbf.
0) `= ‘ =`m + m= =) x= + 01 x + 12
8) `= ‘0;` + =7 1) 2 ‘ 4 x + x=
7) 2m= ‘ 8;`=m + =7`1 4) 1 x= + 1 x`8 + `4
9) 04 + 1; x= + =7 x1 :) x= ∝ =; x + 0;;
2) x4 + 0; x8 + =7 0;) :0 z = ‘ 0:; z + 0;;
00) 0;; x0; ‘ 4;`1 x7 y4 + 2`: y0= 0=) `= ‘ =1`e= x= + 011e1 x1
=;1BIG] ∝ [fnfhjbh
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0.1 Abdgrglhb` ag hu`ar`afs
Xg ng hflfhg hfef abdgrglhb` ag hu`ar`afs ` ul mblfebf gl gn qug sus
târeblfs tbglgl ul` rgst` ag pfr egabf y sg ngs pugag h`nhun`r r`åz hu`ar`a`.
]ghfra`efs qug n` sge`l` `ltgrbfr `prglabefs sfmrg prfauhtfs lft`mngs y
tglå`efs qug3
( + m)( ‘ m) 5 `= ‘ m=
[`r` n` abdgrglhb` ag hu`ar`afs, gn prfhgabebgltf gs gx`ht`egltg gn hfltr`rbf3
abdgrglhb` ag hu`ar`afs
`= ‘ m= 5 ( + m)( ‘ m)
Hfef lft`efs, p`r` n` d`htfrbz`hbøl ag gstg h`sf agmgefs gxtr`gr n` r`åz hu`-
ar`a` ag nfs târeblfs. Nugif, agc`r blabh`a` n` euntbpnbh`hbøl gltrg n` sue` y n`
abdgrglhb` ag gst`s r`åhgs. Gltglagrgefs hn`r`egltg hfl ul gcgepnf3
D`htfrbhgefs gn mblfebf3
• Gxtr`gefs n`s r`åhgs hu`ar`a`s ag `emfs
târeblfs.
• @jfr`, agc`efs blabh`a` n` euntbpnbh`hbøl ag
n` sue` pfr n` abdgrglhb` ag n`s r`åhgs qug gl-
hfltr`efs. [rbegrf n` r`åz agn prbegr târeblf y
nugif n` r`åz agn sgiulaf târeblf.
¡Nbstf!, y` tglgefs n` gxprgsbøl d`htfrbz`a`.
Gl fh`sbflgs sg lfs pugagl prgsglt`r gxprgsbflgs qug lf sfl ul` abdgrglhb` ag
hu`ar`afs gl så, pgrf qug n`s pfagefs d`htfrbz`r hfef t`n. Lf sg `sustg, sfnfagmgefs sgiubr gn ebsef eâtfaf, vg`efs.
D`htfrbhgefs gn pfnblfebf sbiubgltg3
• Lft`efs qug n` gxprgsbøl n` pfagefs tr`m`c`r
hfef ul` abdgrglhb` ag hu`ar`afs, pfr nf qug
sgiubefs gx`ht`egltg gn ebsef prfhgabebgltf.
Gxtr`gefs n` r`åz hu`ar`a` ag `emfs târeb-
lfs.
• Blabh`efs n` euntbpnbh`hbøl ag n` sue` pfr n`abdgrglhb` ag `emfs târeblfs.
• Xbepnbdbh`efs n` gxprgsbøl y ¡nbstf!, fmtglgefs
n` gxprgsbøl d`htfrbz`a`.
¶Nf lftø6, sgiubefs n` ebse` tâhlbh` qug gn
gcgepnf `ltgrbfr.
:0e1 ‘ =7l=
:0e1 5 2e=
=7l= 5 7l
(2e= + 7l)(2e= ‘ 7l)
5 (2e= + 7l)(2e= ‘ 7l)
y= ‘ 2( x ‘ 0)=
y= 5 y2( x ‘ 0)= 5 8( x ‘ 0)
(
y + 8( x ‘ 0))( y ‘ 8( x ‘ 0))
( y + 8 x ‘ 8)(
y ‘ 8 x + 8)
5 ( y + 8 x ‘ 8)( y ‘ 8 x + 8)
=;7E`tgeàtbh` ∝ Xge`l` 08
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Gcgrhbhbf 1
D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gl `niul`s gxprgsbflgs agmgrà `pnbh`r sus hflfhbebgltfs sfmrg
lóegrfs r`hbfl`ngs. [ugag tr`m`c`r gl jfc`s `p`rtg sb gs lghgs`rbf.
0) x4 ‘ 1 =) m= ‘ 0
8) 04 ‘ e1 1) 1 ‘ x=
7) =7`0; y1 ‘ 12m0= x4 4) 04 x1 y: ‘ 0;;
9) 84 x= ‘ `4m1 :) :0m1 x= ‘ 84`=h=
2) 12 x1 y= ‘ 1`1m= 0;) `=
84 ‘ x4
=7
00) 01
‘ 2`= 0=) 0 ‘`=
=7
=;4BIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
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]gsuegl
0. D`htfrbz`r ul` gxprgsbøl sg rgdbgrg n prfhgsf ag sbepnbdbh`rn j`nn`laf afs f eàs d`htfrgs, huyf
prfauhtf gs biu`n ` n` gxprgsbøl prfpugst`. Iglgr`negltg sg gepng` gstg prfhgabebgltf p`r`
tr`m`c`r hfl pfnblfebfs.
J`y abdgrgltgs eâtfafs f "h`sfs" p`r` d`htfrbz`r, agpglabglaf ag n` gxprgsbøl qug agsggefs
sbepnbdbh`r. Gst` sge`l` gstuab`efs hu`trf3
0.0 D`htfr hfeól eflfebf3 gstg eâtfaf hflsbstg gl gxtr`gr ag ul pfnblfebf, ul eflfebf
hfef d`htfr hfeól ` h`a` ulf ag nfs târeblfs agn pfnblfebf gl hugstbøl. Nugif, sg agc`
blabh`a` n` euntbpnbh`hbøl gltrg gn d`htfr hfeól gxtr`åaf y nfs târeblfs qug quga`l ag n`
gxprgsbøl frbibl`n.
0.= D`htfr hfeól pfr `irup`hbøl ag târeblfs3 gepng`efs gstg h`sf sb nfs eflfebfs qug
hfepflgl gn pfnblfebf pugagl rgulbrsg gl irupfs hfl ul d`htfr hfeól abdgrgltg gl h`a`
irupf. Nugif, gepng`efs n` ebse` tâhlbh` qug gl gn d`htfr hfeól.
0.8 Qrblfebf hu`ar`af pgrdghtf3 sg bagltbdbh` pfr tglgr trgs târeblfs, ag nfs hu`ngs afs tbglgl
r`åz hu`ar`a` gx`ht`, y gn rgst`ltg gqubv`ng `n afmng agn prfauhtf ag n`s r`åhgs ag nfs ftrfs
eflfebfs.
( x + y)= 5 x = + = xy + y=
[`r` d`htfrbz`r, agmgefs rgfragl`r nfs târeblfs agc`laf ag prbegrf y ag tgrhgrf nfs târeb-
lfs qug tgli`l r`åz hu`ar`a`, nugif gxtr`gefs n` r`åz hu`ar`a` agn prbegr y tgrhgr târeblf
y nfs gshrbmbefs gl ul p`râltgsbs, sgp`ràlafnfs pfr gn sbilf qug `hfep`ð` `n sgiulaf
târeblf? `n hgrr`r gn p`râltgsbs gngv`efs tfaf gn mblfebf `n hu`ar`af.
0.1 Abdgrglhb` ag hu`ar`afs3 mblfebf gl gn qug sus târeblfs tbglgl ul` rgst` ag pfr egabf ysg ngs pugag h`nhun`r r`åz hu`ar`a`.
[`r` n` d`htfrbz`hbøl ag gstg h`sf agmgefs gxtr`gr n` r`åz hu`ar`a` ag nfs târeblfs. Nugif,
agc`r blabh`a` n` euntbpnbh`hbøl ag n` sue` pfr n` abdgrglhb` ag gst`s r`åhgs.
`= ‘ m= 5 ( + m)( ‘ m)
[ugag vbsbt`r nfs sbiubgltgs sbtbfs wgm p`r` glhfltr`r eàs gcgepnfs sfmrg nfs h`sfs ag d`htfrbz`hbøl3
• H`sfs ag d`htfrbz`hbøl3 iff.in/QIvAiJ
• D`htfrbz`hbøl3 iff.in/uNBJYR
=;9E`tgeàtbh` ∝ Xge`l` 08
Blvgstbiug gl n` rga...
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@htbvba`a 0. Hfeprugmg nf `prglabaf
Hfl sus p`n`mr`s, rgspfla` n`s prgiult`s sbiubgltgs.
0. ¶Puâ gs ul lóegrf prbef6
=. ¶Puâ sbilbdbh` "d`htfrbz`r"6
@htbvba`a =. [r`htbqug nf `prglabaf
@. D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gepngg gn prfhgabebgltf ag ul d`htfr hfeól eflfebf.
[ugag utbnbz`r jfc`s `p`rtg sb gs lghgs`rbf.
0) 0=h8a = + 4;h=a 8 =) 87`=m1 z = ‘ 77`=m8 x
8) =1`= xy= ‘ 84 x= y1 1) 07 y8 + =; y= ‘ 7 y
7) 01 x8 y= ‘ =: x= + 74 x1 y8 4) 1=`4 xz 8 ‘ =;`1 z = + 0:`8 x= z ‘ 1`= z 8
M. D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gepngg gn prfhgabebgltf ag ul d`htfr hfeól pfr irup`hbøl
ag târeblfs. [ugag utbnbz`r jfc`s `p`rtg sb gs lghgs`rbf.
0) 8 x8 ‘ 2`x= ‘ x + 8` =) =`= x ‘ 7`= y + 07my ‘ 4mx
8) = x= y + = xz = + y= z = + xy8 1) l= x ‘ 7`= y= ‘ l= y= + 7`= x
7) 0= x= y8 + 1 x8 ‘ 8 x ‘ 2 y8 4) 1`8 x ‘ 1`=m + 8me ‘ 8`ex
@utfhfltrfn
=;:BIG] ∝ [fnfhjbh
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H. D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gepngg gn prfhgabebgltf ag ul trblfebf hu`ar`af pgrdghtf.
[ugag utbnbz`r jfc`s p`rtg sb gs lghgs`rbf.
0) 84 + 0=e= + e1 =) 0 + 12`= ‘ 01`
8) `0= + 0:`4 + :0 1) 0=0 + 02: x4 + :0 x0=
7) 2`1 z 4 + 0:`= x= y= z 8 + 2 x1 y1 4) 0 + 01 x= y= + 12 x1 y1
9) `1m: + 1`=m1h + 1h= :) 1`4 z 4 ‘ =1`8m= x= z 8 + 84m1 x1
2) 04`4m1 ‘ :`8m=a 8 d 1 + a 4 d : 0;) e: ‘ =;e1 s= x8 + 0;; s1 x4
A. D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gepngg gn prfhgabebgltf ag ul` abdgrglhb` ag hu`ar`afs. [ugag
utbnbz`r jfc`s `p`rtg sb gs lghgs`rbf.
0) 0;;e1l1 ‘ =7r 4 s: =) =74`0= x4 ‘ =:2m1l0;
8) 2`=m1h4a : ‘ 0 1) 2`1m4 ‘ 1v:
7) 04e=l1 ‘ 41a 1v4 4) =7h1e=l4 ‘ 2 x= y1
9) `1 d 4 i 1j0= ‘ c0=o 0;e4r = :) x=
0;; ‘
y= z 1
:0
2) x2
12 ‘ 1`0;
0=0 0;) 0
04 ‘ 1 x=
12
=;2E`tgeàtbh` ∝ Xge`l` 08
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@. ]gsugnv` n`s euntbpnbh`hbflgs ag eflfebfs sbiubgltgs. ]ghugrag n`s rgin`s ag nfs gxpflgltgs. Xb
su rgsunt`af blhnuyg gxpflgltgs lgi`tbvfs, aâcgnfs ag gs` dfre`, lf tr`lsdfreg n` gxprgsbøl `
lóegrf r`hbfl`n.
0) ‘8`(4` ‘=) 5
=) 4`= • 1`8 5
8) :`8(‘8`1) 5
1) ‘1`1(2`1) 5
7) `= • 4`8 5
4) 2` ‘7 • 7`= 5
9) ‘8`1(2`4) 5
:) =` ‘= • 1`= 5
2) 4`8(‘9`) 5
0;) 1` ‘4 • 8`= 5
00) ‘9`7(=`1) 5
0=) (‘1`2)(‘7 ) 5
08) `8(‘1`=) 5
01) 4` ‘ 1 • :`= 5
07) ‘9`8(1`9) 5
04) ‘7`9(4`4) 5
09) =`4 • 2`= 5
0:) :`8(‘9`8) 5
02) 9` ‘8(‘4` ‘8) 5
=;) 4`9 • 4`1 5
=0) 7`8 • 4`8 5
==) 9` ‘4 • 9`1 5
M. ]gsugnv` n`s euntbpnbh`hbflgs ag eflfebfs sbiubgltgs. ]ghugrag n`s rgin`s ag nfs gxpflgltgs.
0) 8 x= y8 • 1`= y8 5
=) ‘7`8r =(7r =t 8) 5
8) :w= x1(‘7 x1 z 1) 5
1) 9 x8 y8 • 4`8 x1 5
7) ‘=`=m1(8m7h=) 5
4) :m
4
a
9
• 9`
=
a
=
5
9) ‘=h1t 8(8h1 z 1) 5
:) 4q=w= • 2w7 z 8 5
2) : ij7(‘1`i =) 5
0;) :u8v= • 2u= x8 5
00) 9el(‘=ew8) 5
0=) 8`=m • 7`h= 5
08) :w1v(‘2wz 1) 5
01) 4 x7 y7 • 1 xz 7 5
07) (‘9`4 z 8)(‘8w8 z =) 5
04) 2`
=
z (‘:`v
8
) 5
09) 8 d =m • 1`=m1 5
0:) ‘:q=r (2r = s) 5
02) 4t 8u • 7ty8 5
=;) (‘1w= z )(‘7 s= z ) 5
=0;BIG] ∝ [fnfhjbh
@ibnba`a ag hànhunf eglt`n
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¡Agspbgrtg sus lgurfl`s! Ng` y rgsugnv` gn sbiubgltg prfmnge` ag nøibh`.
N` h`agl` ag frf
Wl cfvgl gstà gstuab`laf gl ul` hbua`a, ngc`af ag su d`ebnb`. H`a` egs, sus p`args ng glvå`l ablgrf p`r` qug puga` humrbr sus i`stfs.
Gl hbgrt` fh`sbøl, pfr ul` abdhunt`a ghfløebh`, sus p`args ng `vbs`l qug gn ablgrf sg v` ` rgtr`s`rul`s sge`l`s. Gn gstuab`ltg lghgsbt` glhfltr`r n` e`lgr` ag p`i`r gn `nqubngr ag su j`mbt`hbøl,gltflhgs rghugra` qug tbglg ul` h`agl` ag frf hfl =8 gsn`mflgs. Xg ng fhurrg ul` bag` y j`mn` hfl n`augð` ag n` pglsbøl? `emfs hflhnuygl qug ul gsn`møl humrg gn hfstf ag ul aå` ag `nqubngr. R ag gs`dfre` pugag sfnvglt`r su gst`aå` aur`ltg vgbltbtrâs aå`s.
Hfef gn cfvgl s`mg qug gn ablgrf nngi`rà gl niól efegltf gl gsg n`psf ag tbgepf, tbglg n` bltglhbølag rrubl`r su h`agl` nf eglfs pfsbmng j`hbglaf n` eglfr h`ltba`a ag hfrtgs, ag e`lgr` t`n qug h`a`aå` n` sgðfr` tgli` gl su pfagr t`ltfs gsn`mflgs hfef aå`s ân ng `agua`.
[grdghhbfl` ul pfhf n` bag` y sg n` gxpnbh` ` n` augð`, gn `hugraf `n qug nngi` hfl n` augð` gs gnsbiubgltg3 ân pugag a`rng ul gsn`møl pfr aå`, f pugag a`rng ul gsn`møl gn aå` ulf , gn aå` afs pugag
pgabrng gsg gsn`møl y gltrgi`rng ` h`embf ul` pgqugð` h`agl` hfepugst` pfr afs gsn`mflgs. Gn aå`trgs pugag a`rng ul gsn`møl sfnf (qug cultf hfl nfs afs qug gnn` tbglg ng sgrvbrå`l p`r` p`i`r gn tgrhgraå`) f pugag pgabrng qug ng agvugnv` nfs afs qug gnn` y` tbglg y gltrgi`rng ul pgqugðf sgiegltf hfltrgs gsn`mflgs, y `så sbiubglaf, aå` pfr aå`. Nf ólbhf qug agmgrå` bepfrt`rng ` n` augð` gs tglgr gl su
pfagr h`a` aå` n` h`ltba`a ag gsn`mflgs gqubv`ngltg ` n` h`ltba`a ag aå`s qug gn gstuab`ltg ng agmg pfr gst`r gl su jftgn.
@a`pt`af ag iff.in/ElTieM
@jfr`, rgspfla` n`s prgiult`s sbiubgltgs3
0) ¶Huàn gs gn eålbef lóegrf ag sgiegltfs gl qug tbglg qug p`rtbr n` h`agl` gn gstuab`ltg p`r`
`rrubl`rn` nf eglfs pfsbmng y huepnbr hfl su `hugraf nfs vgbltbtrâs aå`s6
=) ¶Høef pfarå` sgr gn prfhgabebgltf qug gn cfvgl agmg sgiubr p`r` huepnbr hfl su `hugraf y qug
n` augð` sbgeprg tgli` n` h`ltba`a ag gsn`mflgs gqubv`ngltg ` nfs aå`s qug ân ng `agua`6
=00E`tgeàtbh` ∝ Xge`l` 08
]`zfl`ebgltf nøibhf
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E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfir`af
glprfhgsf
lfnfir`af
A g s p u â s a g g s t u a b ` r . . .
]ghflfzhf n`s h`r`htgråstbh`s ag ul pfnblfebf d`htfrbz`mng.
[r`htbhf n` d`htfrbz`hbøl ag ul pfnblfebf pfr d`htfr hfeól.
[r`htbhf n` d`htfrbz`hbøl ag ul pfnblfebf pfr irup`hbøl ag târeblfs.
[r`htbhf n` d`htfrbz`hbøl ag ul trblfebf hu`ar`af pgrdghtf.
[r`htbhf n` d`htfrbz`hbøl ag ul` abdgrglhb` ag hu`ar`afs.
[r`htbhf n` euntbpnbh`hbøl ag eflfebfs.
Hu`laf tr`m`c`efs hfl ànigmr`, agmgefs s`mgr høef rgprgsglt`r n`s egaba`s
ag iur`s pn`l`s. Gst` j`mbnba`a rgsunt` dula`eglt`n hu`laf sg lfs prgsglt`l
prfmnge`s qug rgqubgrgl h`nhun`r àrg`s y pgråegtrfs ag `niul` supgrhbg.
Gltflhgs, pr`htbqugefs hfl ul gcgrhbhbf p`r` ags`rrfnn`r gst` agstrgz`3
Agmgefs h`nhun`r y agc`r blabh`a` n` døreun` p`r`
glhfltr`r gn àrg` ag nfs rghtàliunfs sbiubgltgs.
Nfs ólbhfs a`tfs hflfhbafs sfmrg n`s egaba`s
ag sus n`afs sg eugstr`l gl gn ab`ir`e`.
[`r` h`nhun`r nf qug lfs pbagl, lfs a`efs
huglt` qug gn àrg` sbl sfemrg`r gs ul hu`-
ar`af, pfrqug sus n`afs ebagl nf ebsef ( x),
pfr nf qug su àrg` sgrå` ag x • x 5 x=. [`r` gn
rghtàliunf sfemrg`af sgrå` ag 1 • x 5 1 x.
@jfr` ng tfh` ` ustga, gl su hu`agrlf, h`nhung y agcg blabh`a` n` døreun` p`r`
glhfltr`r gn àrg` y pgråegtrf ag n`s iur`s sbiubgltg. Nfs a`tfs hflfhbafs sg
eugstr`l gl n` bnustr`hbøl. H`a` pfnåiflf sg rgprgsglt` hfl ul` ngtr` e`yóshu-
n`, p`r` j`hgr eàs dàhbn su bagltbh`hbøl.
x
x
1
x
x
1
1 x x =
@ M
H A
7 y
7
x
=0=BIG] ∝ [fnfhjbh
]gvbsg su `prglabz`cg
Ags`rrfnng lugv`s j`mbnba`ags
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01 0
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
D`htfrbz`hbøl BB
¶Puâ glhfltr`rà gst` sge`l`6
O`rn Drbgarbhj I`uss, gn prålhbpg ag nfs e`tgeàtbhfs
D`htfrbz`hbøl ag trblfebfs y ag humfs
Gn ngliu`cg `nigmr`bhf
Àrg` ag iur`s igfeâtrbh`s
Gst` sge`l` nfir`rà3
D`htfrbz`r trblfebfs ag n` dfre` x= + mx + h y `x= + mx + h.
D`htfrbz`r n` sue` y abdgrglhb` ag humfs.
[r`htbh`r gn hànhunf eglt`n hfl n` hfepngt`hbøl ag sbilfs y d`htfrgs.
Hflvgrtbr gxprgsbflgs agn ngliu`cg hfeól `n ngliu`cg `nigmr`bhf.
Agtgrebl`r nfs d`htfrgs qug hfepngt`l gn àrg` ag iur`s igfeâtrbh`s.
=08E`tgeàtbh` ∝ Xge`l` 01
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¡[`r` hfeglz`r!O`rn Drbgarbhj I`uss, gn prålhbpg ag nfs e`tgeàtbhfs
¡@ tr`m`c`r!Gl n` sge`l` 9 gstuabø n`s prfirgsbflgs `rbteâtbh`s y n` døreun` p`r` n` sue` ag tfafs nfs târeblfs.Wtbnåhgn` p`r` hflfhgr gn rgsunt`af qug fmtuvf I`uss `n rgsfnvgr gn prfmnge` ag nfs lóegrfs
hflsghutbvfs3 s 5 (`0 + `l )l
=
(0999 ‘ 0:77)
O`rn Drbgarbhj I`uss, e`tgeàtbhf, dåsbhf y `strølfef `ngeàl, l`hbø gl gn sglf
ag ul` d`ebnb` juebnag. Agsag euy tgepr`l` ga`a, abf eugstr`s ag ul` prfab-
ibfs` h`p`hba`a p`r` n`s e`tgeàtbh`s. Huglt` ul` ngygla` qug ` nfs trgs `ðfsbltgrruepbø su p`arg hu`laf gst`m` fhup`af gl n` hflt`mbnba`a ag su lgifhbf
p`r` blabh`rng ul grrfr ag hànhunf.
@ nfs 0; `ðfs, gn e`gstrf ag n` gshugn` prfpusf gn prfmnge` ag sue`r nfs
lóegrfs hflsghutbvfs 0 + = +...+ 0;;. Xg abhg qug gn e`gstrf `pgl`s j`må`
tgrebl`af ag glulhb`r gn prfmnge` hu`laf I`uss y` tglå` n` sfnuhbøl. @n h`mf
ag ul` jfr`, sus hfep`ðgrfs tgrebl`rfl gn hànhunf ag ir`lags sue`s, ebgl-
tr`s qug I`uss sfnf j`må` gshrbtf ul lóegrf y gr` n` ólbh` rgspugst` hfrrght`.
Xus hfltrbmuhbflgs ` n` e`tgeàtbh`, dåsbh` e`tgeàtbh`, `strflfeå`, gst`aåstbh`
y ftr`s r`e`s `pnbh`a`s ag n` hbglhb`, ng abgrfl gl vba` gn lfemrg ag prålhbpg ag
nfs e`tgeàtbhfs.
Ng iust`m` pumnbh`r ul tr`m`cf j`st` gst`r sgiurf ag qug gst`m` pgrdght`egltg
gn`mfr`af? nngiø ` aghbr3 "hu`laf sg dbl`nbz` ul lfmng gabdbhbf lf agmgl quga`r
vbsbmngs nfs `la`ebfs". [grf I`uss lf sfn`egltg rgtbrø nfs `la`ebfs, sblf qug
agstruyø nfs pn`lfs, pfr nf hu`n jfy aå` lf j`y dfre` ag s`mgr høef fmtglå`
sus rgsunt`afs.Qgxtf a`pt`af ag3 jttp3//iff.in/4OBTgj
=01BIG] ∝ [fnfhjbh
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0. Qrblfebf ag n` dfre` x = + mx + h
Gst` sge`l` hfltblu`efs hfl n` d`htfrbz`hbøl, fpgr`hbøl hfltr`rb` ` nfs prf-
auhtfs lft`mngs, y` qug n` d`htfrbz`hbøl pgrebtg gxprgs`r nfs pfnblfebfs a`afsgl târeblfs ag sus d`htfrgs.
[`r` d`htfrbz`r ul trblfebf ag n` dfre` x = + mx + h, rghfragefs gn prfauhtfag afs mblfebfs hfl ul târeblf hfeól. [fr gcgepnf3 ( x + =)( x + 1) 5 x= + 4 x + :.Gstg tbpf ag trblfebfs agmgefs nngv`rnfs ` ul prfauhtf ag afs mblfebfs, hfefgst`m` blbhb`negltg.
[`r` d`htfrbz`r ul trblfebf ag n` dfre` x = + mx + h, agmgefs sgiubr gstfs p`sfs3
`. Agshfepflgr gn trblfebf gl afs d`htfrgs hfl afs târeblfs h`a`ulf. Gn prbegr târeblf ag h`a` d`htfr gs x , qug hfrrgspflag ` n` r`åzhu`ar`a` ag x=.
m. Hfpb`r gn sbilf agn sgiulaf târeblf agn trblfebf y gshrbmbrnf gl gnprbegr d`htfr. Nugif, euntbpnbh`r gn sbilf agn sgiulaf târeblf pfr gnsbilf agn tgrhgr târeblf. Gn rgsunt`af sg gshrbmg gl gn sgiulaf d`htfr.
h. [`r` j`nn`r nfs sgiulafs târeblfs, sg mush`l afs lóegrfs qug su-e`afs f rgst`afs agl gn hfgdbhbgltg m y euntbpnbh`afs agl h.
Xbi`efs nfs p`sfs y d`htfrbhgefs
• Agshfepflgefs gn trblfebf gl afs d`htfrgs(mblfebfs), gxtr`yglaf n` r`åz hu`ar`a` agnprbegr târeblf y gn rgsunt`af nf gshrbmbefs glh`a` ulf ag nfs d`htfrgs.
• Hfpb`efs gn sbilf agn sgiulaf târeblf y nfgshrbmbefs gl gn prbegr d`htfr. Nugif, euntb-pnbh`efs nfs sbilfs agn sgiulaf târeblf pfrgn tgrhgrf y nf gshrbmbefs gl gn sgiulaf d`htfr.
• Mush`efs afs lóegrfs qug sue`afs agl ‘9
y qug euntbpnbh`afs, 0=.• Gn trblfebf d`htfrbz`af gs gn prfauhtf ag nfs
mblfebfs.
x= ‘ 9 x + 0=
x=
5 x
( x )( x )
x= ‘ 9 x + 0=
Ù
( x ‘ )( x ‘ )
x= ‘ 9 x + 0= 5
( x ‘ 1)( x ‘ 8)
@tglhbøl3 sb lf j`y ul` p`rgc` ag lóegrfs gltgrfs qug huepn`hfl n`s hflabhbflgs ag n` sue` f rgst` y n` euntbpnbh`hbøl, gn trblfebflf gs d`htfrbz`mng.
=07E`tgeàtbh` ∝ Xge`l` 01
Gn eulaf ag n` e`tgeàtbh`
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D`htfrbhgefs
• Agshfepflgefs gn trblfebf gl afs d`htfrgs (mblf-ebfs), gxtr`yglaf n` r`åz hu`ar`a` agn prbegr târeblf ygn rgsunt`af nf gshrbmbefs gl h`a` ulf ag nfs d`htfrgs.
• Hfpb`efs gn sbilf agn sgiulaf târeblf y nf gshrbmbefs
gl gn prbegr d`htfr. Nugif, euntbpnbh`efs nfs sbilfs agnsgiulaf târeblf pfr gn tgrhgrf y nf gshrbmbefs gl gnsgiulaf d`htfr.
• Mush`efs afs lóegrfs qug sue`afs agl : y qugeuntbpnbh`afs, 07.
• Gn trblfebf d`htfrbz`af gs gn prfauhtf ag nfs mblfebfs.
D`htfrbhgefs
• Agshfepflgefs gn trblfebf gl afs d`htfrgs (mblf-
ebfs) gxtr`yglaf n` r`åz hu`ar`a` agn prbegr târeblf ygn rgsunt`af nf gshrbmbefs gl h`a` ulf ag nfs d`htfrgs.
• Hfpb`efs gn sbilf agn sgiulaf târeblf y nf gshrbmbefsgl gn prbegr d`htfr. Nugif, euntbpnbh`efs nfs sbilfs agnsgiulaf târeblf pfr gn tgrhgrf y nf gshrbmbefs gl gnsgiulaf d`htfr.
• Mush`efs afs lóegrfs qug rgst`afs agl ‘0 y qugeuntbpnbh`afs agl 4 y=.
• Gn trblfebf d`htfrbz`af gs gn prfauhtf ag nfs mblfebfs.
x= ‘ xy ‘ 4 y=
x= 5 x
( x )( x )
x= ‘ xy ‘ 4 y=
Ù
( x ‘ )( x + )
x= ‘ xy ‘ 4 y= 5
( x ‘ 8 y)( x + = y)
x= + : x + 07
x= 5 x
( x )( x )
x= + : x + 07
Ù
( x + )( x + )
x= + : x + 07 5
( x + 7)( x + 8)
Gcgrhbhbf 0D`htfrbhg abrght`egltg nfs trblfebfs ag n` dfre` x= + mx + h. J`y ul gcgepnf.
;) x= ‘ 7 x + 1 5
8) x= ‘ = x ‘ 8 5
4) x= ‘ : xy + 07 y= 5
0) x= + 8 x + = 5
1) x= + 1 x + 1 5
9) x= + 1 xy ‘ 0= y= 5
=) x= ‘ 00 x + 8; 5
7) x= ‘ =7 x + :1 5
:) x= ‘ 1 xy ‘ 7 y= 5
( x ‘ 1)( x ‘ 0)
=04BIG] ∝ [fnfhjbh
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=. Qrblfebf ag n` dfre` `x = + mx + h
Gl gstg h`sf, gn hfghbgltg ` agn prbegr târeblf gs abdgrgltg ag 0.
[`r` d`htfrbz`r ul trblfebf ag gstg tbpf, agmgefs hflvgrtbrnf ` n` dfre` x= + mx + h, qug vbefs gl gn `p`rt`af `ltgrbfr. Tg`efs nfs p`sfs p`r` d`htfrbz`r
ul trblfebf ag n` dfre` `x=
+ mx + h.
`. Euntbpnbh`r gn trblfebf pfr gn hfgdbhbgltg (`) agn prbegr târeblf. `( x= + mx + h) 5 `= x= + `mx + `h
m. Gxprgs`r gn prbegr târeblf rgsunt`ltg, hfef ( x)= y gl gn sgiulaftâreblf hfleut`r gn fragl ag `mx pfr m`x. ( x)= + m( x) + `h
h. D`htfrbz`r gn pfnblfebf qug sg fmtuvf gl nfs p`sfs ltgrbfrgs hfefsg jbzf hfl n` dfre` x= + mx + h
a. Abvbabr nfs prfauhtfs rgsunt`ltgs gltrg nfs d`htfrgs ag `, qug gs gn
lóegrf pfr gn hu`n sg euntbpnbhø gn trblfebf.
D`htfrbhgefs
• Euntbpnbh`efs gn trblfebf pfr 8.
• Gxprgs`efs 2 x= hfef ul` sfn` pftglhb` yhfleut`efs 09 y 8 gl gn sgiulaf târeblf.
• D`htfrbz`efs gn trblfebf rgsunt`ltg, hfefnf jbhbefs hfl n` dfre` x= + mx + h.
• Gn rgsunt`af p`rhb`n nf abvbabefs gltrg nfsd`htfrgs ag 8 5 8 Ù 0.
• Gn trblfebf d`htfrbz`af gs gn prfauhtf ag nfsmblfebfs.
D`htfrbhgefs
• Euntbpnbh`efs gn trblfebf pfr 4.
• Gxprgs`efs 84 x= hfef ul` sfn` pftglhb` yhfleut`efs 9 y 4 gl gn sgiulaf târeblf.
• D`htfrbz`efs gn trblfebf rgsunt`ltg, hfefnf jbhbefs hfl n` dfre` x= + mx + h.
• Gn rgsunt`af p`rhb`n nf abvbabefs gltrg nfsd`htfrgs ag 4 5 8 Ù =.
• Gn trblfebf d`htfrbz`af gs gn prfauhtf ag nfsmblfebfs.
8 x= ‘ 09 x + 0; 5
8(8 x= ‘ 09 x + 0;) 5
2 x= ‘ 8(09 x) + 8; 5
(8 x)= ‘ 09(8 x) + 8; 5
(8 x ‘ 07)(8 x ‘ =) 5
(8 x ‘ 07)
8
(8 x ‘ =)
0 5
( x ‘ 7)(8 x ‘ =)
4 x= ‘ 9 x ‘ 8 5
4(4 x= ‘ 9 x ‘ 8) 5
84 x= ‘ 4(9 x) ‘ 0: 5
(4 x)= ‘ 9(4 x) ‘ 0: 5
(4 x ‘ 2)(4 x + =) 5
(4 x ‘ 2)
8 (4 x + =)
= 5
(= x ‘ 8)(8 x + 0)
=09E`tgeàtbh` ∝ Xge`l` 01
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Gcgrhbhbf =D`htfrbhg abrght`egltg nfs trblfebfs ag n` dfre` `x= + mx + h. J`y ul gcgepnf.
;) = x= + 9 x + 8 5 0) 4 x= + 00 x + 8 5
1 x= + =(9 x) + 4 5
(= x)= + 9(= x) + 4 5
(= x + 4)(= x + 0) 5
(= x + 4)
= (= x + 0)
0 5
( x + 8)(= x + 0)
=) 4 x= + 09 x + 7 5 8) 0= x= ‘ 08 x + 8 5
D`htfrbhgefs
• Euntbpnbh`efs gn trblfebf pfr 1.
• Gxprgs`efs 04 x= hfef ul` sfn` pftglhb` yhfleut`efs 00 y 1 gl gn sgiulaf târeblf.
• D`htfrbz`efs gn trblfebf rgsunt`ltg hfef nfjbhbefs hfl n` dfre` x= + mx + h.
• Gn rgsunt`af p`rhb`n nf abvbabefs gltrg nfsd`htfrgs ag 1 5 1 Ù 0.
• Gn trblfebf d`htfrbz`af gs gn prfauhtf ag nfsmblfebfs.
1 x= ‘ 00 x + 4 5
1(1 x= ‘ 00 x + 4) 5
04 x= ‘ 1(00 x) + =1 5
(1 x)= ‘ 00(1 x) + =1 5
(1 x ‘ :)(1 x ‘ 8) 5
(1 x ‘ :)
1 (1 x ‘ 8)
0 5
( x ‘ =)(1 x ‘ 8)
=0:BIG] ∝ [fnfhjbh
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Gcgrhbhbf 8D`htfrbhg n` sue` ag humfs sbiubgltgs. Iuågsg pfr gn gcgepnf.
;) w8 + x8 5
0) y8 + z 8 5
=) x8 + : 5
8) =9 + j8 5
1) : + :w8 5
7) x8 + 41 5
8. Xue` ag humfs `8 + m8 Blbhb`efs gn gstuabf ag n` sue` ag humfs, j`hbglaf rgdgrglhb` ` n` euntbpnbh`-hbøl ag nfs pfnblfebfs. Tg`efs.
`= ‘ `m + m=
Ù ` + m
`=m ‘ `m= + m8
`8 ‘ `=m + `m=
`8 + m8
@så qug ( + m)( = ‘ `m + m=) 5 `8 + m8 y hfef blabh`efs `ltgrbfregltg, d`htfrbz`rgs n` fpgr`hbøl fpugst` nfs prfauhtfs lft`mngs? gltflhgs n` sue` ag humfs gspfsbmng gxprgs`rn` hfl n` døreun`3
`8 + m8 5 ( + m)( = ‘ `m + m=)
R sg d`htfrbz` hfl nfs p`sfs sbiubgltgs3
`. Gl gn prbegr d`htfr sg gshrbmg n` sue` agn rgsunt`af ag n` r`åz hómbh`agn prbegr y sgiulaf târeblf ( + m).
m. Gl gn sgiulaf d`htfr agmg br gn prbegr târeblf `n hu`ar`af eglfs gnprbegrf pfr gn sgiulaf, eàs gn sgiulaf `n hu`ar`af ( = ‘ `m + m=).
Gcgepnf3 x8 + y8 5 ( x + y)( x = ‘ xy + y=)
: z 8 + 41 5 (= z + 1)_(= z )= ‘ (= z • 1) + (1)=U
5 (= z + 1)(1 z = ‘ : z + 04)
:e8 + =9l8 5 (=e + 8l)_(=e)= ‘ (=e • 8l) + (8l)=U
5 (=e + 8l)(1e= ‘ 4el + 2l=)
(w + x)_(w)= ‘ (w • x) + ( x)=U 5 (w + x )(w= ‘ wx + x =)
=02E`tgeàtbh` ∝ Xge`l` 01
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Gcgrhbhbf 1D`htfrbhg n` abdgrglhb` ag humfs sbiubgltgs. Iuågsg pfr gn gcgepnf.
;) x8 ‘ 0 5
0) w8 ‘ x8 5
=) y8 ‘ : z 8 5
8) : x8 ‘ 41 5
1) =9 ‘ y8 5
7) 0=7 ‘ :w8 5
1. Abdgrglhb` ag humfs `8 ‘ m8 Gn prfhgabebgltf p`r` d`htfrbz`r ul` abdgrglhb` ag humfs gs p`rghbaf `n quggepng`efs gl n` sue` ag humfs, nf qug h`emb`l sfl nfs sbilfs. ]g`nbhgefs n`euntbpnbh`hbøl ag nfs pfnblfebfs.
`= + `m + m=
Ù ` ‘ m
‘ =m ‘ `m= ‘ m8
`8 + `=m + `m=
`8 ‘ m8
@så qug ( ‘ m)( = + `m + m=) 5 `8 ‘ m8 y su døreun` gs3
`8 ‘ m8 5 ( ‘ m)( = + `m + m=)
R sg d`htfrbz` hfl nfs p`sfs sbiubgltgs3
`. Gl gn prbegr d`htfr sg gshrbmg n` rgst` agn rgsunt`af ag n` r`åz hómbh`agn prbegr y sgiulaf târeblf ( ‘ m).
m. Gl gn sgiulaf d`htfr agmg br gn prbegr târeblf `n hu`ar`af eàs gnprbegrf pfr gn sgiulaf, eàs gn sgiulaf `n hu`ar`af ( = + `m + m=).
Gcgepnf3
x8 ‘ y8 5 ( x ‘ y)( x = + xy + y=)
m8 ‘ : 5 (m ‘ =)_(m)= + (m • =) + (=)=U
5 (m ‘ =)(m= + =m + 1)
0=7w8 ‘ z 8 5 (7w ‘ z )_(7w)= + (7w • z ) + ( z )=U
5 (7w ‘ z )(=7w= + 7wz + z =)
( x ‘ 0)_( x)= + ( x • 0) + (0)=U 5 ( x ‘ 0)( x = + x + 0)
==;BIG] ∝ [fnfhjbh
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]gsuegl
0. D`htfrbz`r ul trblfebf ag n` dfre` x = + mx + h sbilbdbh` gxprgs`rnf hfef gn prfauhtf ag afsd`htfrgs ag n` dfre` ( x + )( x + m). [`r` d`htfrbz`r trblfebfs ag gst` dfre` agmgefs3
• Agshfepflgr gn trblfebf gl afs d`htfrgs (mblfebfs). [`r` gnnf fmtglgefs n` r`åz hu`ar`a` agnprbegr târeblf.
• Hfpb`r gn sbilf agn sgiulaf târeblf gl gn prbegr d`htfr. Nugif, euntbpnbh`r gn sbilf agn sgiul-af târeblf pfr gn sbilf agn tgrhgr târeblf y gshrbmbrnf gl gn sgiulaf d`htfr.
• Gngibr afs lóegrfs qug sue`afs f rgst`afs agl hfef rgsunt`af gn târeblf m y n euntbpnbh`r-nfs agl hfef rgsunt`af gn târeblf h.
=. Nfs trblfebfs ag n` dfre` `x = + mx + h sg d`htfrbz`l hfl nfs p`sfs sbiubgltgs3
• Euntbpnbh`r gn trblfebf pfr gn hfgdbhbgltg ( ).
• Gxprgs`r gn prbegr târeblf rgsunt`ltg, hfef ( x)=
y gl gn sgiulaf târeblf hfleut`r gn fragl`mx pfr m`x.
• D`htfrbz`r gn pfnblfebf qug sg fmtuvf gl nfs p`sfs `ltgrbfrgs, hfef sg jbzf hfl gn trblfebfag n` dfre` x= + mx + h.
• Abvbabr nfs prfauhtfs gltrg nfs d`htfrgs ag `, qug gs gn lóegrf pfr gn hu`n sg euntbpnbhø gntrblfebf.
8. N` sue` ag humfs `8 + m8 sg d`htfrbz` ag `hugraf hfl n` døreun` sbiubgltg3
`8 + m8 5 ( + m)( = ‘ `m + m=)
1. N` abdgrglhb` ag humfs `8 ‘ m8 sg d`htfrbz` ag `hugraf hfl n` døreun` sbiubgltg3
`8 ‘ m8 5 ( ‘ m)( = + `m + m=)
[`r` rgdfrz`r sus hflfhbebgltfs sfmrg n` d`htfrbz`hbøl, vbsbtg3 jttp3//iff.in/LyY9mW
==0E`tgeàtbh` ∝ Xge`l` 01
Blvgstbiug gl n` rga...
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@htbvba`a 0. Ageugstrg nf prglabaf
E`rqug ul` "Z" gl gn hu`arf ag n` fphbøl qug rgspflag hfrrght`egltg h`a` prgiult`.
0) ¶Puâ gxprgsbøl rgprgsglt` n` d`htfrbz`hbøl hfrrght` ag x= + 7 x + 46
=) ¶Puâ sbilfs quga`l gl nfs d`htfrgs `n gxprgs` x= + = x ‘ :, hfeful` euntbpnbh`hbøl ag afs mblfebfs6
8) ¶Puâ gxprgsbøl rgprgsglt` n` d`htfrbz`hbøl hfrrght` ag x=
‘ x ‘ 0=6
1) ¶Huàn gs n` rgspugst` hfrrght` `n d`htfrbz`r 7 x= ‘ 4 x ‘ :6
7) @n d`htfrbz`r (: + `
8
), ¶huàn gs gn sgiulaf târeblf agn prbegr d`htfr6
4) ¶Puâ gxprgsbøl rgprgsglt` ul` abdgrglhb` ag humfs6
9) ¶Huàn gs gn óntbef târeblf qug rgsunt` `n d`htfrbz`r ( x8 ‘ 41)6
:) ¶Huàn gs gn fragl ag nfs sbilfs `n d`htfrbz`r ( x8 + y8)6
@utfhfltrfn
( x ‘ 4)( x ‘ 0) ( x ‘ 8)( x ‘ =)
( x + 4)( x + 0)
( x + 8)( x + =)
( ‘ )( ‘ )
( + )( ‘ )
( ‘ )( + )
( + )( + )
( x ‘ 4)( x ‘ =) ( x ‘ 8)( x ‘ 1)
( x ‘ 1)( x + 8)
( x ‘ 0=)( x ‘ 0)
( x ‘ =)(7 x ‘ 1)
( x + =)(7 x ‘ 1)
( x ‘ =)(7 x + 1)
( x + =)(7 x + 1)
`8 ‘ m8
`8 + m8
( ‘ m)8
( + m)8
( + )(+, +, +)
( + )(‘ , +, ‘)
( ‘ )(+, ‘ , +)
( + )(+, ‘ , +)
` `=
=
=`
1
04
x=
1 x
===BIG] ∝ [fnfhjbh
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@htbvba`a =. [r`htbqug nf prglabaf
Fmsgrvg nfs h`sfs ag d`htfrbz`hbøl qug sg prgsglt`l gl h`a` `p`rt`af. ]g`nbhg nfs gcgrhbhbfs ag dfre`abrght` y gshrbm` gn rgsunt`af sfmrg n` nålg`. Hu`laf sg` lghgs`rbf, rg`nåhgnfs gl su hu`agrlf.
@. Qrblfebf ag n` dfre` x = + mx + h
;) x= + 9 x + 0= 5
0) x= ‘ x ‘ =; 5
=) x= ‘ 4 x ‘ 9 5
8) x= + 4 x + : 5
1) x= + 7 x + 1 5
7) x= ‘ 7 x ‘ 01 5
4) x= ‘ 8 x + = 5
9) x= ‘ 9 x ‘ : 5
:) x= ‘ 1 x ‘ 0= 5
2) x= + 1 x ‘ 17 5
0;) x= ‘ 0= x + =9 5
00) x= + 09 x ‘ 8: 5
M. Qrblfebf ag n` dfre` `x = + mx + h
;) 4 x= + 00 x + 8 5
0) = x= + x ‘ 8 5
=) 4 x= + x ‘ 0= 5
8) 7 x= + 8 x ‘ = 5
1) 4 x= + 00 x + 1 5
7) 8 x= + 04 x ‘ 0= 5
4) 8 x= ‘ = x ‘ 0 5
9) 0= x= ‘ x ‘ 4 5
:) = x= ‘ 7 x + 8 5
2) 8 x= ‘ 00 x + 4 5
0;) 4 x= + 00 x + 8 5
00) 8 x= ‘ 09 x + 0; 5
H. Xue` ag humfs x 8 + m
;) x8 + =9 5
0) x8 + : 5
=) x8 + 0 5
8) x8 + 41 5
1) : x8 + =9 y8 5
7) x8 + y8 5
4) 0=7 + x8 5
9) =9 y8 + : 5
:) =9 x8 + 41 5
2) 0=7 x8 + =9 y8 5
A. Abdgrglhb` ag humfs `8 ‘ m8
;) x8 ‘ =9 5
0) 41 ‘ x8 5
=) 0 ‘ =9 y8 5
8) x8 y8 ‘ 0 5
1) =9 ‘ x8 5
7) x8 ‘ y8 5
4) x8 ‘ 0 5
9) 41 y8 ‘ =9 x8 5
:) =9 x8 ‘ 0=7 5
2) 41 x8 ‘ 0=7 y8 5
( x + 1)( x + 8)
(= x + 8)(8 x + 0)
( x + 8)( x= ‘ 8 x + 2)
( x ‘ 8)( x= + 8 x + 2)
==8E`tgeàtbh` ∝ Xge`l` 01
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Gshrbm` sfmrg n` nålg` nfs târeblfs qug sg ng pbagl gl h`a` `p`rt`af p`r` qug n` d`htfrbz`hbøl sg`hfrrght`. Dåcgsg gl nfs gcgepnfs.
@. Hfepngtg nfs sbilfs agn prbegr y sgiulaf târeblf.;) x= + 7 x + 4 5 ( x + 8)( x + =)
0) x= + 7 x ‘ =1 5 ( x :)( x 8)
=) x= ‘ = x ‘ =1 5 ( x 4)( x 1)
8) x= + 9 x + 0; 5 ( x 7)( x =)
1) x= ‘ 08 x + 84 5 ( x 2)( x 1)
7) x= + 0; x + =0 5 ( x 9)( x 8)
4) x= + 8 x ‘ 1 5 ( x 1)( x 0)
9) x= ‘ 4 x ‘ =9 5 ( x 2)( x 8)
:) x= ‘ : x ‘ =; 5 ( x 0;)( x =)
2) x= + 8 x ‘ 71 5 ( x 2)( x 4)
0;) x= ‘ : x + 0= 5 ( x 4)( x =)
00) x= ‘ 0= x ‘ 17 5 ( x 07)( x 8)
0=) x= ‘ 04 x + 41 5 ( x :)( x :)
08) x= + 08 x + 1= 5 ( x 9)( x 4)
01) x= + 00 x + =1 5 ( x :)( x 8)
07) x= ‘ 04 x + 1: 5 ( x 0=)( x 1)
M. Hfepngtg nfs târeblfs qug d`nt`l gl h`a` mblfebf.
;) x= ‘ 1 x ‘ 8= 5 ( x ‘ :)( x + 1)
0) x= ‘ 9 x + 4 5 ( )( x ‘ 0)
=) x= + x ‘ 9= 5 ( )( x ‘ :)
8) x= ‘ 8 x ‘ =: 5 ( )( x + 1)
1) x= + 00 x + 8; 5 ( )( x + 7)
7) x= + 08 x + 1; 5 ( )( x + 7)
4) x= ‘ 08 x + 84 5 ( )( x ‘ 1)
9) x= ‘ 4 x ‘ 1; 5 ( )( x + 1)
:) x= + 8 x ‘ =: 5 ( x + 9)( )
2) x= + : x + 07 5 ( x + 7)( )
0;) x= ‘ 0: x + :0 5 ( x ‘ 2)( )
00) x= ‘ 0= x ‘ =: 5 ( x ‘ 01)( )
0=) x= ‘ 08 x ‘ 1: 5 ( x ‘ 04)( )
08) x= + 0; x + =7 5 ( x + 7)( )
01) x= ‘ 07 x + 11 5 ( x ‘ 00)( )
07) x= ‘ 07 x ‘ 0;; 5 ( x ‘ =;)( )
H. Gshrbm` gn mblfebf f trblfebf qug hfepngt` h`a` d`htfrbz`hbøl.
;) `8 + : 5 ( + =)( = ‘ =` + 1)
0) =9 + m8 5 ( )(2 ‘ 8m + m=)
=) x8 + 41 5 ( )( x= ‘ 1 x + 04)
8) y8 + =9 5 ( )( y= ‘ 8 y + 2)
1) 41 + : x8 5 ( )(04 ‘ : x + 1 x=)
7) h8 ‘ =9 5 (h ‘ 8)( )
4) : x8 ‘ 0 5 (= x ‘ 0)( )
9) 0=7 ‘ : y8 5 (7 ‘ = y)( )
:) 41w8 ‘ z 8 5 (1w ‘ z )( )
2) x8 y8 ‘ : 5 ( xy ‘ =)( )
==1BIG] ∝ [fnfhjbh
@ibnba`a ag hànhunf eglt`n
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Gn ngliu`cg `nigmr`bhf sg utbnbz` p`r` rgprgsglt`r v`nfrgs agshflfhbafs y pn`ltg`r ghu`hbflgs quggxprgsgl n` bldfre`hbøl prfpfrhbfl`a`. Gn rgtf qug ng prfpflgefs gs qug hflvbgrt` gxprgsbflgs agnngliu`cg hfeól, qug gstàl gl n` t`mn` ag `m`cf, gl ngliu`cg `nigmr`bhf.
Ngliu`cg hfeól Ngliu`cg nigmr`bhf
N` sue` ag afs lóegrfs abdgrgltgs.
Gn humf ag ul lóegrf.
N` sue` ag nfs hu`ar`afs ag afs lóegrfs.
N` sue` ag afs lóegrfs abvbaba` gltrg su abdgrglhb`.
N`s afs qublt`s p`rtgs ag ul lóegrf `ueglt`af gl ul tgrhbf.Wl lóegrf abseblubaf gl 4.
Wl lóegrf p`r hu`nqubgr`.
Wl lóegrf hu`nqubgr` `ueglt`af gl abgz.
N` abdgrglhb` ag afs lóegrfs hu`ngsqubgr`.
N` sgebsue` ag afs lóegrfs.
Qrgs lóegrfs l`tur`ngs hflsghutbvfs.
Gn afmng ag ul lóegrf gxhgabaf gl fhjf.
Gn hu`ar`af ag ul lóegrf eàs n` ebt`a ag gsg lóegrf.
Gn humf ag ul lóegrf eàs gn trbpng agn hu`ar`af ag abhjf lóegrf.
Gn aglfebl`afr ag ul` dr`hhbøl gs trgs ulba`ags e`yfr qug suluegr`afr.
Wl tgrrglf ag dfre` rght`liun`r, su `lhjf ebag n` ebt`a ag sun`rif.
Gn `lhjf ag ul rghtàliunf gs biu`n ` n`s trgs hu`rt`s p`rtgs ag sunflibtua.
Gn luegr`afr ag ul` dr`hhbøl gxhgag `n aglfebl`afr gl trgsulba`ags.
Lóegrf qug `n `irgi`rnf `n 2 sue` =1.
N` abdgrglhb` gltrg ul lóegrf y su `ltgrbfr.
==7E`tgeàtbh` ∝ Xge`l` 01
]`zfl`ebgltf nøibhf
8/20/2019 Libro Polochic Matemática 1er. Sem
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E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfir`af gl
prfhgsflf
nfir`af
A g s p u â s a g g s t u a b ` r . . .
D`htfrbzf trblfebfs ag n` dfre` x= + mx + h y `x= + mx + h.
D`htfrbzf n` sue` y abdgrglhb` ag humfs.
[r`htbhf gn hànhunf eglt`n hfl n` hfepngt`hbøl ag sbilfs y d`htfrgs.
Hflvbgrtf gxprgsbflgs agn ngliu`cg hfeól `n ngliu`cg `nigmr`bhf.
Agtgreblf nfs d`htfrgs qug hfepngt`l gn àrg` ag iur`s igfeâtrbh`s.
[`r` d`htfrbz`r glhugltrg gn n`af qug d`nt` gl h`a` iur` y ageugstrg qug àrg`sg fmtbglg `n euntbpnbh`r nfs mblfebfs. [rgstg `tglhbøl `n gcgrhbhbf ;.
;)
x ‘ =
6
@ 5 x= + 0; x ‘ =1
Àrg` 5 x= + 0; x ‘ =1 5 ( x + 0=)( x ‘ =)
]/ Gn ftrf n`af agn rghtàliunf ebag x + 0=.
0)
x ‘ 1
6@ 5 x= ‘ 04 x + 1:
=)
: x
6
@ 5 1: x=
@ 5 =1 x=
8)
@ 5 7 x= + 08 xy + 4 y= x + = y
6
==4BIG] ∝ [fnfhjbh
]gvbsg su `prglabz`cg
Ags`rrfnng lugv`s j`mbnba`ags
8/20/2019 Libro Polochic Matemática 1er. Sem
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070
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
Dr`hhbflgs `nigmr`bh`s B
¶Puâ glhfltr`rà gst` sge`l`6
]ghfragefs n`s fpgr`hbflgs hfl dr`hhbflgs
Dr`hhbflgs `nigmr`bh`s
T`nu`r gxprgsbflgs `nigmr`bh`s
E`tgeàtbh`s `pnbh`a`s ` n` blvgstbi`hbøl ag h`sfs
Gst` sge`l` nfir`rà3
Agshrbmbr g bagltbh`r quâ gs ul` dr`hhbøl `nigmr`bh`.
]gsfnvgr sbepnbh`hbflgs, euntbpnbh`hbflgs y abvbsbflgs hfl dr`hhbflgs `nigmr`bh`s.
H`nhun`r hfl `ibnba`a gn v nfr ag ul` gxprgsbøl `nigmr`bh` a`af gn v nfr ag su blhøilbt`.
==9E`tgeàtbh` ∝ Xge`l` 07
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¡[`r` hfeglz`r!]ghfragefs n`s fpgr`hbflgs hfl dr`hhbflgs
@ltgs ag hfeglz`r, agmgefs rghfra`r y tglgr hn`rf gn prfhgabebgltf qug sgiub-
efs p`r` rg`nbz`r niul`s fpgr`hbflgs hfl dr`hhbflgs. Gstf gs lghgs`rbf pfrqug
n`s dr`hhbflgs `nigmr`bh`s sbiugl ul hfepfrt`ebgltf euy sbebn`r.
Xbepnbdbh`hbøl ag dr`hhbflgs3 sbepnbdbh`r ul` dr`hhbøl gs tr`lsdfre`rn` gl ul`
dr`hhbøl gqubv`ngltg eàs rgauhba`.
[`r` sbepnbdbh`r gepgz`efs abvbabglaf gltrg nfs prbegrfs lóegrfs prbefs3 =,
8, 7, 9... Gs aghbr, prfm`efs ` abvbabr luegr`afr y aglfebl`afr gltrg = ebgl-
tr`s sg puga`, agspuâs p`s`efs `n 8 y `så suhgsbv`egltg. Wl gcgepnf3
84
4; 5
0:
8; 5
2
07 5
8
7
Euntbpnbh`hbøl ag dr`hhbflgs3 p`r` glhfltr`r gn prfauhtf ag afs dr`hhbflgs,euntbpnbh`efs "gl nålg`", gs aghbr, luegr`afr pfr luegr`afr y aglfebl`afr pfr
aglfebl`afr. Nugif, sbepnbdbh`efs.
07
=7 •
4
2 5
07 • 4
=7 • 2 5
2;
==7
2;
==7 5
=
7
Abvbsbøl ag dr`hhbflgs3 p`r` abvbabr afs dr`hhbflgs, j`hgefs ul` euntbpnbh`hbøl
hruz`a`, gn luegr`afr ag n` prbegr` dr`hhbøl pfr gn aglfebl`afr ag n` sgiula`,
agspuâs, gn aglfebl`afr ag n` prbegr` dr`hhbøl pfr gn luegr`afr ag n` sgiula`.
=
1 ù
7
0; 5
= • 0;
1 • 7 5
=;
=;=;
=; 5 0
¡@ tr`m`c`r!
]gsugnv` n`s sbiubgltgs euntbpnbh`hbflgs y abvbsbflgs ag dr`hhbflgs lueârbh`s.
0)1
4 •
7
: 5 =)
:
2 ù
1
7 5
8):
0; ù
8
4 5 1)
4
0= •
7
2 5
==:BIG] ∝ [fnfhjbh
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0. Dr`hhbflgs `nigmr`bh`sWl` dr`hhbøl nigmr`bh` gs ul` gxprgsbøl dr`hhbfl`rb` gl n` qug gn luegr`afr y
gn aglfebl`afr sfl gxprgsbflgs `nigmr`bh`s, iglgr`negltg pfnblfebfs. x= + x
1 x= ‘ 8?
1 y8 ‘ : y= + y ‘ 8
7 y= + y ‘ 1
Dr`hhbflgs nigmr`bh`s gqubv`ngltgs
Aghbefs qug afs dr`hhbflgs `nigmr`bh`s sfl gqubv`ngltgs sb gn prfauhtf agn lu-egr`afr ag n` prbegr` pfr gn aglfebl`afr ag n` sgiula` gs biu`n `n prfauhtfagn aglfebl`afr ag n` prbegr` pfr gn luegr`afr ag n` sgiula`.
[( x)P( x)
y]( x)X( x)
? sfl gqubv`ngltgs sb [( x) • X( x) 5 P( x) • ]( x)
Tg`efs ul gcgepnf3
• N`s dr`hhbflgs ltgrbfrgs sfl gqubv`ngltgs pfrquggn prfauhtf hruz`af gltrg gnn`s gs biu`n.
@jfr` qug y` s`mgefs quâ sfl n`s dr`hhbflgs `nigmr`bh`s pfagefs gstuab`r n`sfpgr`hbflgs qug sg pugagl rg`nbz`r hfl gnn`s.
0.0 Xbepnbdbh`hbøl ag dr`hhbflgs `nigmr`bh`s
N`s dr`hhbflgs `nigmr`bh`s sg sbepnbh`l ag dfre` euy p`rghba` ` n`s dr`hhbflgs`rbteâtbh`s.
[`r` sbepnbh`r ul` gxprgsbøl r`hbfl`n, d`htfrbz`efs luegr`afr y aglfebl`-afr y nugif h`lhgn`efs d`htfrgs hfeulgs. [`r` gstf `pnbh`rgefs nfs h`sfs agd`htfrbz`hbøl qug jgefs `prglabaf gl n`s sge`l`s `ltgrbfrgs. [fli` `tglhbøl`n gcgepnf.
Xbepnbqugefs n` dr`hhbøl3
• Lft`efs qug gl gn luegr`afr j`y ul trblfebf hu`ar`-
af pgrdghtf y gl gn aglfebl`afr j`y ul` abdgrglhb` aghu`ar`afs. @pnbh`laf nfs prfhgabebgltfs qug `prglab-efs gl n` sge`l` 08, d`htfrbz`efs nfs afs pfnblfebfs.
• @jfr`, h`lhgn`efs nfs d`htfrgs hfeulgs, gs aghbr, nfsqug sfl "biu`ngs". Gl gstg h`sf, sgrå` ( x ‘ =) qug sg rgpb-tg gl luegr`afr y aglfebl`afr.
• Nngi`efs ` n` gxprgsbøl sbepnbdbh`a`.
x + =
x= ‘ 1 y 0
x ‘ =
( x + =)( x ‘ =) 5 ( x = ‘ 1)(0)
x = ‘ 1 5 x = ‘ 1
x= ‘ 1 x + 1
x= ‘ 1 5
( x ‘ =)=
( x ‘ =)( x + =) 5
( x ‘ =)=
( x ‘ =)( x + =) 5
( x ‘ =)
( x + =)
==2E`tgeàtbh` ∝ Xge`l` 07
Gn eulaf ag n` e`tgeàtbh`
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0.= Euntbpnbh`hbøl ag gxprgsbflgs `nigmr`bh`s
Gn prfauhtf ag afs dr`hhbflgs `nigmr`bh`s gs ftr` dr`hhbøl `nigmr`bh` gl n`qug gn luegr`afr gs gn prfauhtf ag nfs luegr`afrgs y gn aglfebl`afr gs gnprfauhtf ag nfs aglfebl`afrgs. Gs aghbr, agmgefs euntbpnbh`r "gl nålg`", `nbiu`n qug nf j`hgefs hfl n`s dr`hhbflgs lueârbh`s.
[( x)P( x)
• ]( x)X( x)
5 [( x) • ]( x)P( x) • X( x)
Gl aflag [( x), P( x), ]( x) y X( x) rgprgsglt`l `niul` gxprgsbøl pfnblfeb`n.
[`r` euntbpnbh`r afs f eàs gxprgsbflgs `nigmr`bh`s r`hbfl`ngs3
`. D`htfrbz`efs nfs luegr`afrgs y nfs aglfebl`afrgs j`st` aflag sg`pfsbmng.
m. Blabh`efs n` euntbpnbh`hbøl agn luegr`afr pfr luegr`afr y aglfebl`-afr pfr aglfebl`afr.
h. Xbepnbdbh`efs gn rgsunt`af j`st` rgauhbrnf ` su eålbe` gxprgsbøl.
Gn prfhgabebgltf gs euy sglhbnnf, pfli`efs `tglhbøl ` ul gcgepnf.
Euntbpnbqugefs n`s dr`hhbflgs sbiubgltgs3
• D`htfrbz`efs nfs luegr`afrgs y nfs aglfebl`-afrgs ag `em`s dr`hhbflgs. [`r` gstf `pnbh`efsnfs prfhgabebgltfs qug jgefs `prglabaf gl n`ssge`l`s `ltgrbfrgs.
• Blabh`efs n` euntbpnbh`hbøl agn luegr`afr pfrluegr`afr y aglfebl`afr pfr aglfebl`afr.
• Xbepnbdbh`efs gn rgsunt`af h`lhgn`laf nfs d`h-tfrgs hfeulgs qug j`y gl gn luegr`afr y gnaglfebl`afr.
Gl gstg h`sf, gstfs d`htfrgs sfl ( x ‘ 0) y ( x + =).
• Qgrebl`efs ag rgsfnvgr sb quga`r` niul` fpgr`-hbøl pglabgltg.
• Fmtglgefs gn rgsunt`af dbl`n.
Hfef pfagefs lft`r, gs gsglhb`n gn afeblbf ag nfs abdgrgltgs h`sfs ag d`h-tfrbz`hbøl qug jgefs gstuab`af ` nf n`rif ag n`s sge`l`s `ltgrbfrgs. ]gsunt`bepfrt`ltg t`embâl, pfagr bagltbh`r n`s h`r`htgråstbh`s ag h`a` gxprgsbøl`nigmr`bh` p`r` `pnbh`r gn prfhgabebgltf hfrrgspflabgltg.
8 x ‘ 8
= x + 1 •
x= + 1 x + 1
x= ‘ x 5
8( x ‘ 0)
=( x + =) •
( x + =)=
x( x ‘ 0) 5
8( x ‘ 0)( x + =)=
= x( x + =)( x ‘ 0) 5
8( x ‘ 0)( x + =)=
= x( x + =)( x ‘ 0) 5
8( x + =)
= x 5
8 x + 4
= x
=80E`tgeàtbh` ∝ Xge`l` 07
8/20/2019 Libro Polochic Matemática 1er. Sem
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Gcgrhbhbf =]gsugnv` n`s euntbpnbh`hbflgs sbiubgltgs. [ugag utbnbz`r jfc`s `p`rtg f su hu`agrlf sb gs lghgs`rbf.
0) x + 8
x ‘ = •
x= + = x ‘ :
x= ‘ 2 =)
x= ‘ = x + 0
x= ‘ 2 •
x= + x ‘ 4
x= ‘ 0
8)e + l
el ‘ l= •
l=
e= ‘ l= 1)
x= ‘ = x
x= ‘ 7 x + 4 •
x= + 1 x + 1
x= ‘ 1
7) xy ‘ = y=
x= + xy •
x= + = xy + y=
x= ‘ = xy 4)
2 ‘ 4 x + x=
x= ‘ 2 •
x= ‘ 7 x + 4
8 x= ‘ 2 x
9) x= ‘ 1 xy + 1 y=
x= ‘ = xy • x=
x= ‘ 1 y= :) =` ‘ =
=`= ‘ 7; • `= ‘ 1` ‘ 7
8` + 8
2) x= ‘ 2
x= ‘ 1 •
x= ‘ 9 x + 0;
x= ‘ : x + 07 0;)
= x= ‘ 8 x ‘ =
4 x + 8•
8 x + 4
x= ‘ 1
Ftrf gcgepnf, euntbpnbqugefs3
• D`htfrbz`efs nfs luegr`afrgsy nfs aglfebl`afrgs ag n`strgs dr`hhbflgs.
• Blabh`efs n` euntbpnbh`hbøl ag
nfs d`htfrgs ag n`s dr`hhbflgs.
• Xbepnbdbh`efs gn rgsunt`afh`lhgn`laf nfs d`htfrgs hfeu-lgs. Gl gstg h`sf, nfs d`htfrgssfl ( + 0), ( ‘ 0), ( ‘ 8), ( + =) y (8` + 1).
• R lf j`y lbliul` fpgr`hbølsbl rgsfnvgr, pfr nf qug nngi`-efs `n rgsunt`af dbl`n.
`= ‘ 0
`= + =` •
`= ‘ ` ‘ 4
8`= + 9 + 1 •
8` + 1
`= ‘ 1` + 8 5
( + 0)( ‘ 0)
`( + =) •
( ‘ 8)( + =)
( + 0)(8` + 1) •
8` + 1
( ‘ 0)( ‘ 8) 5
( + 0)( ‘ 0)(` ‘ 8)( + =)(8` + 1)`( + =)(` + 0)(8` + 1)(` ‘ 0)(` ‘ 8)
5
( + 0)( ‘ 0)( ‘ 8)( + =)(8` + 1)
`( + =)( + 0)(8` + 1)(` ‘ 0)( ‘ 8) 5
0
`
=8=BIG] ∝ [fnfhjbh
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0.8 Abvbsbøl ag dr`hhbflgs `nigmr`bh`s
[`r` abvbabr afs dr`hhbflgs `nigmr`bh`s, sgiubefs ul prfhgsf m`st`ltg sbebn`r `nqug j`hgefs hfl n`s dr`hhbflgs lueârbh`s.
Agmgefs euntbpnbh`r n` prbegr` dr`hhbøl pfr gn blvgrsf ag n` sgiula` dr`hhbøl.Aghbefs gltflhgs qug n` abvbsbøl gs ul` euntbpnbh`hbøl hruz`a` y pfr t`ltf,utbnbz`efs n` ebse` tâhlbh` qug `prglabefs gl gn `p`rt`af `ltgrbfr.
[( x)P( x)
ù]( x)X( x)
5[( x) • X( x)P( x) • ]( x)
Gl aflag [( x), P( x), ]( x) y X( x) rgprgsglt`l `niul` gxprgsbøl pfnblfeb`n.
[fr gcgepnf.
]gsfnv`efs n` abvbsbøl3
• Blvgrtbefs n` sgiula` dr`hhbøl y h`emb`efs
gn sbilf ag abvbsbøl pfr gn ag euntbpnbh`hbøl.
• D`htfrbz`efs nfs luegr`afrgs y nfs aglfeb-l`afrgs.
• Blabh`efs n` euntbpnbh`hbøl ag nfs d`htfrgs.
• Xbepnbdbh`efs gn rgsunt`af h`lhgn`laf nfsd`htfrgs hfeulgs. Gl gstg h`sf, gstfs d`htfrgssfl ( x + =) y ( x ‘ =).
• Nngi`efs `n rgsunt`af dbl`n. ¶Tg nf sglhbnnfqug gs6
Ftrf gcgepnf, abvba`efs n`s gxprgsbflgs sbiubgltgs3
• Blvgrtbefs n` sgiula` dr`hhbøl y h`emb`efsgn sbilf ag abvbsbøl pfr gn ag euntbpnbh`hbøl.
• D`htfrbz`efs nfs luegr`afrgs y nfs aglfeb-l`afrgs. Gl gstg h`sf 8`= y 7`8 lf sg pugaglrgauhbr eàs, pfr nf qug nfs agc`efs `så.
• Blabh`efs n` euntbpnbh`hbøl ag nfs d`htfrgs.
• Xbepnbdbh`efs gn rgsunt`af.
• ]gsfnvgefs n`s fpgr`hbflgs qug quga`l ynngi`efs `n rgsunt`af dbl`n.
x= + = x
x= ‘ 7 x + 4 ù
x= + 1 x + 1
x= ‘ 1 5
x= + = x
x= ‘ 7 x + 4 •
x= ‘ 1
x= + 1 x + 1 5
x( x + =)
( x ‘ =)( x ‘ 8)•
( x + =)( x ‘ =)
( x + =)= 5
x( x + =)( x + =)( x ‘ =)
( x ‘ =)( x ‘ 8)( x + =)= 5
x( x + =)( x + =)( x ‘ =)
( x ‘ =)( x ‘ 8)( x + =)= 5
x x ‘ 8
8`=
`= + 4`m + 2m= ù
7`8
`=m + 8`m= 5
8`=
`= + 4`m + 2m= •
`=m + 8`m=
7`8 5
8`=
( + 8m)= •`m( + 8m)
7`8 5
8`8m( + 8m)
7`8( + 8m)= 5
8`8m( + 8m)
7`8( + 8m)= 5
8m
7( + 8m) 5
8m
7` + 07m
]ghugrag3 gn blvgrsf
ag ul` dr`hhbøl gssbepngegltg n` dr`hhbøl"vfntg`a`".
`m
m`
=88E`tgeàtbh` ∝ Xge`l` 07
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Gcgrhbhbf 8]gsugnv` n`s abvbsbflgs sbiubgltgs. [ugag utbnbz`r jfc`s `p`rtg f su hu`agrlf sb gs lghgs`rbf.
0) x8 ‘ x
= x= + 4 x ù
7 x= ‘ 7 x
= x + 4 =)
: x= + =4 x + 07
04 x= ‘ 2 ù
4 x= + 08 x ‘ 7
2 x= ‘ 0
8)=; x= ‘ 8; x
07 x8 + 07 x= ù
1 x ‘ 4
x + 0 1)
`x= + 7
1`= ‘ 0 ù
`8 x= + 7`=
=` ‘ 0
7)`= ‘ 4` + 7
`= ‘ 07` + 74 ù
`= + =` ‘ 87
`= ‘ 7` ‘ =1 4)
`= ‘ 4`
`8 + 8`= ù
`= + 8` ‘ 71
`= + 2`
=81BIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
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]gsuegl
0. Wl` dr`hhbøl nigmr`bh` gs ul` gxprgsbøl dr`hhbfl`rb` gl n` qug gn luegr`afr y gn aglfebl`afrsfl gxprgsbflgs `nigmr`bh`s, iglgr`negltg pfnblfebfs.
[fr gcgepnf3 x= + 0; x + =7
x= ‘ y=
Afs dr`hhbflgs sfl gqubv`ngltgs sb gn prfauhtf hruz`af gltrg `em`s gs biu`n.
0.0 Xbepnbdbh`hbøl ag gxprgsbflgs nigmr`bh`s3 p`r` sbepnbdbh`r ul` gxprgsbøl r`hbfl`n, d`htfrb-z`efs gn luegr`afr y gn aglfebl`afr, y nugif h`lhgn`efs d`htfrgs hfeulgs.
0.= Euntbpnbh`hbøl ag dr`hhbflgs nigmr`bh`s3 gn prfauhtf ag afs dr`hhbflgs nigmr`bh`s gs ftr`dr`hhbøl `nigmr`bh` gl n` qug gn luegr`afr gs gn prfauhtf ag nfs luegr`afrgs y gn aglfeb-l`afr gs gn prfauhtf ag nfs aglfebl`afrgs. Gs aghbr, agmgefs euntbpnbh`r "gl nålg`".
Gn prfhgabebgltf gs gn sbiubgltg3`. D`htfrbz`efs nfs luegr`afrgs y nfs aglfebl`afrgs j`st` aflag sg` pfsbmng.
m. Blabh`efs n` euntbpnbh`hbøl agn luegr`afr pfr luegr`afr y aglfebl`afr pfr aglfeb-l`afr.
h. Xbepnbdbh`efs gn rgsunt`af j`st` rgauhbrnf ` su eålbe` gxprgsbøl.
0.8 Abvbsbøl ag dr`hhbflgs `nigmr`bh`s3 sg euntbpnbh` n` prbegr` dr`hhbøl pfr gn blvgrsf ag n`sgiula` dr`hhbøl. Aghbefs gltflhgs qug n` abvbsbøl gs ul` euntbpnbh`hbøl hruz`a` y pfrt`ltf, utbnbz`efs n` ebse` tâhlbh` qug `prglabefs gl n` euntbpnbh`hbøl.
Ng gxjfrt`efs ` qug vg` y pfli` `tglhbøl ` nfs sbiubgltgs vbagfs. Gl gnnfs glhfltr`rà gcgepnfs ygxpnbh`hbflgs sfmrg høef tr`m`c`r hfl dr`hhbflgs `nigmr`bh`s.
• Xbepnbdbh`hbøl ag dr`hhbflgs `nigmr`bh`s3 iff.in/;YHtjh
• Euntbpnbh`hbøl y abvbsbøl ag dr`hhbflgs `nigmr`bh`s3 iff.in/0]2l@7
• Abvbsbøl ag dr`hhbflgs `nigmr`bh`s3 iff.in/Y=ZERA
=87E`tgeàtbh` ∝ Xge`l` 07
Blvgstbiug gl n` rga...
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@htbvba`a 0. Ageugstrg nf prglabaf
Hfl sus p`n`mr`s rgspfla` n`s prgiult`s sbiubgltgs.
0) ¶Puâ sfl n`s dr`hhbflgs `nigmr`bh`s6
=) ¶Høef s`mgefs sb afs dr`hhbflgs `nigmr`bh`s sfl gqubv ngltgs6
@htbvba`a =. [r`htbqug nf `prglabaf@. Xbepnbdbqug n`s dr`hhbflgs `nigmr`bh`s sbiubgltgs. Jài`nf agsp`hbf y hfl h`ne`, sbgeprg pugag `pf-
y`rsg gl nfs prfhgabebgltfs ag d`htfrbz`hbøl qug jgefs vbstf gl n`s sge`l`s `ltgrbfrgs.
0)0= x= + 2 x
04 x= ‘ 2 =)
x= ‘ 12
x= ‘ 7 x ‘ 01
8) ex ‘ 7e + 1 x ‘ =;e= + :e + 04 1) 4 x
=
+ 7 x ‘ 407 x= ‘ 9 x ‘ =
7)=0 + 8m ‘ 9 ‘ `m
2 ‘ `= 4)
= x= + 9 x + 8
= x= ‘ 9 x ‘ 1
9)2 y= ‘ =7
8 y= + : y + 7 :)
8 z 8 + 2 z =
z = + 4 z + 2
@utfhfltrfn
=84BIG] ∝ [fnfhjbh
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M. ]gsugnv` n`s sbiubgltgs euntbpnbh`hbflgs hfl dr`hhbflgs `nigmr`bh`s. [ugag tr`m`c`r gl jfc`s `p`rtgsb gs lghgs`rbf.
0)81`x= y + 70`= y ‘ 4:`y=
4=7 x= ‘ 12 •
4=7 x= ‘ 87; x + 12
= x= ‘ 1 y + 8` =)
1 x= ‘ 2
= x= + 9 x + 4 •
1 x1 + 4 x8 + 2 x=
: x9 ‘ =9 x1
8)2 x= ‘ 1
8 x= + 7 x + = •
2 x1 ‘ 4 x8 + 1 x=
=9 x1 + : x 1)
1`e8 ‘ 0=`el ‘ e= + 8l
e= + 0=e + 84 •
e= ‘ e ‘ 1=
e1 ‘ 2l=
7)w= ‘ 9w ‘ 0:
‘9 =w ‘ 01`= + 8w= + 4w •
8w ‘ 9 =
7w= ‘ 1=w ‘ =9 4)= x= + 00 x + 07
x= + 4 x + 2 • x= ‘ 2
9 x= + = xy= ‘ =0 x ‘ 4 y=
H. ]gsugnv` n`s sbiubgltgs abvbsbflgs hfl dr hhbflgs `nigmr`bh`s. [ugag tr`m`c`r gl jfc`s `p`rtg sb gslghgs`rbf.
0) y= ‘ 4 y + 7
y= ‘ 07 y + 74 ù
y= + = y ‘ 87
y= ‘ 7 y ‘ =1 =)
:0`= ‘ 1
74`8 + 87` ù
:0`= ‘ 84` + 1
48` ‘ 01
8)=1e= + 04e
1e ù
8e= + =;e + 0=
=e + = 1)
=7w= ‘ x=
0;w= x ‘ = x= ù
0=7w8 + x8
=7w1 ‘ 0;w= x + x=
7)e=l= ‘ =elxy + x= y=
el ‘ xy ù
el + xy
e=l= ‘ x= y= 4)
8`mx= ‘ = y= ‘ = x= + 8`my=
`= ù
07`8m ‘ 0;`=
x1 ‘ y1
=89E`tgeàtbh` ∝ Xge`l` 07
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@. Xustbtuy` gn v`nfr ag n`s blhøilbt`s gl n`s gxprgsbflgs sbiubgltgs y fpgrg p`r` glhfltr`r gn rgsun-t`af dbl`n. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.
0) ‘: x + 7, aflag x 5 7
=) 1 x ‘ 2, aflag x 5 ‘9
8) 4 x + 7, aflag x 5 8
1) : x ‘ 9, aflag x 5 9
7) ‘7 x ‘ 1, aflag x 5 8
4) 2 x + 8, aflag x 5 ‘0=
9) = x + ;, aflag x 5 8
:) 2 x + 4, aflag x 5 ‘7
2) 7 x ‘ :, aflag x 5 4
0;) 9 x + 0, aflag x 5 2
00) = x ‘ 8, aflag x 5 ‘9
0=) 7 x + 9, aflag x 5 ;
08) ‘2 x ‘ =, aflag x 5 1
01) 8 x + 1, aflag x 5 8
07) ‘7 x ‘ :, aflag x 5 ‘8
04) 2 x + 9, aflag x 5 ;
09) ‘8 x ‘ 1, aflag x 5 0
0:) 8 x + =, aflag x 5 1
02) ‘1 x + 8, aflag x 5 ‘9
=;) 1 x + 4, aflag x 5 0;
=0) 9 x ‘ 4, aflag x 5 00
==) ‘: x + 9, aflag x 5 ;
=8) ‘9 x + 4, aflag x 5 ‘0
=1) 4 x ‘ 8, aflag x 5 :
M. Xustbtuy` gn v`nfr ag n`s blhøilbt`s gl n`s gxprgsbflgs sbiubgltgs y fpgrg p`r` glhfltr`r gn rgsun-t`af dbl`n. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.
0) ‘1 z = ‘ 8, aflag z 5 7
=) = z = + 8, aflag z 5 ‘8
8) 9 z = + 7, aflag z 5 4
1) : z = ‘ 1, aflag z 5 ‘8
7) z = + 9, aflag z 5 1
4) = z = ‘ 4, aflag z 5 8
9) ‘1z= + 9, aflag z 5 1
:) 4 z = ‘ 0, aflag z 5 ‘0
2) ‘= z = + :, aflag z 5 8
0;) 9 z = + 2, aflag z 5 ;
00) z = ‘ 1, aflag z 5 ‘1
0=) 7 z = ‘ 9, aflag z 5 9
08) : z = + 8, aflag z 5 9
01) ‘ z = ‘ :, aflag z 5 ‘4
07) 8 z = + 2, aflag z 5 ‘:
04) 2 z = + :, aflag z 5 0
09) 8 z = + 1, aflag z 5 =
0:) ‘ z = ‘ 8, aflag z 5 ;
02) z = + 7, aflag z 5 7
=;) 4 z = + 1, aflag z 5 4
=0) ‘= z = ‘ 2, aflag z 5 ‘9
==) ‘8 z = + =, aflag z 5 :
=8) 8 z = + 8, aflag z 5 0
=1) ‘= z = ‘ =, aflag z 5 =
=8:BIG] ∝ [fnfhjbh
@ibnba`a ag hànhunf eglt`n
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¡[fli` ` tr`m`c`r sus lgurfl`s! [bglsg y sfnuhbflg nfs prfmnge`s ag nøibh` sbiubgltgs.
N` hfsghj` agn ir`lcgrf
Wl ir`lcgrf tglå` ul tgrrglf gl gn qug hfsghj`m` tfe`tg. Gn tgrrglf gr` hustfab`af pfr hu`trf pgrrfsqug j`må`l sbaf hfnfh`afs gstr`tâibh`egltg. H`a` pgrrf vbvå` gl ul` h`sbt` qug gn ir`lcgrf j`må`hflstrubaf, sg abstrbmuå`l ag n` dfre` gl qug eugstr` n` be`igl.
Gn ir`lcgrf tuvf mugl`s vglt`s y aghbabø `ueglt`r n`s abeglsbflgs ag su tgrrglf. [grf sg ng prgsgl-tø ul prfmnge`3 qugrå` qug nfs hu`trf pgrrfs sbiubgr`l vbibn`laf n` hfsghj` agsag sus h`sbt`s, sblh`emb`r su umbh`hbøl. ¶Høef nfir`r gstf6 Abmucg gl n` be`igl n` dfre` gl qug sg pugag gxp`labr gntgrrglf huepnbglaf hfl n` hflabhbøl eglhbfl`a`.
Blvgrsbøl hfeulbt`rb`
Nfs vghblfs @, M, H y A aghbabgrfl j`hgr ul `hugraf hfeulbt`rbf y sgemr`rfl e`åz gl ul` p`rhgn`
qug sg glhugltr` `n egabf ag sus h`s`s.[fhf agspuâs, h`a` vghblf hflstruyø ul i`nnblgrf (G, D, I y J), tfafs gnnfs eàs hgrh` ag n` pn`lt`hbøl.Xbl gem`rif, n`s i`nnbl`s gepgz`rfl ` hfegrsg y ` `rrubl`r n`s e`zfrh`s.
@ltg gst` sbtu`hbøl, nfs vghblfs aghbabgrfl ngv`lt`r ul` v`nn` qug ngs pgrebt` `hhgagr `n e`bz`n, pgrfqug bepba` qug n`s i`nnbl`s puga`l gltr`r.
Fmsgrvg n` bnustr`hbøl y abmucg pfr aølag agmg p`s`r n` v`nn` p`r` huepnbr hfl gst` hflabhbøl.
M
A
H@ G I
J
D
=82E`tgeàtbh` ∝ Xge`l` 07
]`zfl`ebgltf nøibhf
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¡[`r` hfeglz`r!
]ghfragefs n` sue` y rgst` ag dr`hhbflgs
Gst` sge`l` hfltblu`efs hfl n`s dr`hhbflgs `nigmr`bh`s, `jfr` hfl n` sue` y
rgst` ag dr`hhbflgs `nigmr`bh`s. @ltgs ag gepgz`r hfl gn tge`, rgdrgsh`rgefs
lugstrfs hflfhbebgltfs sfmrg n` sue` y rgst` ag dr`hhbflgs `rbteâtbh`s.
]ghfragefs qug sg pugagl prgsglt`r afs h`sfs3 hfl biu`n y hfl abdgrgltg
aglfebl`afr. [r`htbqugefs ul gcgepnf hfl h`a` ulf.
• Dr`hhbflgs hfl biu`n aglfebl`afr
Hu`laf sue`efs f rgst`efs dr`hhbflgs hfl biu`n aglfebl`afr, fpgr`efs
nfs luegr`afrgs y ólbh`egltg hfpb`efs gn aglfebl`afr.
Gcgepnf37
8 +
0
8 5
4
8 5 =
4:
‘ 8:
5 8:
• Dr`hhbflgs hfl abdgrgltg aglfebl`afr
Hu`laf sue`efs f rgst`efs dr`hhbflgs hfl abdgrgltg aglfebl`afr,
rg`nbz`efs nfs p`sfs sbiubgltgs3
`. Agshfepflgefs nfs aglfebl`afrgs gl sus d`htfrgs prbefs p`r` j`nn`r
gn eålbef hfeól eóntbpnf (e.h.e.) qug sgrà gn aglfebl`afr hfeól.
m. Abvbabefs gn e.h.e. gltrg gn aglfebl`afr ag h`a` dr`hhbøl y gn rgsunt`af
nf euntbpnbh`efs pfr su rgspghtbvf luegr`afr.
[r`htbqugefs gstfs p`sfs hfl n` sue` sbiubgltg.
¡@ tr`m`c`r!
]g`nbhg n`s sue`s y n`s rgst`s dr`hhbfl`rb`s. Nugif, gshrbm` gn rgsunt`af sfmrg n` nålg`.
0)0
= +
7
= 5
=)2
9 ‘
8
9 5
8)4
2 ‘
00
2 +
8
2 5
1) 0
= +
0
8 5
7) 7
8 ‘
0
= +
9
4 5
4) =
8 ‘
8
: ‘
7
0= 5
7
8 +
=
2 5
8(7) + 0(=)
2 5
07 + =
2 5
09
2
e.h.e. 5 8 Å 8 5 8= 5 2
2 8 8
8 0 8
0 0
=1=BIG] ∝ [fnfhjbh
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0. Xue` y rgst` ag dr`hhbflgs`nigmr`bh`s hfl biu`n aglfebl`afr
[`r` sue`r f rgst`r dr`hhbflgs `nigmr`bh`s hfl biu`n aglfebl`afr, sg `pnbh`n` ebse` rgin` ag n` sue` f rgst` ag dr`hhbflgs `rbteâtbh`s hfl aglfebl`afr
hfeól3 sue`efs f rgst`efs nfs luegr`afrgs y hfpb`efs gn aglfebl`afr.
Tg`efs n` rgprgsglt`hbøl sbemønbh` gl gn rghu`arf ag m`cf y niulfs gcgepnfs.
`
h +
m
h 5
` + m
h
Xuegefs
• Xue`efs nfs luegr`afrgs y hfpb`efs
gn aglfebl`afr.
• Xbepnbdbh`efs gn rgsunt`af.
]gstgefs
• ]gst`efs nfs luegr`afrgs y hfpb`efs gn
aglfebl`afr.
• ]gauhbefs târeblfs sgegc`ltgs y gshrb-
mbefs gn rgsunt`af.
Xuegefs
• Xue`efs nfs luegr`afrgs y hfpb`efs
gn aglfebl`afr.
• ]gauhbefs târeblfs sgegc`ltgs.
• D`htfrbz`efs gn luegr`afr y rgauhbefs
târeblfs.
• Gshrbmbefs gn rgsunt`af.
7`=
1` +
9`=
1` 5
7`= + 9`=
1`5 0=`=
1` 5
8`
: x x + 7
‘ x + 1 x + 7
5
: x ‘ ( x + 1) x + 7
5: x ‘ x ‘ 1
x + 7 5
9 x ‘ 1 x + 7
x= + : x + 04( x + 8)( x ‘ 7)
+8 x + :
( x + 8)( x ‘ 7) 5
x= + : x + 8 x + 04 + :( x + 8)( x ‘ 7) 5
x= + 00 x + =1( x + 8)( x ‘ 7) 5
( x + :)( x + 8)( x + 8)( x ‘ 7)
5
x + : x ‘ 7
=18E`tgeàtbh` ∝ Xge`l` 04
Gn eulaf ag n` e`tgeàtbh`
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Gcgrhbhbf 0
]g`nbhg n`s fpgr`hbflgs `nigmr`bh`s sbiubgltgs. Qgli` gl huglt` qug gn rgsunt`af sg rgauhg ` su eålbe`
gxprgsbøl, gs aghbr, sg sbepnbdbh`.
0) x
x + = +
4 x
x + = 5 =)
7 x= ‘ 1
: x8 +
x= + 1
: x8 5
8)4 x=
4 x ‘ 9 ‘
9 x
4 x ‘ 9 5 1)
x= + 8 x
x= ‘ x ‘ 4 ‘
x= ‘ 4
x= ‘ x ‘ 4 5
]gstgefs
• ]gst`efs nfs luegr`afrgs y hfpb`efs
gn aglfebl`afr.
• ]gauhbefs târeblfs sgegc`ltgs.
• D`htfrbz`efs gn luegr`afr y rgauhbefs
târeblfs.
• Gshrbmbefs gn rgsunt`af.
7 x= + 1 x( x + 8)( x ‘ =)
‘8 x= ‘ = x
( x + 8)( x ‘ =) 5
7 x= + 1 x ‘ (8 x= ‘ = x)
( x + 8)( x ‘ =) 5
7 x= + 1 x ‘ 8 x= + = x
( x + 8)( x ‘ =)
5
= x= + 4 x
( x + 8)( x ‘ =) 5
= x ( x + 8)
( x + 8)( x ‘ =) 5
= x
x ‘ =
=11BIG] ∝ [fnfhjbh
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=. Xue` y rgst` ag dr`hhbflgs`nigmr`bh`s hfl abdgrgltg aglfebl`afr
[`r` sue`r y rgst`r dr`hhbflgs `nigmr`bh`s hfl abdgrgltg aglfebl`afr, agmgefs
sgiubr gstfs p`sfs3
`. D`htfrbz`r nfs aglfebl`afrgs p`r` j`nn`r gn eålbef hfeól eóntbpnf
(e.h.e.) qug sgrà gn aglfebl`afr hfeól.
m. Abvbabr gn e.h.e. gltrg gn aglfebl`afr ag h`a` târeblf. Nugif, euntbpnbh`r
gn rgsunt`af pfr h`a` luegr`afr.
h. Xbepnbdbh`r gn rgsunt`af.
[r`htbh`rgefs gn e.h.e. ag pfnblfebfs, hfl nfs gcgepnfs sbiubgltgs.
0. Gl nfs pfnblfebfs3 y( y + =), ( y + 8)( y + =) y ( y + =)=
N` e`yfr pftglhb` ag y gs 0, ag ( y + =) gs = y n` ag ( y + 8) gs 0.
Gltflhgs gn e.h.e. gs3 y( y + =)=( y + 8)
=. Gl nfs pfnblfebfs3 x= + 1 x + 1 y = x= + 8 x ‘ =
D`htfrbz`efs x= + 1 x + 1 5 ( x + =)( x + =) 5 ( x + =)=
= x= + 8 x ‘ = 5 (= x ‘ 0)( x + =)
Gltflhgs gn e.h.e. gs3 ( x + =)= (= x ‘ 0)
@jfr` qug y` s`mgefs høef fmtglgr gn e.h.e. ag pfnblfebfs, pfagefs sue`r
y rgst`r dr`hhbflgs `nigmr`bh`s.
Xuegefs
• Abvbabefs gn e.h.e. ( x= y) gltrg h`a` aglfebl`-
afr. Nugif, euntbpnbh`efs gn rgsunt`af pfr h`a`
luegr`afr.
• D`htfrbz`efs gn luegr`afr y rgauhbefs târeblfs.
• Gshrbmbefs gn rgsunt`af.
Xuegefs
• Abvbabefs gn e.h.e. (4 x=) gltrg h`a` aglfebl`afr.
Nugif, euntbpnbh`efs gn rgsunt`af pfr h`a`
luegr`afr.
• ]gauhbefs târeblfs sgegc`ltgs.
• D`htfrbz`efs gn luegr`afr y sbepnbdbh`efs gn
rgsunt`af.
= x x=
+ 8 xy
5
= x( y) + 8( x)
x= y 5
= xy + 8 x
x= y
x (= y + 8)
x= y5
= y + 8 xy
8
= x +
x ‘ =
4 x=
5
8(8 x) + ( x ‘ =)(0)
4 x= 5
2 x + x ‘ =
4 x= 50; x ‘ =
4 x= 5
=(7 x ‘ 0)
4 x= 57 x ‘ 0
8 x =
=17E`tgeàtbh` ∝ Xge`l` 04
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8. Dr`hhbflgs hfepngc`s
Wl` dr`hhbøl hfepugst` f hfepngc gs hu`nqubgr dfre` dr`hhbfl`rb` qug tgli`
dr`hhbflgs gl gn luegr`afr y gl gn aglfebl`afr, f gl `emfs. Xbemønbh`egltg
sg rgprgsglt` `så3
` + m`
` ‘ m`
Nålg` prblhbp`n agn` dr`hhbøl
Luegr`afr ag n`dr`hhbøl hfepngc`
Aglfebl`afr ag n`dr`hhbøl hfepngc`
Nf qug sg mush` gl tfa` dr`hhbøl hfepngc` gs sbepnbdbh`rn`, gs aghbr, tr`lsdfr-
e`rn` ` ul` dr`hhbøl sbepng, rgauhba` ` sus târeblfs eàs sglhbnnfs y qug sg`
gqubv`ngltg ` n` gxprgsbøl frbibl`n.
[`r` sbepnbdbh`r dr`hhbflgs hfepngc`s, agmgefs sgiubr gstfs p`sfs3
`. Glhfltr`r gn eålbef hfeól eóntbpnf e.h.e. ag tfafs nfs
aglfebl`afrgs.
m. Euntbpnbh`r gn luegr`afr y gn aglfebl`afr ag n` dr`hhbøl
hfepngc` pfr gn e.h.e.
h. Xbepnbdbh`r gn rgsunt`af hu`laf sg` pfsbmng.
Xbepnbdbqugefs
• Gxprgs`efs tfafs nfs târeblfs hfef dr`hhbøl.
• Abvbabefs gn e.h.e. (0)( ) 5 ` gltrg h`a` aglf-
ebl`afr. Nugif, euntbpnbh`efs gn rgsunt`af pfr
h`a` luegr`afr.
• ]g`nbz`efs n` abvbsbøl ag dr`hhbflgs y sbepnbdb-
h`efs gn rgsunt`af.
• D`htfrbz`efs gn rgsunt`af p`rhb`n y lugv`egltg
sbepnbdbh`efs.
0
`
+ 0
5` ‘ 0
`
0`
+00
5`0
‘ 0`
0(0) + 0( )
`
`( ) ‘ 0(0)
`
5
0 + `
`
`= ‘ 0
`
5
`(0 + )`( = ‘ 0)
5(0 + )( = ‘ 0)
5
(0 + )
( + 0)( ‘ 0)5
0
` ‘ 0
]ghugrag3
`m
ùha
5`m
•a h
=19E`tgeàtbh` ∝ Xge`l` 04
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]gsuegl
Gl n` sue` y rgst` ag dr`hhbflgs `nigmr`bh`s `n biu`n qug gl n`s dr`hhbflgs `rbteâtbh`s pfagefs
fpgr`r dr`hhbflgs hfl biu`n y abdgrgltg aglfebl`afr.
0. Gl n` sue` f rgst` ag dr`hhbflgs `nigmr`bh`s hfl biu`n aglfebl`afr, sg sue`l f rgst`l nfs
luegr`afrgs y sg hfpb` gn aglfebl`afr hfeól.
=. [`r` sue`r f rgst`r dr`hhbflgs `nigmr`bh`s hfl abdgrgltg aglfebl`afr, prbegrf sg agmg
j`nn`r gn eålbef hfeól eóntbpnf (e.h.e.) ag nfs aglfebl`afrgs, d`htfrbz`efs h`a` pfnblfebf y
tfe`efs tfafs nfs d`htfrgs abstbltfs gngv`afs ` n` eàxbe` pftglhb`. Nugif, abvbabefs gn e.h.e.
gltrg h`a` aglfebl`afr y gn rgsunt`af sg euntbpnbh` pfr h`a` luegr`afr. Agspuâs sg sbepnbdbh`
n` dr`hhbøl.
8. N`s dr`hhbflgs hfepugst`s f hfepngc`s sfl n`s qug tbglgl dr`hhbflgs gl gn luegr`afr f gn
aglfebl`afr, f gl `em`s p`rtgs ag n` dr`hhbøl. [`r` sbepnbdbh`rn` f rgauhbrn` ` sus târeblfs eàs
sglhbnnfs y qug sg` gqubv`ngltg ` n` gxprgsbøl frbibl`n, agmgefs sgiubr gstfs p`sfs3
`) Glhfltr`r gn eålbef hfeól eóntbpnf ag tfafs nfs aglfebl`afrgs.
m) Euntbpnbh`r gn luegr afr y gn aglfebl`afr ag n` dr`hhbøl hfepngc` pfr gn e.h.e.
h) Xbepnbdbh`r nfs rgsunt`afs hu`laf sg` pfsbmng.
[`r` `db`lz`r sus hflfhbebgltfs sfmrg dr`hhbflgs hfepngc`s, vbsbtg n` abrghhbøl sbiubgltg3
jttp3//iff.in/[uPHRA
=7;BIG] ∝ [fnfhjbh
Blvgstbiug gl n` rga...
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@utfhfltrfn
@htbvba`a 0. Ageugstrg nf `prglabaf
E`rqug ul` "Z" gl gn hu`arf ag n` fphbøl qug hfepngt` h`a` glulhb`af.
0) N`s dr`hhbflgs dfre`a`s pfr ftr` dr`hhbøl gl gn luegr`afr f gl gnaglfebl`afr rghbmgl gn lfemrg ag…
=) Gn aglfebl`afr hfeól ag n`s dr`hhbflgs7 x
, :
x=,
4 x8
gs…
8) Gn rgsunt`af ag n` fpgr`hbøl :8
+=8
‘78
gs...
1) N` d`htfrbz`hbøl hfepngt` ag x8 ‘ x gs…
7) @n euntbpnbh`r gn mblfebf (0; ‘ y) pfr ‘0 sg fmtbglg…
4) N` d`htfrbz`hbøl hfrrght` agn trblfebf x= ‘ : x + 04 gs…
9) [fr sbepng blspghhbøl, gn târeblf qug sg gnbebl` gl n` abvbsbøl
` + m
`
` ‘ m
`
5 gs…
:
x
x=
x8
78
74
72
70=
( x= ‘ x)=
x( x= ‘ 0)
x( x=
+ x) x( x ‘ 0)( x + 0)
y ‘ 0;
y + 0;
‘ y + 0;
‘0; ‘ y
( x ‘ 1)=
( x + 1)=
( x ‘ 1)( x + 1)
( x ‘ =)( x + :)
`
`=
` + m
` ‘ m
prfpb`
sbepng
beprfpb`
hfepngc`
=70E`tgeàtbh` ∝ Xge`l` 04
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@htbvba`a =. [r`htbqug nf `prglabaf
@. Xueg n`s dr`hhbflgs `nigmr`bh`s hfl biu`n aglfebl`afr y rgaózh`n`s ` su eålbe` gxprgsbøl.
0) x
= x ‘ 0 +
=
= x ‘ 0 5
8)8 x + =
= x + 8 +
x ‘ =
= x + 8 5
=)= x
8 x ‘ 7 +
7
8 x ‘ 7 5
1)= x ‘ 0
x= ‘ 8 x ‘ 1 +
8
x= ‘ 8 x ‘ 1 5
M. ]gstg n`s dr`hhbflgs nigmr`bh`s hfl biu`n aglfebl`afr y rgaózh`n`s su eålbe` gxprgsbøl.
0) x
= x + 9 ‘
=
= x + 9 5
8)8 x= ‘ 0
8 x8 ‘
4 x= ‘ 0
8 x8 5
=) x
x ‘ 7 ‘
7
x ‘ 7 5
1)9 x=
2 x= ‘ 1 ‘
4 x ‘ = x=
2 x= ‘ 1 5
=7=BIG] ∝ [fnfhjbh
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H. @ltgs ag rgsfnvgr n`s fpgr`hbflgs, prgstg `tglhbøl ` nfs aglfebl`afrgs. Nugif, gl jfc`s `p`rtg
sueg n`s dr`hhbflgs `nigmr`bh`s, rgaózh`n`s ` su eålbe` gxprgsbøl y gshrbm` gn rgsunt`af sfmrg n`
nålg`. Qbglg ul gcgepnf.
;)= x + 7
4 x +
x + 4
1 x 5
0) x ‘ = x + 1
+ 4 x + 1
5
=) x ‘ 8
= x + 0 +
= x
= x + 0 5
8) x= + 1 x
x= ‘ x ‘ 4 +
1 x ‘ 1
x= ‘ x ‘ 4 5
1) x ‘ =
1 x +
x + 7
0; x 5
7) 8 x + =4 x
+ 1 x ‘ 0: x
5
4)8
x + 8 +
7
x ‘ 1 5
9)8 x + 4
x= + x ‘ =; +
x
x= ‘4 x + : 5
9x + =:
0=x
A. ]gstg n`s dr`hhbflgs `nigmr`bh`s hfl abdgrgltg aglfebl`afr y rgaózh`n`s ` su eålbe` gxprgsbøl
Qr`m`cg gl jfc`s `p`rtg. Nugif, gshrbm` n` rgspugst` gl n` nålg` hfrrgspflabgltg.
0) x + 0
x ‘ = ‘
x
x ‘ = 5
=) x
= x + 9 ‘
=
= x + 9 5
8)2 x= + 9
4 x8 ‘
9 ‘ 8 x=
4 x8 5
1) x8 ‘ 8
= x1 ‘
9 x8 ‘ 8
= x1 5
7) 8 x + 0
1 x ‘ = ‘
x + 0
1 x ‘ = 5
4) x + 8
0= x ‘
x + 1
04 x 5
9) x + 0
x= + x ‘ 0= ‘
=
x= +7 x ‘ =1 5
:)7 x ‘ 0
x= += x ‘ 8 ‘
= x + =
x= +7 x + 4 5
G. Xbepnbdbqug n`s dr`hhbflgs hfepngc`s gl jfc`s `p`rtg y gshrbm` gn rgsunt`af sfmrg n` nålg`.
0)
2`
+8`=
8 +0 x
5
=)8 y
+ 0= y
y + y
=
5
8) x +
0 x
0 x=
5
1)
01 x=
‘0=7
‘
0= x
‘07
5
7)4
x= ‘ 9
x ‘ =1
‘ 8
x= ‘
7 x
‘ :
5
4)
x + 8 +4
x ‘ 1
x + 2 +1=
x ‘ 1
5
=78E`tgeàtbh` ∝ Xge`l` 04
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Fmsgrvg h`a` ghu`hbøl y rgsuânv`n` eglt`negltg. Nugif, gshrbm` su rgsunt`af. J`y ul gcgepnf gl
h`a` `p`rt`af.
@. ;) : + y 5 07 y 5
0) 7 + y 5 08 y 5
=) 2 + y 5 04 y 5
8) 8 + y 5 0: y 5
1) 2 + y 5 =1 y 5
7) 0 + y 5 12 y 5
4) 0= + y 5 =7 y 5
9) =1 + y 5 8; y 5
:) 82 + y 5 71 y 5
2) 0; + y 5 1; y 5
0;) x ‘ 2 5 4 x 5
00) x ‘ 8 5 0= x 5
0=) x ‘ 4 5 07 x 5
08) x ‘ 1 5 =0 x 5
01) x ‘ : 5 0; x 5
07) x ‘ ; 5 8= x 5
04) x ‘ 01 5 4 x 5
09) x ‘ 0; 5 2; x 5
0:) x ‘ =7 5 7; x 5
02) x ‘ 4; 5 =; x 5
9
M.
;) 1 x 5 84 x 5
0) 7 x 5 0; x 5
=) 2 x 5 :0 x 5
8) 4 x 5 0: x 5
1) 9 x 5 ; x 5
7) 2 x 5 71 x 5
4) 8 x 5 8; x 5
9) 1 x 5 =: x 5
:) 0; x 5 0; x 5
2) 07 x 5 17 x 5
0;)
w
1 5 7
w 5
00) w
4 5 0: w 5
0=) w
07 5 = w 5
08) w
1 5 0; w 5
01) w
7 5 0= w 5
07) w
0 5 2; w 5
04) w
7 5 7 w 5
09) 1:w
5 =1 w 5
0:) 4;w
5 0= w 5
02) :;
w 5 =; w 5
2
=71BIG] ∝ [fnfhjbh
@ibnba`a ag hànhunf eglt`n
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@. [r`htbqug n` rgsfnuhbøl ag ghu`hbflgs ag prbegr ir`af hfl nfs gcgrhbhbfs sbiubgltgs.
0) = x + 8 5 7
=) 1 x + 8 5 07
8) = x ‘ 2 5 ‘00
1) 9 x + 00 5 1
7) 7 x ‘ = x ‘ x 5 =;
4) = x + 7 x ‘ 8 x 5 8=
9) 9 x 5 4 + 1 x
:) 8 x 5 : ‘ 8 x
2) = x + 0= 5 9 x + =
0;) 00 x ‘ 4 x ‘ 4 5 =; ‘ : x
00)7 x
1 5 4;
0=) 4 x9
5 ‘1=
08) 8 x1
5 07=:
01) ‘ = x2
5 ‘ 90=
07) 7 x ‘: 5 ‘ 04=7
04) 8 x:
+ x1
5 0;
09) x8
‘ x:
5 7
0:) 8 x1
‘ : x2
5 70:
02) =( x + 1) 5 04
=;) 1(8 x + 9) + 7 5 88
=0) : + =(8 x ‘ 0) 5 =0
==) 8 ‘ =(8 x ‘ 1) 5 01
=8) = ‘ 7(= ‘ x) 5 ‘7
=1) 7(= x ‘ 0) 5 =7 + 8( x ‘ 8)
=7) 4(= x ‘ 8) 5 = ‘ 9(8 ‘ x)
=4) 8(7 ‘ = x) 5 : + 9(0 ‘ = x)
M. [n`ltgg ul` ghu`hbøl qug agshrbm` nf glulhb`af gl h`a` prfmnge`. Nugif, rgsuânv`n` y hfeprugmg
su rgsunt`af.
0) @n trbpng ag ul lóegrf ng rgst`efs 04 y sg fmtbglg =;. ¶Huàn gs gsg lóegrf6
=) @ ul lóegrf ng sue`efs 0: y sg fmtbglg 20. ¶Huàn gs gsg gn lóegrf6
8) @n trbpng ag ul lóegrf ng rgst`efs =1 y sg fmtbglg =2. ¶Huàn gs gsg lóegrf6
1) N` sue` ag afs lóegrfs hflsghutbvfs gs 20. ¶Huàngs sfl gsfs lóegrfs6
7) N` sue` ag n`s ga`ags ag Gstgn` y E`ri`rbt` gs 11 `ðfs. Xb Gstgn` gs = `ðfs e`yfr qug
E`ri`rbt`, ¶quâ ga`a tbglg h`a` ul`6
4) J`nn`r n` egaba` ag nfs n`afs ag ul rghtàliunf qug tbglg =9 he ag pgråegtrf, y n` m`sg gs=9 ag n` `ntur`.
9) Xgm`stbàl i`st` P0=;.;; gl ul p`lt`nøl y ul` h`ebs`. Lf s`mg gn prghbf ag h`a` prgla`,
pgrf n` h`ebs` v`ng afs hu`rt`s p`rtgs qug gn p`lt`nøl. ¶Huàltf v`ng h`a` prgla`6
:) Gl gn s`nøl ag n` hfeulba`a j`y 7;; pgrsfl`s. Xb s`mgefs qug j`y 4: eucgrgs eàs qug
jfemrgs, ¶huàltfs jfemrgs y huàlt`s eucgrgs j`y6
2) @e`nb` i`stø gl n` tbgla` ul tgrhbf agn ablgrf qug nngv`m`. Xb sg qugaø hfl P=1.;;, ¶hfl
huàltf ablgrf nngiø ` n` tbgla`6
=77E`tgeàtbh` ∝ Xge`l` 04
]`zfl`ebgltf nøibhf
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E`rqug hfl ul hjgqug n` h`sbnn` qug egcfr blabqug su rglabebgltf.nfir`af
glprfhgsf
lfnfir`af
A g s p u â s a g g s t u a b ` r . . .
Xuef y rgstf dr`hhbflgs `nigmr`bh`s hfl prghbsbøl.
Xbepnbhf dr`hhbflgs hfepngc`s.
]gsugnvf eglt`negltg ghu`hbflgs ag prbegr ir`af.
]gsugnvf prfmnge`s hftbab`lfs hfl ghu`hbflgs ag prbegr ir`af.
Qfe` ag aghbsbflgs
[r`htbqug n` ibnba`a ag hànhunf y n` tfe` ag aghbsbflgs hfl n` htbvba`a sbiubgltg.
Wstga j` sbaf agsbil`af prgsbagltg agn hfebtâ ag su hfeulba`a y ng prgfhup`
n` s`nua ag nfs lbðfs, pfr nf qug aghbag rg`nbz`r ul` glhugst` p`r` agtgrebl`r
n`s gldgrega`ags eàs drghugltgs. [`r` rg`nbz`r n` glhugst`, sfnbhbt` n` `yua` ag
84 pgrsfl`s ` n`s hu`ngs, `ageàs ag ftrfs ótbngs, ngs prfpfrhbfl`rà n`pbhgrfs.
Nfs prghbfs ag nfs n`pbhgrfs sg eugstr`l gl n` t`mn` ag `m`cf. J`i` nfs hànhunfs
sfnbhbt`afs p`r` agtgrebl`r n` h`ltba`a ag n`pbhgrfs ` hfepr`r y `n eglfr hfstf
pfsbmng.
0) H`nhung gn hfstf ag h`a` fphbøl. Ng `yua`efs hfl n` prbegr` dbn`.
=) Wstga hflsbagr` qug n` ebt`a ag glhugst`afrgs lghgsbt`rà ul lugvf n`pbhgrf.
H`nhung lugv`egltg gn hfstf y gshråm`nfs ag e`yfr ` eglfr gl n` nålg`.
Nugif, rfagg hfl ul hårhunf n` egcfr fphbøl.
8) ¶Puâ p`qugtg f p`qugtgs ag n`pbhgrfs agmg hfepr`r y pfr quâ6
N`pbhgrfs Hfstf pfr p`qugtg Hfstf tft`n
[`qugtg ag 0; ulba`ags P00.;; P11.;;
[`qugtg ag =1 ulba`ags P=1.;;
[`qugtg ag 1; ulba`ags P8:.;;
=74BIG] ∝ [fnfhjbh
]gvbsg su `prglabz`cg
Ags`rrfnng lugv`s j`mbnba`ags
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090
= 8 1
8
1
7
=
0
;
x
y
‘ 0 5 b
x = +
y = 5
r =
]gp`sf3 sge`l`s 2 ` 04
Gst` sge`l` nfir`rà3 ]gp`s`r nfs hfltglbafs ag n` sge`l` 2 ` n` 04.
]gsfnvgr nfs gcgrhbhbfs ag rgp`sf p`r` gv`nu`rsg gl n` prugm`
l`n.
[rgp`r`rsg mbgl p`r` n` prugm` l`n.
=79E`tgeàtbh` ∝ Xge`l` 09
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Pugrba` y qugrbaf gstuab`ltg3
Xg `prfxbe` n` prugm` l`n y agmg prgp`r`rsg `aghu`a`egltg, rgp`s`laf nfs hfltglbafs ag n`
sge`l` 2 ` n` 04.
[`r` `prfvghj`r gstg rgp`sf sbi` gst`s rghfegla`hbflgs3
• J`i` ul pn`l ag nf qug gstuab`rà h`a` aå` y tr`tg ag huepnbrnf. Agabqug eàs tbgepf `
nfs tge`s qug ng rgsuntgl abdåhbngs.
• Musqug ul nui`r tr`lqubnf, bnuebl`af y sbnglhbfsf p`r` gstuab`r.
• Ng` nfs rgsóeglgs ag h`a` sge`l` y gshrbm` n`s bag`s eàs bepfrt`ltgs gl su hu`agrlf.
• Gshuhjg n` hn`sg r`ab`n. Xus e`gstrfs nfhutfrgs ng `hfep`ð`ràl gl gstg rgp`sf y ng `yu-
a`ràl ` rgsfnvgr `niulfs gcgrhbhbfs.
• Hfeprugmg qug j`y` rg`nbz`af mbgl nfs `utfhfltrfngs. Xb tbglg aua`s, vugnv` ` nggr n`s
sge`l`s, `jå glhfltr`rà gxpnbh`hbflgs y gcgepnfs.
¶Høef sgrà n` prugm` ag gv`nu`hbøl6
N` prugm` l`n gv`nó` nfs ebsefs hfltglbafs y ag n` ebse` dfre` gl qug nfs j` tr`m`c`af
sge`l` ` sge`l`.
Gl n` prugm` glhfltr`rà3
• Wl` sgrbg ag `ibnba`a eglt`n p`r egabr su agstrgz` y r`pbagz ag hànhunf gl ul tbgepf
nåebtg ag trgs eblutfs.
• Abdgrgltgs gcgrhbhbfs qug gv`nó`l nf `prglabaf gl n`s fhjf sge`l`s. Gstfs gcgrhbhbfs
sgràl sgegc`ltgs nfs qug ustga rgsfnvbø gl n`s htbvba`ags agn utfhfltrfn. Xg ng pgabrà3
rgspflagr prgiult`s,
e`rh`r hfl ul` gqubs "Z" gn hu`arf ag n` fphbøl hfrrght`,
rgsfnvgr fpgr`hbflgs y
rgsfnvgr prfmnge`s.
• Hu`laf rgsugnv` gcgrhbhbfs y prfmnge`s, agmg agc`r hflst`lhb` ag su prfhgabebgltf.
• Euy bepfrt`ltg3 h`a` sgrbg hfltbglg blstruhhbflgs gx`ht`s ag nf qug agmg rg`nbz`r gl
h`a` `p`rt`af, `så hfef n` v`nfr`hbøl `sbil`a`.
Xb ustga sg prgp`r` hfl tbgepf y agabh`hbøl, gn rgsunt`af sgrà s`tbsd`htfrbf.
Ng rghfra`efs qug agmg prgp`r`rsg p`r` n` prugm` l`n y p`r` n`
prugm` agn EBLGAWH pr`htbh`laf hfl nfs gcgrhbhbfs qug sg glhugltr`l
`n l`n ag su nbmrf gl gn Q`nngr e`tgeàtbhf.
=7:BIG] ∝ [fnfhjbh
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Gcgrhbhbf 0@. @pnbqug n`s rgin`s ag pftglhb`hbøl, ag dfre` abrght`. ]g`nbhg nf qug sg ng pbag gl h`a` p`rt`af.
Agcg tfafs nfs gxpflgltgs pfsbtbvfs. Qbglg ul gcgepnf.
;) (e8l) ‘8(e8l)= 5
0) (‘=m8)8
(‘=m8)= 5
=) y ‘2 5
8) 2w8 z 1(4 y= z 8); 5
1) ( 9 x8 y= z =
: x8 z = );
5
7) 0:(m1h=); 5
4) x8
x= 5
9) m ‘8 • m7 • m= 5
:)(9e=l=)7
(9e=l=)8 5
2) (1`8m=)1(1`8m=)8 5
0;) (w= z =)(w= z =)8 5
00) (‘0=r 8
s1
);
5
0=) (8 x7 y8); 5
08)t 7
t 9 5
01) ( w=h=
1`l8 ) ‘=
5
07) (m9h4a =)4 5
04) (e1lw7
8r = z ) ‘0 5
09) ‘7(7mh8w1)7 5
0:) (0;e1l)2 5
02)`9
`= 5
=;) :`8m( x= y=)1 5
=0) (7r = s8
4e8l=u )=
5
==) ( x8 y= z 4)= 5
=8) m ‘0 5
=1) ( ‘=h= d =
2`m8 )8
5
=7) ( uw=
8 x7 y8 )1
5
=4) (e1l= s8
i =j7r 7 )1
5
=9) z 8 • z 4 • z ‘= 5
=:) ( ‘= x8 yz 1
rs=t = ) ‘7
5
=2) w ‘= 5
M. Xbepnbdbqug n`s gxprgsbflgs `nigmr`bh`s ag dfre` abrght`.
0)
(0:`a 1 s8
8`=a = s )
8 5 4)
(8;m7a 9 d 4
4m9a 8 d )
‘= 5
=) (2`1mh1)(0=`8m8h=) 5 9)(00 x= yz 7)
(4 xy: z 9) 5
8)7m8h=v=
=7m=h=v= 5 :)
=`8e1l=
0;`7el= 5
1) (0:e=lr 8)(8e7l1r 7) 5 2) (9 =e8)(:`7e4) 5
7) ( :0h8 i
8h= i 8 )1
5 0;) (:`1m8)(8`=m=) 5
0
(e8l)
=4;BIG] ∝ [fnfhjbh
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Gcgrhbhbf =@. @pnbqug n`s rgin`s ag n` r`abh`hbøl p`r` rgsfnvgr nfs gcgrhbhbfs sbiubgltgs.
0) (7 x= y8)= 5
=) 8 41 x4 y4 5
8)1`=m1
x= y4 5
1)4=7w1
:0 x: y0=1 5
7)7 (e7l7 x7)1 5
4)8
(=18e07l7) 5
9) (=74`:m1
04h0= y1 ) 5
:)1 04 x: 5
2)8 =9w0= x2 y4 z 8 5
M. Hflvbgrt` nfs gxpflgltgs dr`hhbfl`rbfs gl r`abh`ngs y sbepnbdbqug n` gxprgsbøl ag sgr pfsbmng.
0) ( xy)0/= 5
=) ( 1m:h4)=/1 5
8) e0/8 5
1) (m8 z 4)=/8 5
7) ( 1)0/1 5
4) (=9e4l8
w= )0/8 5
9) ( 2j8o 8e4)1/8 5
:) (rs)0/7 5
2) ( = x=
y4 z 8 )4/8 5
H. ]gsugnv` ag dfre` abrght` n`s sue`s y rgst`s ag r`abh`ngs sgegc`ltgs.
0) :8 00m ‘ 7
8 00m 5
=) 07` 0= xy + 7` 0= xy 5
8) 0=m18 04e=l= + 0:m18
04e=l= 5
1) =0m8h1 4`8 ‘ 8=m8h
1 4`8 5
7) ‘01 d =8wx + =7 d =8wx 5
4) =9e= 7 12r 8 s= ‘ 0:e= 7
12r 8 s= 5
A. ]gsugnv` n`s euntbpnbh`hbflgs y abvbsbflgs sbiubgltgs. Xbepnbdbqug su rgspugst`.
0)8 4m2h4
•7 4m2h4 5
=)7 8=m7 x7 z 0;
7 r 0; s07
5
8) `=ex8 •1 =ex8 5
1) x= y8 •
1 x= y8 5
7)2 x1 y4
41 x0= y: 5
4)8 41`8e4
8 70=`4l8
5
G. ]`hbfl`nbhg n`s gxprgsbflgs sbiubgltgs. Xbepnbdbqug su rgspugst`.
0): x
4`m 5
=)2 y
0;mex ‘ 1w 5
8):`
9mz + : x 5
1)01el
09`x 5
7)00 x
0=`y ‘ =h 5
4)0= xy
4w 5
=4=BIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
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Fpgr`hbflgs pfnblfeb`ngs
0. Wl pfnblfebf gs ul` gxprgsbøl e`tgeàtbh` dfre`a` ` p`rtbr ag n` ulbøl ag afs f eàs v`rb`mngs y
hflst`ltgs, qug sg vblhun`l ` tr`vâs ag n`s fpgr`hbflgs `rbteâtbh`s ag sue`, rgst` y euntbpnbh`hbøl.
Gl ftr`s p`n`mr`s, ul pfnblfebf gs n` ulbøl ag afs f eàs eflfebfs.
0.³ târeblf =.³ târeblf 8.³ târeblf
2 x= ‘ 8; xy + =7 y=
0.0 Nfs pfnblfebfs sg pugagl hn`sbdbh`r sgiól su lóegrf ag târeblfs, sgiól nfs târeblfs qug
hfeprglag y sgiól gn ir`af ag sus v`rb`mngs.
=. [`r` n` `abhbøl y n` sustr`hhbøl gltrg afs f eàs pfnblfebfs, sg sue`l f rgst`l nfs hfgdbhbgltgs ag
nfs târeblfs sgegc`ltgs, gs aghbr, ag nfs târeblfs qug tgli`l n` ebse` v`rb`mng hfl gn ebsef ir`af.
8. [`r` euntbpnbh`r pfnblfebfs sg utbnbz` n` prfpbga`a abstrbmutbv`? gn prfauhtf ag pfnblfebfs pugag
prgsglt`r abdgrgltgs h`sfs3
8.0 Euntbpnbh`hbøl ag ul lóegrf pfr ul pfnblfebf3 sg euntbpnbh` gn lóegrf pfr h`a` ulf ag nfs
târeblfs ag n` gxprgsbøl pfnblfeb`n.
8.= Euntbpnbh`hbøl ag ul eflfebf pfr ul pfnblfebf3 sg euntbpnbh` gn eflfebf pfr tfafs y h`a`
ulf ag nfs târeblfs qug dfre`l gn pfnblfebf, gs aghbr, hfgdbhbgltg pfr hfgdbhbgltg y v`rb`mng pfr
v`rb`mng.
8.8 Euntbpnbh`hbøl gltrg pfnblfebfs3 sg euntbpnbh` h`a` târeblf agn prbegr pfnblfebf pfr tfafs
y h`a` ulf ag nfs târeblfs agn sgiulaf pfnblfebf.
1. [`r` abvbabr ul pfnblfebf pfr ul eflfebf, sg abvbag h`a` ulf ag nfs târeblfs agn pfnblfebf
gltrg gn eflfebf.
Gcgrhbhbf 8@. ]gsugnv` n`s sbiubgltgs sue`s y rgst`s ag pfnblfebfs. Qgli` huba`af hfl nfs sbilfs.
0) ‘(0=`1 + 9`= ‘ : ) + (:`1 + 2`= ‘ 08) =) (:e8 ‘ e= + =e) + (0:e1 + 09e8 ‘ =7e)
8) (: y8 ‘ 00 y= + 01) ‘ (: y8 + 0; y= ‘ y) 1) (=`x= + 01`8e) + ( x= ‘ 00`8e + e8)
7) (9 x= y= + 2 x8 y= + 00) ‘ (0= x= y= + 8 x8 y= ‘ y=) 4) ‘(=7o 7 + =0o 8 + 8=) ‘ (09o 7 ‘ :o 8 + 9o = + o )
=48E`tgeàtbh` ∝ Xge`l` 09
8/20/2019 Libro Polochic Matemática 1er. Sem
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M. ]gsugnv` n`s euntbpnbh`hbflgs pfnblfeb`ngs sbiubgltgs. Xbepnbdbqug sus rgspugst`s. [ugag utbnbz`r
jfc`s `p`rtg p`r` gn prfhgabebgltf.
0) (‘9)(: x8 + 4 x= ‘ x) 5
=) (08`m1)(:t = x + ty= + 2t ) 5
8) (=rt 8 + t 8w=)(:o =e ‘ 2l=) 5
1) (‘1)(00m8 x ‘ 08`= y ‘ = z ) 5
7) (7`=)(4`8 ‘ :`= ‘ 00` + =) 5
4) (:o 8 + 4o = ‘ 1o )(2o = + o ‘ 8) 5
9) 0=(:m8 + 9h= ‘ 0=m=h ‘ 0=) 5
:) :(0= x8 y= z + 8 x= y= ‘ 4 x + z ) 5
2) (:wx
8
)(2w
8
+ 9 x
=
+ 00wx ‘ =) 5
0;) (7 x1 ‘ 8 x= + = x)(01 x8 + 0= x ‘ 7) 5
00) (00m=h= ‘ 2m + 7h)(=m= + 7m + = ) 5
0=) (9e8l= + 2el)(0;ex8 ‘ =e8l= + el) 5
H. ]gsugnv` n`s abvbsbflgs pfnblfeb`ngs sbiubgltgs. Xbepnbdbqug sus rgspugst`s. [ugag utbnbz`r jfc`s
`p`rtg p`r` gn prfhgabebgltf.
0) (12m1 ‘ =0m8 + 9m) ù (9m=) 5
=) (2`h= + =1`= ‘ 4h=) ù (8`=h) 5
8) (0;l9 ‘ 0=l8 ‘ 71l= x) ù (=l=) 5
1) (‘84 p4 + 4 p1 + 1: p8 ‘ p) ù (4 p8) 5
7) (‘41a 4 + :`a 8 ‘ 04a + 1) ù (:a 8) 5
4) (09e=l= ‘ 9l8 + 7e= ‘ :) ù (el=) 5
9) (8r 1 s + 1: s8 ‘ 71r 8 ‘ =9r = s=) ù (8r =s) 5
:) (‘0=7 y= z 8 ‘ 0;; yz = + =; yz ) ù (7 y= z =) 5
2) (0=e: ‘ 8e9 ‘ 9e7 + :e= + :) ù (e8) 5
0;) (12r 1w1 ‘ =:r 8w= + 01rw= + 4r =w) ù (9rw) 5
00) (07o 4e4 ‘ 02o 7e8 + ==o =e= ‘ o =) ù (oe=) 5
0=) (7;a 1 i 7 ‘ 41a 8 i 8 + 0=a = i = ‘ 011a = i ) ù (=a = i ) 5
=41BIG] ∝ [fnfhjbh
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[rfauhtfs lft`mngs
Nfs prfauhtfs lft`mngs sfl h`sfs gspghb`ngs ag n` pftglhb`hbøl y euntbpnbh`hbøl ag pfnblfebfs qug
huepngl hfl rgin`s c`s y qug sg pugagl rgsfnvgr pfr sbepng blspghhbøl, gs aghbr, rgsfnvgr eglt`negltg
n`s fpgr`hbflgs, sbl lghgsba`a ag gshrbmbr tfaf gn prfhgabebgltf.
Nfs prfauhtfs lft`mngs qug gstuab`efs sg rgsuegl gl n` t`mn` sbiubgltg3
Gcgrhbhbf 1@. Ags`rrfnng gn hu`ar`af ag n` sue` y ag n` abdgrglhb` ag nfs mblfebfs ag dfre` abrght`. Ng a`efs
ul gcgepnf.
;) (7 = + 8m8)= 5
0) ( x + 2)= 5
=) ( y + 4)= 5
8) (7 x + 8)= 5
1) (= y + 8 z )= 5
7) (: x + 7 y)= 5
4) (2 x + = y)= 5
9) (1 x= + 7 y)= 5
:) (: x8 + 8 y=)= 5
2) ( 8
7j8 +
=
2o 8)
=
5
0;) ( 1
8w= +
2
= x=)
=
5
00) (9j ‘ o )= 5
0=) (m ‘ h)= 5
08) (7 x ‘ y)= 5
01) (2 ‘ 7 z )= 5
07) (4w ‘ 4)= 5
04) (7 p ‘ =r )= 5
09) (2 ‘ 7 z )= 5
0:) (=j= ‘ 8o =)= 5
02) (7 x= ‘ y8)= 5
=;) ( 7
8 x8 ‘
8
1 y7)
=
5
=0) ( 8
7w= ‘
9
2 x=) 5
Lfemrg [rfauhtf lft`mng Døreun` iglgr`n
Hu`ar`af ag n` sue` ag ul mblfebf ( + m)= 5 `= + =`m + m=
Hu`ar`af ag n` abdgrglhb` ag ul mblfebf ( ‘ m)= 5 `= ‘ =`m + m=
[rfauhtf ag n` sue` pfr n` abdgrglhb` ( + m)( ‘ m) 5 `= ‘ m=
[rfauhtf ag afs mblfebfs hfl ul târeblf
hfeól y sbilfs biu`ngs
( + m)( + h) 5
( ‘ m)( ‘ h) 5
`= + `(m + h) + mh
`= + `(‘ m ‘ h) + (‘ m)(‘ h)
Humf ag n` sue` ag ul mblfebf ( + m)8 5 `8 + 8`=m + 8`m= + m8
Humf ag n` abdgrglhb` ag ul mblfebf ( ‘ m)8 5 `8 ‘ 8`=m + 8`m= ‘ m8
=7`1 + 8;`=m8 + 2m4
=47E`tgeàtbh` ∝ Xge`l` 09
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M. Ags`rrfnng gn prfauhtf ag n` sue` pfr n` abdgrglhb` ag dfre` abrght`. Qbglg ul gcgepnf.
;) (7e + 8l)(7e ‘ 8l) 5
0) (7 + t )(7 ‘ t ) 5
=) (= x + y)(= x ‘ y) 5
8) (: y + x)(: y ‘ x) 5
1) (= x + 1)(= x ‘ 1) 5
7) (1 z + 1)(1 z ‘ 1) 5
4) (8 p + 4)(8 p ‘ 4) 5
9) (:e + 7)(:e ‘ 7) 5
:) ( 7
8e + 1)( 7
8e ‘ 1) 5
2) ( 0=
w + 8 x)( 0=
w ‘ 8 x) 5
0;) (: z + :)(: z ‘ :) 5
00) (7w + 4)(7w ‘ 4) 5
0=) (1q + 7)(1q ‘ 7) 5
08) (8 x= + y)(8 x= ‘ y) 5
01) (1 x= + 0)(1 x= ‘ 0) 5
07) (0; + 1 )(0; ‘ 1 ) 5
04) (8 x1 + = y=)(8 x1 ‘ = y=) 5
09) (7e= + :l=)(7e= ‘ :l=) 5
0:) ( 8
1e + 8l)( 8
1e ‘ 8l) 5
02) ( 12 y + 0
7 z )( 1
2 y ‘ 0
7 z ) 5
H. Ags`rrfnng gn prfauhtf ag afs mblfebfs hfl ul târeblf hfeól. Qbglg ul gcgepnf.
;) ( + :)( + 7) 5
0) (h + 2)(h + :) 5
=) ( x + :)( x + 4) 5
8) (m + 4)(m + 1) 5
1) ( y= + 8)( y= + 8) 5
7) (a + 1)(a + 0;) 5
4) ( z 8 + 0)( z 8 + 0;) 5
9) (w= + 7)(w= + 1) 5
:) (e ‘ 7)(e ‘ =) 5
2) (= x ‘ 1)(= x ‘ 0) 5
0;) (: y ‘ =)(: y ‘ 8) 5
00) (= x ‘ 7)(= x ‘ 9) 5
0=) (1 y ‘ 4)(1 y ‘ :) 5
08) (= x= ‘ 2)(= x= ‘ 0) 5
01) (9w= ‘ 1)(9w= ‘ =) 5
07) (8 y= ‘ 0)(8 y= ‘ 7) 5
A. Ags`rrfnng n` sue` y abdgrglhb` ag humfs. Qbglg ul gcgepnf.
;) (=` + 7)8 5
0) ( x + 8)8 5
=) (e + l)8 5
8) (w + 4)8 5
1) (7 y + =)8 5
7) (8 z + 8)8 5
4) (1l + 7)8 5
9) (1e + 0)8 5
:) (=r ‘ 0)8 5
2) (7 x ‘ =)8 5
0;) (8 y ‘ 7)8 5
00) (1 z = ‘ 1)8 5
0=) (=m ‘ 4)8 5
08) (7o = ‘ 1)8 5
=7e= ‘ 2l=
`= + 08` + 1;
:`8 ‘ 4;`= + 87;` + 0=7
=44BIG] ∝ [fnfhjbh
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D`htfrbz`hbøl B
0. D`htfrbz`r ul` gxprgsbøl sg rgdbgrg `n prfhgsf ag sbepnbdbh`rn j`nn`laf afs f eàs d`htfrgs, huyf
prfauhtf gs biu`n ` n` gxprgsbøl prfpugst`. Iglgr`negltg sg gepng` gstg prfhgabebgltf p`r` tr`-
m`c`r hfl pfnblfebfs.
J`y abdgrgltgs eâtfafs f "h`sfs" p`r` d`htfrbz`r, agpglabglaf ag n` gxprgsbøl qug agsggefs sbe-pnbdbh`r. Gstuab`efs hu`trf3
0.0 D`htfr hfeól eflfebf3 gstg eâtfaf hflsbstg gl gxtr`gr ag ul pfnblfebf, ul eflfebf
hfef d`htfr hfeól ` h`a` ulf ag nfs târeblfs agn pfnblfebf gl hugstbøl. Nugif, sg agc`
blabh`a` n` euntbpnbh`hbøl gltrg gn d`htfr hfeól gxtr`åaf y nfs târeblfs qug quga`l ag n`
gxprgsbøl frbibl`n.
0.= D`htfr hfeól pfr `irup`hbøl ag târeblfs3 gepng`efs gstg h`sf sb nfs eflfebfs qug
hfepflgl gn pfnblfebf pugagl rgulbrsg gl irupfs hfl ul d`htfr hfeól abdgrgltg gl h`a`
irupf. Nugif, gepng`efs n` ebse` tâhlbh` qug gl gn d`htfr hfeól.
0.8 Qrblfebf hu`ar`af pgrdghtf3 sg bagltbdbh` pfr tglgr trgs târeblfs, ag nfs hu`ngs afs tbglglr`åhgs hu`ar`a`s gx`ht`s, y gn rgst`ltg gqubv`ng `n afmng agn prfauhtf ag n`s r`åhgs ag nfs ftrfs
eflfebfs.
( x + y)= 5 x = + = xy + y=
[`r` d`htfrbz`r, agmgefs rgfragl`r nfs târeblfs agc`laf ag prbegrf y ag tgrhgrf nfs târeblfs
qug tgli`l r`åz hu`ar`a`, nugif gxtr`gefs n` r`åz hu`ar`a` agn prbegr y tgrhgr târeblf y nfs
gshrbmbefs gl ul p`râltgsbs, sgp`ràlafnfs pfr gn sbilf qug `hfep`ð` `n sgiulaf târeblf? `n
hgrr`r gn p`râltgsbs gngv`efs tfaf gn mblfebf `n hu`ar`af.
0.1 Abdgrglhb` ag hu`ar`afs3 mblfebf gl gn qug sus târeblfs tbglgl ul` rgst` ag pfr egabf y sgngs pugag h`nhun`r r`åz hu`ar`a`.
[`r` n` d`htfrbz`hbøl ag gstg h`sf agmgefs gxtr`gr n` r`åz hu`ar`a` ag nfs târeblfs. Nugif,
agc`r blabh`a` n` euntbpnbh`hbøl gltrg n` sue` y n` abdgrglhb` ag gst`s r`åhgs.
`= ‘ m= 5 ( + m)( ‘ m)
Gcgrhbhbf 7@. Glhugltrg gn d`htfr hfeól ag nfs pfnblfebfs sbiubgltgs. Gshrbm` gn rgsunt`af sfmrg n` nålg`.
0) xy= ‘ y=w 5
=) 7 xy= ‘ 07 y 5
8) =1`8m= ‘ 0=`8m8 5
1) 07 y8 + =; y= ‘ 7 y 5
7) 1 x= y ‘ : xy= ‘ 0= xy8 5
4) =`= x + =`x= ‘ 8`x 5
9) 2 x8 ‘ 4 x= + 0= x7 ‘ 0: x9 5
:) 2 x=`m ‘ 8 x`=m8 + 4 x=`m 5
2) 04`1m7 ‘ =;`8m= ‘ =1`=m4 5
0;) =7 x9 ‘ 0; x7 + 0= x8 ‘ 0: x= 5
=49E`tgeàtbh` ∝ Xge`l` 09
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M. D`htfrbhg nfs pfnblfebfs pfr `irup`hbøl ag târeblfs.
H. D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gepngg gn prfhgabebgltf ag ul trblfebf hu`ar`af pgrdghtf.
0) y= + 0= y + 84 5
=) 1`= + 0=` + 2 5
8) =7 x= ‘ 1; x + 04 5
1) 41 + 8= x + 1 x= 5
7) 1 x= ‘ 0= xy + 2 y= 5
4) 2e= ‘ 4ey + y= 5
9) 84 x= + =1 xy + 1 y= 5
:) 2e= ‘ 0=el + 1l= 5
2) 12r = + 00=rs + 41 s= 5
0;)2
1 x= ‘
07
1 x +
=7
04 5
00)=7
84 y= ‘
7
4 y +
0
1 5
0=) 84 x4 + =1 x8 y= + 1 y1 5
08) =7w: ‘ 0;;w1 + 0;; 5
01) 1 x1 y4 ‘ : x= y8 z = + 1 z 1 5
A. D`htfrbhg nfs pfnblfebfs sbiubgltgs. Gl `niulfs blhbsfs agmgrà gepng`r eàs ag ul h`sf ag
d`htfrbz`hbøl.
0) w= ‘ x= 5
=) x= ‘ 04 5
8) y= ‘ z = 5
1) z = ‘ 04 5
7) 1 x= ‘ 0;; 5
4) =7 y= ‘ 1 5
9) 1`= ‘ 04m= 5
:) x= y= ‘ =7 z = 5
2)2
1 x= ‘
0
=7 5
0;) 04w1 x= ‘ 41 5
00) y=
1 ‘ 04 z = 5
0=) `:
=7 ‘
m=h=
2 5
08) 0 ‘e=l=
84 5
01) :0w
1 x
= ‘ 12 y4 z = 5
0) `x ‘ x ‘ ` + 0 5
=) xy + 7 x ‘ y ‘ 7 5
8) xy + y= ‘ x ‘ y 5
1) xy ‘ x + 7 y ‘ 7 5
7) 8 x + 0= ‘ xy ‘ 1 y 5
4) 7`x + `y ‘ 7mx ‘ my 5
9) =er + es + 4lr + 8ls 5
:) 7 y ‘ 0; ‘ xy + = x 5
2) xy + 8 y + zx + 8 z 5
0;) 4e= x + 1e= y + 8lx + =ly 5
00) px ‘ pq ‘ = yx + = yq 5
0=) 4` ‘ 4m + 8`x ‘ 8 xm 5
08) 9 xe ‘ =0 x ‘ 1e + 0= 5
01) =lx + 01 x + 8lx= + =0 x= 5
=4:BIG] ∝ [fnfhjbh
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D`htfrbz`hbøl BB
0. D`htfrbz`r ul trblfebf ag n` dfre` x = + mx + h sbilbdbh` gxprgs`rnf hfef gn prfauhtf ag afs d`htfrgs
ag n` dfre` ( x + `)( x + m). [`r` d`htfrbz`r trblfebfs ag gst` dfre` agmgefs3
`. Agshfepflgr gn trblfebf gl afs d`htfrgs (mblfebfs). [`r` gnnf fmtglgefs n` r`åz hu`ar`a` agn
prbegr târeblf.
m. Hfpb`r gn sbilf agn sgiulaf târeblf gl gn prbegr d`htfr. Nugif, euntbpnbh`r gn sbilf agn sgiulaf
târeblf pfr gn sbilf agn tgrhgr târeblf y gshrbmbrnf gl gn sgiulaf d`htfr.
h. Gngibr afs lóegrfs qug sue`afs f rgst`afs agl hfef rgsunt`af gn târeblf m y `n euntbpnbh`rnfs
agl hfef rgsunt`af gn târeblf h.
=. Nfs trblfebfs ag n` dfre` `x = + mx + h sg d`htfrbz`l hfl nfs p`sfs sbiubgltgs3
`. Euntbpnbh`r gn trblfebf pfr gn hfgdbhbgltg ( ).
m. Gxprgs`r gn prbegr târeblf rgsunt`ltg, hfef ( x)= y gl gn sgiulaf târeblf hfleut`r gn fragl `mx
pfr m`x.
h. D`htfrbz`r gn pfnblfebf qug sg fmtuvf gl nfs p`sfs `ltgrbfrgs, hfef sg jbzf hfl gn trblfebf ag
n` dfre` x= + mx + h.
a. Abvbabr nfs prfauhtfs gltrg nfs d`htfrgs ag `, qug gs gn lóegrf pfr gn hu`n sg euntbpnbhø gn trblf-
ebf.
8. N` sue` ag humfs `8 + m8 sg d`htfrbz` ag `hugraf hfl n` døreun` sbiubgltg3
`8 + m8 5 ( + m)( = ‘ `m + m=)
`. Gl gn prbegr d`htfr sg gshrbmg n` sue` agn rgsunt`af ag n` r`åz hómbh` agn prbegr y sgiulaftâreblf ( + m).
m. Gl gn sgiulaf d`htfr agmg br gn prbegr târeblf `n hu`ar`af eglfs gn prbegrf pfr gn sgiulaf,
eàs gn sgiulaf `n hu`ar`af ( = ‘ `m + m=).
1. N` abdgrglhb` ag humfs `8 ‘ m8 sg d`htfrbz` ag `hugraf hfl n` døreun` sbiubgltg3
`8 ‘ m8 5 ( ‘ m)( = + `m + m=)
`. Gl gn prbegr d`htfr sg gshrbmg n` rgst` agn rgsunt`af ag n` r`åz hómbh` agn prbegr y sgiulaf
târeblf ( ‘ m).
m. Gl gn sgiulaf d`htfr agmg br gn prbegr târeblf `n hu`ar`af eàs gn prbegrf pfr gn sgiulaf, eàsgn sgiulaf `n hu`ar`af ( = + `m + m=).
=42E`tgeàtbh` ∝ Xge`l` 09
8/20/2019 Libro Polochic Matemática 1er. Sem
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Gcgrhbhbf 4@. D`htfrbhg nfs trblfebfs ag n` dfre` x= + mx + h y gshrbm` gn rgsunt`af sfmrg n` nålg`.
0) `= + 00` + =1 5
=) m= + 0=m + 87 5
8) w= ‘ 0;w + =1 5
1) x= ‘ 00 x + 0: 5
7) y= + 1 y ‘ 8= 5
4) z = + 9 z ‘ 4; 5
9) e= + 8e ‘ 71 5
:) l= ‘ 8l ‘ 9; 5
2) p= ‘ 2 p ‘ 84 5
0;) r = ‘ 9r ‘ 0: 5
00) s= ‘ 7 s ‘ 07; 5
0=) x= + = x ‘ 1: 5
M. D`htfrbhg nfs trblfebfs ag n` dfre` `x= + mx + h y gshrbm` gn rgsunt`af sfmrg n` nålg`.
0) :`= + 0:` + 8 5
=) 4e= + =8e + 9 5
8) 7l= + 09l + 4 5
1) 8w= + 1w ‘ 1 5
7) 4 p= + 09 p ‘ =: 5
4) 4 x= ‘ 08 x + 4 5
9) 7q= ‘ =8q ‘ 0; 5
:) 4 y= + 00 y + 8 5
2) :r = ‘ 89r ‘ 07 5
0;) : z = + 7 z + 0 5
00) 4j= ‘ 07j + 4 5
0=) 8o = + =8o + 8; 5
08) 0= x= ‘ 9 x + 0 5
01) 1 y= + : y + 8 5
07) =h= ‘ 7h + 8 5
04) 4`= + 9` + = 5
09) 7w= ‘ 9w + 0= 5
0:) 4 x= + 09 x ‘ 8 5
H. D`htfrbhg n` sue` y abdgrglhb` ag humfs y gshrbm` gn rgsunt`af sfmrg n` nålg`.
0) `8 + m8 5
=) : + h8 5
8) e8 + 41 5
1) : x8 + =9 y8 5
7) z 8 + 0=7 5
4) 41o 8 + 0 5
9) :w8 + x8 5
:) x8 y8 + =9 z 8 5
2) :`8m8 + 41h8a 8 5
0;) j8 ‘ o 8 5
00) =9 ‘ z 8 5
0=) :`8 ‘ m8h8 5
08) 0=7 ‘ `8 5
01) :e8 ‘ l8 5
07) 41 x8 ‘ : y8 5
04) =04 ‘ m8 5
=9;BIG] ∝ [fnfhjbh
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Dr`hhbflgs `nigmr`bh`s B
0. Wl` dr`hhbøl `nigmr`bh` gs ul` gxprgsbøl dr`hhbfl`rb` gl n` qug luegr`afr y aglfebl`afr sfl
gxprgsbflgs `nigmr`bh`s, iglgr`negltg pfnblfebfs.
[fr gcgepnf3 x= + 0; x + =7
x= ‘ y=
Afs dr`hhbflgs sfl gqubv`ngltgs sb gn prfauhtf hruz`af gltrg `em`s gs biu`n.
0.0 Xbepnbdbh`hbøl ag gxprgsbflgs nigmr`bh`s3 p`r` sbepnbdbh`r ul` gxprgsbøl r`hbfl`n, d`htfrbz`-
efs luegr`afr y aglfebl`afr, y nugif h`lhgn`efs d`htfrgs hfeulgs.
0.= Euntbpnbh`hbøl ag dr`hhbflgs `nigmr`bh`s3 gn prfauhtf ag afs dr`hhbflgs `nigmr`bh`s gs ftr`
dr`hhbøl `nigmr`bh` gl n` qug gn luegr`afr gs gn prfauhtf ag nfs luegr`afrgs y gn aglfebl`-
afr gs gn prfauhtf ag nfs aglfebl`afrgs. Gs aghbr, agmgefs euntbpnbh`r "gl nålg`".
Gn prfhgabebgltf gs gn sbiubgltg3
• D`htfrbz`efs nfs luegr`afrgs y nfs aglfebl`afrgs j`st` aflag sg` pfsbmng.
• Blabh`efs n` euntbpnbh`hbøl agn luegr`afr pfr luegr`afr y aglfebl`afr pfr aglfebl -
afr.
• Xbepnbdbh`efs gn rgsunt`af j`st` rgauhbrnf ` su eålbe` gxprgsbøl.
[( x)
P( x)•
]( x)
X( x)5
[( x) • ]( x)
P( x) • X( x)
0.8 Abvbsbøl ag dr`hhbflgs `nigmr`bh`s3 sg euntbpnbh` n` prbegr` dr`hhbøl pfr gn blvgrsf ag n` sg-
iula` dr`hhbøl. Aghbefs gltflhgs qug n` abvbsbøl gs ul` euntbpnbh`hbøl hruz`a` y pfr t`ltf,
utbnbz`efs n` ebse` tâhlbh` qug `prglabefs gl n` euntbpnbh`hbøl.
[( x)
P( x)ù
]( x)
X( x)5
[( x) • X( x)
P( x) • ]( x)
Gcgrhbhbf 9@. Xbepnbdbqug n`s gxprgsbflgs r`hbfl`ngs gl su hu`agrlf. Nugif, gshrbm` n` rgspugst` sfmrg n` nålg`.
0)1` + 1
1
5
=)8 y
8 y= + 4 y 5
8) x= ‘ 2 y=
( x + 8 y)= 5
1) ( ‘ m)=
`= ‘ m= 5
7) y= ‘ 8 y + 1 y ‘ 0=
y ‘ 8
5
4) e= + 8e ‘ 0:
8e ‘ 2 5
9) 4 z = ‘ 9 z ‘ 7
= z + 0 5
:) w= + =wx ‘ 8 x=
w= + 7wx + 4 x= 5
=90E`tgeàtbh` ∝ Xge`l` 09
8/20/2019 Libro Polochic Matemática 1er. Sem
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M. Euntbpnbqug n`s gxprgsbflgs r`hbfl`ngs y sbepnbdbqug gn rgsunt`af gl su hu`agrlf. Nugif gshrbm` n`
rgspugst` sfmrg n` nålg`.
0)8 x ‘ =
8 x + = •
1 x ‘ 0
0 ‘ 1 x 5
8) x
x= ‘ y= •
x + y
x= + xy 5
7)
y= ‘ 84
y= + y ‘ 8; •
y ‘ 7
8 y 5
=) x8 + :
x= ‘ x ‘ 4 •
x + 8
x= ‘ = x + 2 5
1) x= + 8 x + =
x1 y •
x8 y=
x= + 1 x + 8 5
4)
x= + x ‘ 4
x= ‘ 7 x + 4 •
x= ‘ = x ‘ 8
x= ‘ 1 x ‘ 7 5
H. Abvba` n`s gxprgsbflgs r`hbfl`ngs y sbepnbdbqug gn rgsunt`af gl su hu`agrlf. Nugif, gshrbm` n`
rgspugst` sfmrg n` nålg`.
0)
x= + = x ‘ :
x= ‘ 8 x ‘ 1 ù
x= ‘ 1 x + 1
x= + 4 x + : 5
8) x= ‘9 x + 0;
x= ‘ 4 x + 7 ù
x= + 7 x ‘ 01
x= + : x + 9 5
7) y= + 8 y + =
y= ‘ 7 y + 1 ù
y= + 4 y ‘ 04
y= + y ‘ =; 5
=)
x= ‘ 1 x + 8
8 x= ‘ x ‘ = ù
x= + 0; x + =1
x= + 8 x ‘ 0: 5
1)0
x= + 9 x ‘ 0: ù
0
x= ‘ 09 x + 8; 5
4)9 x= ‘ 07 x + =
x= + x ‘ 4 ù
x= ‘ 8 x ‘ 0;
x= ‘ = x ‘ 07 5
=9=BIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
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Dr`hhbflgs `nigmr`bh`s BB
Gl n` sue` y rgst` ag dr`hhbflgs nigmr`bh`s n biu`n qug gl n`s dr`hhbflgs rbteâtbh`s pfagefs fpgr`r
dr`hhbflgs hfl biu`n y abdgrgltg aglfebl`afr.
0. Gl n` sue` f rgst` ag dr`hhbflgs `nigmr`bh`s hfl biu`n aglfebl`afr, sg sue`l f rgst`l nfs
luegr`afrgs y sg hfpb` gn aglfebl`afr hfeól.
`
h +
m
h 5
` + m
h
=. [`r` sue`r f rgst`r dr`hhbflgs `nigmr`bh`s hfl abdgrgltg aglfebl`afr, sgiubefs nfs p`sfs
sbiubgltgs3
`. D`htfrbz`r nfs aglfebl`afrgs p`r` j`nn`r gn eålbef hfeól eóntbpnf (e.h.e.) gn hu`n sgrà gn
aglfebl`afr hfeól.
m. Abvbabr gn e.h.e. gltrg gn aglfebl`afr ag h`a` târeblf. Nugif, euntbpnbh`r gn rgsunt`af pfr h`a`
luegr`afr.
h. Xbepnbdbh`r gn rgsunt`af.
8. N`s dr`hhbflgs hfepugst`s f hfepngc`s sfl n`s qug tbglgl dr`hhbflgs gl gn luegr`afr f gn
aglfebl`afr, f gl `em`s p`rtgs ag n` dr`hhbøl.
` + m`
` ‘ m`
Nålg` prblhbp`n agn` dr`hhbøl
Luegr`afr ag n`dr`hhbøl hfepngc`
Aglfebl`afr ag n`dr`hhbøl hfepngc`
[`r` sbepnbdbh`r f rgauhbr ul` dr`hhbøl hfepngc` ` sus târeblfs eàs sglhbnnfs y qug sg` gqubv`ngltg
` n` gxprgsbøl frbibl`n, agmgefs sgiubr gstfs p`sfs3
`. Glhfltr`r gn eålbef hfeól eóntbpnf ag tfafs nfs aglfebl`afrgs.
m. Euntbpnbh`r gn luegr`afr y gn aglfebl`afr ag n` dr`hhbøl hfepngc` pfr gn e.h.e.
h. Xbepnbdbh`r nfs rgsunt`afs hu`laf sg` pfsbmng.
=98E`tgeàtbh` ∝ Xge`l` 09
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Gcgrhbhbf :@. Xueg n`s gxprgsbflgs r`hbfl`ngs hfl biu`n y abdgrgltg aglfebl`afr gl su hu`agrlf. Nugif,
gshrbm` n` rgspugst` sfmrg n` nålg`.
0) x
x + =
+8
x + =
5
=) x
x + 8 +
0
x + 8 5
8)= x ‘ 8
= x + 7 +
:
= x + 7 5
1)1 x= + 8 x
7 x + = +
x= ‘ x
7 x + = 5
7) = x ‘ 0
x= ‘ 8 x ‘ 1 + 8
x= ‘ 8 x ‘ 1 5
4)7 x ‘ 9
4 x= ‘ 00 x + 8 +
0 ‘ x
4 x= ‘ 00 x + 8 5
9)8
1 x ‘ 8
+0
8 x + 0
5
:) x
x + = +
8
= x ‘ 8 5
2) x
= x + 8 +
0
x ‘ = 5
0;) x ‘ =
1 x= + 7 x + 0 +
1
1 x= + 2 x + = 5
00) ‘1 x + 9
= x= + 7 x + = + x=
8 x= + 1 x ‘ 1 5
0=) x ‘ =
x= ‘ 7 x ‘ =1 +
8
x= + 00 x + =1 5
M. ]gstg n`s gxprgsbflgs r`hbfl`ngs hfl biu`n y abdgrgltg aglfebl`afr gl su hu`agrlf. Nugif, gshrbm`
n` rgspugst` sfmrg n` nålg`.
0)1 x
1 x ‘ 8 ‘
8
1 x ‘ 8 5
=)= x + 0
8 x ‘ 9 ‘
x + :
8 x ‘ 9 5
8)= x= + 0
1 x= ‘
= x= ‘ 0
1 x= 5
1)4 x=
4 x ‘ 9
‘9 x
4 x ‘ 9
5
7) x=
1 x= ‘ 0 ‘
x ‘ x=
1 x= ‘ 0 5
4) x= + 8 x
x= + x ‘ 0= ‘
x ‘ 0=
x= + x ‘ 0= 5
9) x + 0
4 x ‘
= x + 8
1 x 5
:) x + 8
8 x ‘
7 x ‘ 0
7 x= 5
2)1 x ‘ 7
= x= ‘ 7 x ‘ 8 ‘
x
= x= ‘ 7 x ‘ 8 5
0;) x ‘ =
1 x=
+ 7 x + 0
‘1
1 x=
+ 2 x + =
5
00)8 x ‘ =
x= ‘ 8 x ‘ 1 ‘
x + =
x= ‘ 7 x + 1 5
0=)9 x + 01
8 x= + 0; x ‘ : ‘
x + 4
8 x= + x ‘ = 5
=91BIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
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@. ]gsugnv` n`s fpgr`hbflgs hfembl`a`s sbiubgltgs. ]ghugrag sbgeprg n` cgr`rquå` ag n`s fpgr`hbflgs.
0) ‘2 + 7 Ù 0 + 8 5
=) 7 Ù 4 ù (‘=) + 4 5
8) 4 Ù 7 ‘ =; ù 7 5
1) 1 Ù 4 ù = + 0 5
7) 8 ù 8 + 1 ‘ 4 5
4) ‘8 ‘ 4 ù = ‘ 9 5
9) 2 Ù : ‘ 9 Ù 4 5
:) 8 + ; ‘ 7 ‘ 0 5
2) 9 Ù : ‘ 1 ‘ 0 5
0;) ‘= + 0; ù 7 5
00) = ‘ 8 + : ù (‘:) 5
0=) 1 ‘ 0 Ù ; ‘ 8 5
08) ‘: ‘ 1 + 4 ù = 5
01) ‘0 Ù 4 + = ‘ 8 5
07) 2 Ù 9 + 2 ù 8 5
04) ; ù 0 + : Ù 2 5
M. ]gsugnv` n`s fpgr`hbflgs hfembl`a`s sbiubgltgs. ]ghugrag n` cgr`rquå` ag nfs sbilfs ag irup`hbøl.
0) _(2 + :) ‘ 1U Ù 8 5
=) (=; ù 1) ‘ 4 Ù = 5
8) 0 + 8 Ù (0 + 1) 5
1) 0; ù 7 ‘ (7 ‘ 1) 5
7) _(4 ‘ 9) Ù 7U + 1 5
4) (8 Ù 0) + = ‘ 0 5
9) ; ‘ _(1 + 1) ‘ 4U 5
:) = Ù 0 + (8 ‘ =) 5
2) (= Ù 8) ‘ (= + 8) 5
0;) _(2 ‘ 4) + 9U Ù 1 5
00) (7 ‘ 4) + 1 ‘ = 5
0=) _7 + (: ù 1)U ‘ 1 5
08) 9 ‘ _2 ù (4 ‘ 8)U 5
01) = Ù = ‘ (0 + :) 5
07) 7 Ù 1 ‘ (; + 0) 5
04) (9 + 8) Ù (0; ù 7) 5
H. ]gsugnv` n`s ghu`hbflgs sbiubgltgs. Gshrbm` su rgspugst` gl n` nålg` hfrrgspflabgltg.
0) ‘8 + e 5 0; e 5
=) : ‘ e 5 ‘0: e 5
8) 2 + e 5 ; e 5
1) 0 ‘ e 5 04 e 5
7) : + e 5 : e 5
4) ‘2 ‘ e 5 =0 e 5
9) 2 + e 5 02 e 5
:) ‘9 ‘ e 5 ‘09 e 5
2) ‘ e + 0: 5 01 e 5
0;) e ‘ 9 5 ‘=8 e 5
00) e + 00 5 == e 5
0=) ‘ e ‘ 4 5 =9 e 5
08) e + : 5 0= e 5
01) e + ; 5 ‘2 e 5
07) ‘ e + : 5 ‘= e 5
04) e ‘ : 5 08 e 5
=97E`tgeàtbh` ∝ Xge`l` 09
@ibnba`a ag hànhunf eglt`n
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Hn`vgs
=99E`tgeàtbh` ∝ Hn`vgs
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Xge`l` 0¡@ tr`m`c`r!0) 04 ag cunbf
=) Ng prgsglt`efs n` dfre` eàs abrght` ag rgsfnvgr gn prfmnge`3
@nmgrtf s`mg gn egs y Mgrl`raf gn aå`. Gl n` prbegr``dbre`hbøl, @nmgrtf gstà sgiurf qug Mgrl`raf lf s`mghuàlaf gs gn huepng`ðfs. Gstf pgrebtg agsh`rt`re`yf y culbf, pfrqug sb Mgrl`raf s`mg gn aå`, nfsólbhfs lóegrfs qug lf sg rgpbtgl sfl gn 02 y gn 0:,sb dugr` ulf ag gsfs aå`s Mgrl`raf så nf s`mrå`.
Hfl e`yf y culbf agsh`rt`afs, `jfr` Mgrl`rafabhg qug ân så s`mg huàlaf gs gn huepng`ðfs. Gstfagsghj` ul` dghj` qug nngvg gn 01, pfrqug `p`rghgafs vghgs, gn 01 ag cunbf y gn 01 ag `ifstf.
Gltflhgs n`s dghj`s pfsbmngs sfl gn 04 ag cunbf, 07 ag`ifstf, y 09 ag `ifstf.
Nugif, @nmgrtf abhg qug sb Mgrl`raf s`mg, ân t`embâlnf s`mg. Gltflhgs sb `emfs gstàl sgiurfs lf pugagsgr gl `ifstf pfrqug pfarå` sgr gn 07 f gn 09.
[fr nf t`ltf, gn huepng`ðfs ag @ngc`lar` gs gn 04 ag cunbf.
Gcgrhbhbf 0@.;) Xå 4) Lf0) Lf 9) Xå
=) Lf :) Xå8) Xå 2) Xå1) Xå 0;) Lf7) Lf
M. ]gspugst`s abstblt`s. Hfeprugmg qug nfs glul-hb`afs gshrbtfs huepn`l hfl nfs hrbtgrbfs p`r` sgr
prfpfsbhbflgs. ]gvbsg t`embâl nfs v`nfrgs nøibhfs.
Gcgrhbhbf =Gn gstuab`ltg puaf j`mgr gepng`af abvgrs`s dr`sgs
p`r` gxprgs`r n` lgi`hbøl (gs d`nsf qug, lf gs hbgrtfqug, gth). Ng prgsglt`efs ulfs gcgepnfs3
;) ~t3 N` nnuvb` lf h`g agn hbgnf. D
0) ~ c 3 Gn `iu` agn e r lf gs aunhg. T=) ~ p3 Gl gn `brg lf j`y fxåiglf. D8) ~v 3 Gs d`nsf qug ul` mbhbhngt` tbglg afs ruga`s. D1) ~s3 Nfs r`yfs ag sfn lf a`l h`nfr. D7) ~r 3 N`s rfh`s lf sfl sgrgs vbvfs. T
Gcgrhbhbf 8@.;) Gxbstg `n eglfs ul` x qug pgrtglghg ` M y t`embâl
pgrtglghg ` P.0) [`r` tfa` x qug pgrtglghg ` W, sg huepng qug 7 x
pgrtglghg ` W.=) Gxbstg `n eglfs ul x qug pgrtglghg D y pgrtglghg
t`embâl ` E.
M.;) ∋ x ∊ L , :x ∊ L0) ∈ x ∊ @ ∩ x ∊ M=) ∋ x ∊ ] , 4 x ∊ ] 8) ∈ x ∊ F ∩ x ∊ [
Ags`rrfnng lugv`s j`mbnba`ags0) m=) a8) h1) m7) h
=9:BIG] ∝ [fnfhjbh
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Xge`l` 8¡@ tr`m`c`r!0) Bldblbt`s hbdr`s aghbe ngs lf pgrbøabh`s.
=) Lf gs pfsbmng, pfrqug lf sg fmtbglg ul` r`åz hu`-ar`a` gx`ht`.
Gcgrhbhbf 0;) 1 9) 1.84=1
0) : :) ;
=) =7 2) 8.=:41
8) 72
0;) 8 :
1) 04 00) 1.897
7) =2 0=) =74) :
09
08) 8
7
Gcgrhbhbf =;) hfleut`tbv`0) hn`usur`tbv`=) blvgrsf `abtbvf u fpugstf8) hfleut`tbv` ag n` euntbpnbh hbøl1) blvgrsf euntbpnbh`tbvf7) gngegltf lgutrf ag n` euntbpnbh`hbøl4) `sfhb`tbv`9) abstrbmutbv` agn prfauhtf rgspghtf ` n` sue`
Gcgrhbhbf 8;) :8 + 4 5 :2
0) 0
=) ( 0= )
= + =
5 ( 0= )
1
8) 81
91
1) 7: ‘ 8 5 77
7) 0
4) 29 ‘ 4 5 2
9) =8
78
:) 0:=
2) 0(‘=)1
Gcgrhbhbf 1;) 4 4/8 5 4 = 5 84
0) 411 5 := 5 1
=) 0;= 5 0;
8) 1 04 •1 04 5 = • = 5 1
1) 8•= 8 5 4 8
7)041
5 1= 5 =
4) 7 17 5 1
9) :4/= 5 :8 5 70=
Ags`rrfnng lugv`s j`mbnba`ags
7
9
1 01Å 10 5 791
=:;BIG] ∝ [fnfhjbh
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Xge`l` 1¡@ tr`m`c`r!N`s rgspugst`s pugagl v`rb`r.
Gcgrhbhbf 0@.;) 04? 04 = 5 =74
0) 8? 8= 5 2 =) 1? 1= 5 04 8) 2? 2= 5 :0 1) 0=? 0== 5 011 7) 0:? 0:= 5 8=1 4) 07? 07= 5 ==7 9) =;? =;= 5 1;; :) 8;? 8;= 5 2;; 2) 00? 00= 5 0=0
0;) 4;? 4;
=
5 84;;00) 7;? 7;= 5 =7;;
M.;) 8? 88 5 =9
0) 1? 18 5 41 =) =? =7 5 8= 8) 7? 78 5 0=7 1) 0;? 0;8 5 0;;; 7) 8? 89 5 =0:9 4) ‘8? (‘8)8 5 ‘=9 9) ‘1? (‘1)8 5 ‘41 :) ‘=? (‘=)7 5 ‘8=
2) ‘7? (‘7)8
5 ‘0=70;) ‘0;? (‘0;)8 5 ‘0;;;00) ‘8? (‘8)9 5 ‘=0:9
H.;) 1;;(‘0) 5 1;;b = 5 1;; • b = 5 =;b
0) 04(‘0) 5 04b= 5 04 • b= 5 1b
=) =7(‘0) 5 =7b= 5 =7 • b= 5 7b
8) 84(‘0) 5 84b= 5 84 • b= 5 4b
1) 41(‘0) 5 41b= 5 41 • b= 5 :b
7) 042(‘0) 5 042b= 5 042 • b= 5 08b
4) 0;;(‘0) 5 0;;b= 5 0;; • b= 5 0;b
9) 024(‘0) 5 024b=
5 024 • b=
5 01b
Gcgrhbhbf =0) 8 + 8b 7) = + 0;b
=) : + 4b 4) ‘ 0= + b
8) ‘7 + =b 9) ‘: ‘ 1b
1) 0; ‘ 4b :) ‘07 + =.:8b
Gcgrhbhbf 80) : + 8b + = + 9b 5 : + = + 8b + 9b 5
0; + 0;b
8) ‘1 + 9b ‘ 0 + 8b 5 ‘1 ‘ 0 + 9b + 8b 5 ‘7 + 0;b
7) 8= + 1;b ‘ 1:b ‘ 4;b= 5 8= ‘ :b ‘ 4;b= 5 8= ‘ :b ‘ 4;(‘0) 5 8= ‘ :b + 4; 5 8= + 4; ‘ :b 5 2= ‘ :b
Gcgrhbhbf 1
0) ( 2 + b
b )( b
b ) 5 (2b + b=
b= ) 5 2b + (‘0)
‘0 5 2b
‘0 ‘ 0
‘0 5
‘2b + 0 5 0 ‘ 2b
=) (0; ‘ 7b
b )( b
b ) 5 (0;b ‘ 7b=
b= ) 5
0;b + 7(‘0)‘0 5 0;b
‘0 ‘ 7
‘0 5
‘0;b + 7 5 7 ‘ 0;b
Ags`rrfnng lugv`s j`mbnba`ags
=) 8 + 7b ‘ 2 + =b 5 8 ‘ 2 + 7b + =b 5
‘4 + 9b
1) 07b ‘ 1;b= 5 07b ‘ 1;(‘0) 5 07b + 1; 5 1; + 07b
4) 14
‘ 7b
= + :b
8 ‘ 0;b= 5
=8
+ b
4 ‘ 0;b= 5
=8
+ b
4 ‘ 0;(‘0) 5
=8 + b4 + 0; 5
=8
+ 0; + b
4 5
8=8
+ b
4
: Ù = + 1 5 =;
+ Ù Ù
8 Ù 9 ‘ 0 5 =;
+ + Ù
2 + 4 + 7 5 =;
5 =; 5 =; 5 =;
=:0E`tgeàtbh` ∝ Hn`vgs
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Xge`l` 7¡@ tr`m`c`r!
9 8 = 1 7 0 4 : 2
0 7 4 9 : 2 = 1 8
: 2 1 4 = 8 9 0 7
4 = 2 0 8 : 7 9 1
8 0 9 = 1 7 : 2 4
1 : 7 2 9 4 0 8 =
7 4 : 8 0 1 2 = 9
= 1 0 7 2 9 8 4 :
2 9 8 : 4 = 1 7 0
Gcgrhbhbf 0;) 8 Ù 80) 8 Ù ==) = Ù =8) 0 Ù 11) 8 Ù =7) 1 Ù 1
Gcgrhbhbf =
0) ‘0 ‘12 ‘=
=) : 28 ‘08
8) 0 ‘4= 7
1) 08 == ‘08
7) 0 4 =‘: ‘8 11 1 4
4) 00 0 0‘= ‘8 :1 = 1
9) 9 ; 0=
01 0 1: ‘0 4
:) 0= ‘4 0;
08 = 17 ‘0 =
2) ‘0 01 ‘4‘9 ‘9 :7 0 :
0;) ‘9 01 ‘4= ‘7 11 ‘8 4
Gcgrhbhbf 8[rfpbga`a
;) 00 ‘8 ‘00; 9 ‘0
8 0; ‘0=
`sfhb`tbv`
0) ; ; ;
; ; ;
; ; ;
Gngegltffpugstf
=) 9 : ‘1
0 ; 7
8 4 9
Gngegltflgutrf
Ags`rrfnng lugv`s j`mbnba`ags]gvbsg qug j`y` abmuc`af tfafs nfs agt`nngs ag n`bnustr`hbøl. N` be`igl hfepngt` agmgrå` vgrsg `så3
=:=BIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
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Xge`l` 4 Xge`l` 9¡@ tr`m`c`r!]gspugst`s abstblt`s. Gn gstuab`ltg puaf j`mgr gshrbtf
hu`nqubgr rghfepgls` ag su prgdgrglhb`.
Gcgrhbhbf 00) ‘0; 17 ‘7 ; =) ‘: 84
1 04
8) ‘1 ‘=;=: ;
‘04 ‘0=
1) ‘02 ‘0;; 0
01 ‘=
7) : = ; 1
‘1 4 = =
4) 84 1: ‘4; 0:
4 1= ; 4
9) 4 ‘8; 0=‘4 ; ‘84
‘1= ‘=1 ‘71
:) :0 ‘84 71; 2 ‘9=2 ‘2; 48
Gcgrhbhbf =0) 00 =7
‘8 ‘7
=) 02 ‘=
8) 0 0‘2 8
1) ‘8 8‘1 0;0: ‘2
7) ‘0= 4‘=1 ‘0:0= 4
4) 2 ‘7‘4 =
9) 00 ‘4 144 =4 =20 01 ‘4
:) ‘0; ‘02 ‘=701 ‘=0 8= ‘02 ‘9
2) 7 9 ‘088 4 ‘0=‘08 =7 ‘=:
0;) 4= 8; =7; =; 1‘1 ‘= ;
Ags`rrfnng lugv`s j`mbnba`ags0) `=) h8) `1) a
¡@ tr`m`c`r!]gspugst`s abstblt`s. Gn gstuab`ltg puaf j`mgr gshrbtf
sfmrg gn nfirf ag Ngfl`raf ag [bs` qug eàs ng j`y`bltgrgs`af.
Gcgrhbhbf 0@.;) a 5 ‘4 , gs aghrghbgltg 7) a 5 9, gs hrghbgltg0) a 5 08, gs hrghbgltg 4) a 5 ;, gs hflst`ltg=) a 5 ‘ ;.7, gs aghrghbgltg 9) a 5 ‘7.7, gs aghrghbgltg8) a 5 ;, gs hflst`ltg :) a 5 4, gs hrghbgltg1) a 5 ‘04, gs aghrghbgltg
M.0) `18 5 02= 7) a 5 0=.7
=) `22 5 ‘2=8 4) `49 5 1248) `0;0 5 ‘187 9) a 5 ‘:
1) `0 5 ‘49: :) `0 5 090;.7
H.0) `=;; 5 =108? s 5 =18 :;;
=) `07; 5 0127? s 5 00= 7;;
8) `97 5 00=8? s 5 1= 4;;
1) `8;; 5 =104? s 5 844 ;;;
7) `0=7 5 =:;? s 5 02 7;;
4) `7; 5 284? s 5 =8 7=7
9) `==7 5 497? s 5 94 =97
:) `0;; 5 9;8? s 5 87 47;
Gcgrhbhbf =@.
;) r 50
7 , aghrghbgltg 7) r 5 0, hflst`ltg
0) r 5 1, hrghbgltg 4) r 5 0
7, aghrghbgltg
=) r 5 8, hrghbgltg 9) r 5 4, hrghbgltg
8) r 5 0
=, aghrghbgltg :) r 5 0
0;, aghrghbgltg
1) r 5 7, hrghbgltg
M.0) `0; 5 0;=1 4) `0 5 9
=) `07 5 1 9:= 242 9) r 5 8
8) `9 5 8= ;;; :) r 5 4
1) `7 50;2
≊ 0.00 2) `9 5 0 :87 ;;:
7) `08 50 721 8=8 0;) `0 5 7
=:8E`tgeàtbh` ∝ Hn`vgs
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Xge`l` :Gcgrhbhbf 0@. N` rga`hhbøl ag n`s prfpfsbhbflgs lgi`a`s pugag
v`rb`r. N` bag` iglgr`n agmg e`ltglgrsg.
;) ~r3 Pubhjâ lf gs ul agp`rt`egltf ag Iu`tge`n`. D 0) ~w 3 Hum` gs ul p`ås qug lf pgrtglghg Gurfp`. T=) ~j3 Gn `iu` agn e`r lf gs aunhg. T8) ~ i3 Lf gs hbgrtf qug gn babfe` fdbhb`n ag Iu`tge`n`
gs gn `ngeàl. T1) ~s3 8 lf gs abvbsfr ag 2. D7) ~ p3 Gn vgrag lf gs ul hfnfr sghula`rbf. D4) ~q3 Gs d`nsf qug Eâxbhf sg glhugltr` `n sur
ag Iu`tge`n`. T
M.;) Gxbstg `n eglfs ul` x qug pgrtglghg ` Q y pgrtglghg
t`embâl ` I.
0) [`r` tfa` x qug pgrtglghg ` G, sg huepng qug : xt`embâl pgrtglghg J.
=) Gxbstg `n eglfs ul` x qug pgrtglghg ` H y t`embâlpgrtglghg ` M.
8) [`r` tfa` x qug pgrtglghg ` @, sg huepng qug x + 8pgrtglghg ` M.
1) Gxbstg `n eglfs ul` x qug pgrtglghg ` ] y t`embâlpgrtglghg ` Y.
7) Gxbstg `n eglfs ul` x qug pgrtglghg ` O y t`embâlpgrtglghg ` H.
Gcgrhbhbf =@. [`r` rgprgsglt`r n`s prfpfsbhbflgs sg gepng`rfl n`s
ngtr`s p y q, sbl gem`rif gn gstuab`ltg puaf j`mgrutbnbz`af hu`nqubgr ftr` hfembl`hbøl ag ngtr`s.
0) p ∪ q=) p ↝ q8) p ∩ q1) p ↑ q7) p ↝ q
M. p q
;) T T p ↝ q3 T
0) T D p ↝ q3 D=) T D p ∩ q3 D8) T D p ∪ q3 D1) T T p ∩ q3 T7) T D p ∩ q3 D4) T T p ↑ q3 T
H.0) `2 5 17 2=9? s 5 4: ::9
=) `: 5 9:0.=7? s 5 022 =0:.97
8) `07 5 019 174? s 5 =21 2;8
1) `7 5 4; ;=7? s 5 9; ;=7
7) `4 5 ==2 894? s 5 =4= 018
4) `0= 5 9;: 7::? s 5 0 ;4= ::;
Gcgrhbhbf 8N` rga`hhbøl ag n`s rgspugst`s pfarà v`rb`r. N` rgspugst`agmg hfblhbabr.
0) `0 5 7;; ei? a 5 ‘07 ei `. `
=7 5 01; ei
Gn p`hbgltg tfe`rà 01; ei ag egabh`egltfgl gn óntbef aå` agn tr`t`ebgltf.
m. s 5 :;;; ei Gn p`hbgltg j`mrà bligrbaf :;;; ei ag egab-
h`egltf ` nf n`rif ag tfaf gn tr`t`ebgltf.
=) r 5 =? `0 5 0;;;? `
0; 5 70= ;;;
Nugif ag 7 jfr`s, j`mrà 70= ;;; m`htgrb`s.
8) `0 5 P07;.;; , r 5 =, `9 5 P2,4;;.;; Gl gn sâptbef aå` agmgràl agpfsbt`r P2,4;;.;;.
1) `0 5 P1,;;;.;;, a 5 P1.7;? `0:; 5 P1,:;7.7; Nugif ag 0:; aå`s, j`mrå` agpfsbt`afs P1,:;7.7;.
7) `7 5 1784, r 5 8? `0 5 74 Gl gn prbegr `ðf j`må` 74 pfnnfs .
4) `0 5 P77.;;? r 5 = `. `4 5 P0,94;.;; Gn sàm`af i`lâ P0,94;.;;.
m. s 5 P8,147.;; Gl tft`n, ag nulgs ` sàm`af, i`lâ P8,147.;;.
Ags`rrfnng lugv`s j`mbnba`ags:
4 1
9 7 2
41
0
09
0==
9
:
00
0108
028
7
04
0; 2
07
0:
0) Gn gcgrhbhbf tbglg eàs ag ul`sfnuhbøl. Nf bepfrt`ltg gshuepnbr hfl n` hflabhbøla`a`. Wl` ag n`s pfsbmngsrgspugst`s p`r` gstg gcgrhbhbfsg ng prgsglt` hfef gcgepnf.
=) Gn gcgrhbhbf tbglg eàs ag ul`sfnuhbøl. Nf bepfrt`ltg gshuepnbr hfl n` hflabhbøla`a`. Wl` ag n`s pfsbmngsrgspugst`s p`r` gstg gcgrhbhbfsg ng prgsglt` hfef gcgepnf.
=:1BIG] ∝ [fnfhjbh
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Gcgrhbhbf [email protected]) 0;0 7) :9
=) 9; 4) ‘:78) :1 9) =;1) ‘009 :) ‘02
M.0) gngegltf lgutrf euntbpnbh`tbvf=) blvgrsf euntbpnbh`tbvf8) gngegltf lgutrf `abtbvf1) hfleut`tbv`7) `sfhb`tbv`4) abstrbmutbv` agn prfauhtf rgspghtf ` n` sue`9) hn`usur`tbv`:) blvgrsf `abtbvf
H. @niulfs blhbsfs sg eugstr`l hfl afs rgspugst`s qugsfl gqubv`ngltgs. @em`s sfl hfrrght`s.
0) 0 :) ‘ 08=
=) 77 f 80=7 2) 4‘0 f 04
8) ( =8 )
8 f :
=9 0;) 0
041) (‘8)7 f ‘=18 00) 0
7) 078 f 0
0=7 0=) :
4) 84 f 9=2 08) 1=79) 48 f =04 01) =4 f 70=
A.0) 4 :) :
=) 1 2) 0=
8) 8=
0;) 7
1) 8 00) 0=7) 8 0=) 74) 7 08) 849) 1 01) 01
Gcgrhbhbf [email protected]) 8b 9) b
=) :b :) b =8) 0;b 2) 4b
1) =;b 0;) 7b
7) 2b 00) 0=b
4) 9b 0=) 00b
M.0) 0; ‘ 00b 4) 104 + =b
=) =9 + 4b 9)
728
88 ‘
12b
888) ‘: + 8b :) =9
0; ‘ 82b
=;
1) ‘4; + 44b 2) 709
‘ 8b
09
7) 1: ‘ :1b 0;) ‘ 47
‘ 8b
7
Gcgrhbhbf 7@.
0) [uaf j`mgr dfre`af hu`nqubgr e`trbz qug huepn`hfl n` hflabhbøl ag tglgr 1 dbn`s y 8 hfnuel`s.
=) B 50 ; ;; 0 ;; ; 0
8) [uaf j`mgr dfre`af hu`nqubgr e`trbz qug huepn` hfln` hflabhbøl ag sbegtrå`. Ng prgsglt`efs ul gcgepnf3
@ 57 4 94 : ‘09 ‘0 2
@t57 4 94 : ‘09 0 2
1) [uaf j`mgr dfre`af hu`nqubgr e`trbz y su rgspghtbv`e`trbz tr`lspugst`. Ng a`efs ul gcgepnf3
E 5‘= 4 :1 ‘0 ;; ; ‘=
Et5‘= 1 ;4 ‘0 ;: ; ‘=
M.
0) 4 0 00 7: 07 ‘2 ‘0=
=) ‘1 ‘4 ‘0‘4 1 9‘9 ; ‘:
8) ; ‘1 9 7: 0; ; ‘02
1) ‘: 9 ‘0=4 8 ‘=
‘0 ‘2 07
7) 2 ‘0 ‘0 1‘1 ‘8 ; 0=
4) ‘1 0; 08‘8 00 22 0: =
=:7E`tgeàtbh` ∝ Hn`vgs
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Gcgrhbhbf [email protected]) 41 :; ‘::
‘9= ; :74 8= ‘=1
=) ; =9 0==9 ‘07 ‘84‘4 ; 8
M.0) ‘77 ‘1; =) 79 ‘87
8) ‘: :0 ‘0
1) 9 ‘87‘00 8;
7) 84 04‘7 ‘2
4) ‘41 ‘8 11: 7: ‘49
9) 07 9 ‘8=11 87 ‘74088 9: ‘=;
:) ‘19 ‘01 ‘82‘0: = =99: 8= 04
2) 8 ‘:0 ‘:1; 8; 7420 ‘49 44
0;) 070 :: ‘110;: 48 ‘8478 00 ‘2:
Gcgrhbhbf [email protected]) `18 5 ‘=:4=) `74 5 :8.78) `81 5 =14
1) a 5 97) a 5 ‘0=
4) `0 5 7:
9) `0 5 :1.7:) `44 5 48=?
s 5 == 1;9
M.0) `0= 5 0= =::=) `00 5 1 021 8;18) `07 5 =887.18
1) r 5 77) r 5 84) `0 5 0=
9) `0 5 0;:) `0; 5 7=1 =::? s 5 422 ;7;
H.0) `0 5 P8;.;;? a 5 P:.;;`. `4 5 P9;.;;? s 5 P8;;.;; N` sgxt` jfr` tglarà ul hfstf ag P9;.;; y gl tft`n,
nugif ag gs`s 4 jfr`s ag `nqubngr, agmgrgefs p`i`r
P8;;.;;m. Døreun` agn târeblf iglgr`n3 `
l 5 == + :l
=) `0 5 P1:,;;;.;;? r 5 0=
`. `4 5 P0,7;;.;; @n sgxtf prfpbgt`rbf, n` eàqubl` ng hfstø P0,7;;.;;m. `: 5 P897.;;? s 5 P27,4=7.;; Xb gn prfpbgt`rbf lóegrf fhjf sg quga` hfl n`
eàqubl`, n` sue` tft`n p`i`a` gl tfa`s n`s vglt`ssgrà ag P27,4=7.;;
Xge`l` 2¡@ tr`m`c`r!;) 79 8) (‘=):
0) 80; 1) (2)8
=) (‘:)0;
Gcgrhbhbf 0@.;) 7 8) ‘7(0) 5 ‘7
0) 0 1) :j(0) 5 :j
=) 0
M.;) `0= 8) (el)9
0) x2 1) (=wy)1
=) e07 7) (rs):
H. ;) e1 1) 0
0) m4 7) ( pqr )=
=) w9 4) (7el)=
8) y0= 9) 0
(‘8 xyz )0;
A. ;) `1m0;h0= 1) p0;r 01
0) x04 7) 0
=) e1l04 4) ‘ 1e1j4o :
8) w4 y0: 9) ‘7`8m07h8a 4
G. ;) :0 1
04m01 8) =7e4l1
1w: z 4
0) x4
y4 1) 2j
:o :
=7 p:q
0=
=) `4m4
h
4a 4 7) `04m
=:h0=
w: x
0= y:
Gcgrhbhbf =0) ( m
` )7
5m7
`7
=) (1 y
8 x)8
541 y8
=9 x8
8) (8h8
7m=)= 5 2h4
=7m1
1) ( h1a 7
`=m8 )8
5h0=a 07
`4m2
7) ( = p1q8
1e8l= )8
50 p0=q2
:e2l4
4) ( y8 z 9
‘7w1 x7 )8
5 y2 z =0
‘0=7w0= x07
=:4BIG] ∝ [fnfhjbh
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Gcgrhbhbf 8;) (1w8 x= y4 )8 5 41w2 x4 y0:
0) 4`4m2h7
=) (=`1m8h=)(=9`2m8h07) 5 71`08m4h09
8) ‘0= x7 y0;
1) (7w8 x= y1)(=9 x4 y2) 5 087w8 x: y0;
7) 8 p8q1r 1
4) :`4m7h7
9) (8 x= y1)1 5 :0 x: y04
:) (w= x1 y=)7 5 w0; x=; y0;
2) 0;
0= x01 y4 z 9 5
7
4 x01 y4 z 9
Ags`rrfnng lugv`s j`mbnba`ags0) 08 5 0 Gn humf gstà dfre`af pfr ul` pbgz`.
=) =8 5 : Gn humf gstà dfre`af pfr fhjf pbgz`s.
8) 88 5 =9 Gn humf gstà dfre`af pfr vgbltbsbgtg pbgz`s.
1) 18 5 41 Gn humf gstà dfre`af pfr sgsglt` y hu`trf pbgz`s.
Xge`l` 0;¡@ tr`m`c`r!
@ 5 4=7e=
x= 5 4=7e=
x= 5 4=7e=
x 5 =7e
Xg pugagl sgemr`r =7 pn`lt`s gl h`a` dbn` y hfnuel` `0 egtrf ag abst`lhb` h`a` ul`.
Gcgrhbhbf 0;) ` • m
0)8
h •8
a
=) el
8) =; x
1) 8 e= •
8 l=
7) p
q
4)8 r 8 s
9) :
: 5 `
:) 4 4 5 `
Gcgrhbhbf =@. ;) `4/8 5 `=
0) `:/7
=) a :/= 5 a 1
8) (=; x)1/8
1) (:l)=/1 5 (:l)0/=
7) ( p
q )0/04
4) j
o
9) ( yz )1/: 5 ( yz )0/=
:) ( x= y8)0/7
M. ;) 7
(`m)= 7) 1 ( pq)8
0) 1 x8 4) ( jo )
72
=) 9 w7 9) ( r
s )8
7
8) 8 z = :) 8
( xyz )
1) 2 (el)=
=:9E`tgeàtbh` ∝ Hn`vgs
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Xge`l` 00¡@ tr`m`c`r!]gspugst`s v`rb`mngs. [uaf j`mgr gshrbtf n` `pnbh`hbølqug eàs ng j`y` bltgrgs`af. @niul`s àrg`s aflag sg
utbnbz` gn ànigmr` sfl n` bliglbgrå`, n` hflstruhhbøl, gl n`mfns` ag v`nfrgs, gltrg euhj`s ftr`s.
Gcgrhbhbf [email protected]) tgtr`lfebf 1) mblfebf=) mblfebf 7) mblfebf8) trblfebf 4) trblfebf
M. 0) tgrhgr ir`af f hómbhf =) hu`rtf ir`af 8) hu`rtf ir`af 1) qubltf ir`af 7) tgrhgr ir`af f hómbhf 4) prbegr ir`af f nblg`n 9) hu`rtf ir`af :) tgrhgr ir`af f hómbhf 2) sgiulaf ir`af f hu`aràtbhf0;) sgiulaf ir`af f hu`aràtbhf
H. 0) x8 + 0= x= ‘ x
=) ‘0; x= ‘ 7 x + 7
8) 4 y8 + 7 y ‘ =0
1) ‘0;`t 8
+ =;t + 0 7) 8=`1 + ==`8 + `=
4) ‘=4r = + 1r ‘ :
9) 8; x= y + 02 xy ‘ 08
:) 84w8 ‘ 8w= + 00w ‘ 0
2) ‘=1 i = ‘ =8 i ‘ 0
0;) ‘7wy8 + wy + 1w + 1 y=
Gcgrhbhbf = 0) 7; x8 + 08; x ‘ 8;
=) 07`7m= + 2`1m1 ‘ =;`=m ‘ 0=`m8
8) 87w7 x + 07w8 x8 ‘ =7w8 x
1) ‘: x8 ‘ 9= xy= ‘ : y8
7) 84 i 1j7 + 084 i 8j1 ‘ :8 i =j8 ‘ 0= i =j= ‘ =;0 ij= ‘ 84 ij +
0=;j + 17
4) 2:`m7h8 + 071`m=h7 ‘ 00=m=h8
9) 70l8e ‘ 084l=e= + 0;=le8 + 084
:) ‘=;w1 x8 y= + 4;w1 x= y + 17w8 x1 y8 + =:w8 x8 y8 ‘ :1w8 x= y=
‘ 48w= x1 y1 ‘ 8=w= x8 y7 + 24w= x= y1 + 9=wx1 y4
2) ‘17 x7 y= + =9 x1 y8 + 48 x1 y
0;) ‘8=w8 y= z + =;w= x8 z 8 + 1:w= y= z ‘ 8;wx8 z 8
Gcgrhbhbf 80) = x8 ‘ 8 x= y + 1 xy= ‘ = y8
=) = x= + 8 x ‘0
x ‘ 1
8) ‘ 8`m ‘1
` + 7
1) 8w + 1 x ‘7
x
7) 2`
m +
4m
2 ‘ 8
4) ‘ 1
( xy=) + 8 x + y
9) ‘=;m= ‘2
mh= +
00
h
:) x=
y8 ‘
=
x= y8 ‘
9 x
y= +
=
x +
8
y
2) 0`m=
+ 4`m ‘ 1m
Ags`rrfnng lugv`s j`mbnba`ags0) TB + BT 5 Z=) Z 5 BB + TBB
=:2E`tgeàtbh` ∝ Hn`vgs
8/20/2019 Libro Polochic Matemática 1er. Sem
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Xge`l` 0=¡@ tr`m`c`r!N`s rgspugst`s pugagl v`rb`r. @niulfs gcgepnfs sfl3Guhnbags, [btàifr`s, Lgwtfl, gth.
Gcgrhbhbf 0;) (x
= )= + =(x= • m8 ) + (m8 )= 5 x1 + =x=m8 + m4
0) (`)= + =(` • 8) + (8)= 5 `= + 4` + 2
=) (=h=)= + =(=h= • 1a =) + (1a =)= 5 1h1 + 04h=a = +04a 1
Gcgrhbhbf =@. ;) x
4 ‘ 0= x
8 + 84
0) x
= ‘ = xy + y=
=) w= ‘ =wz + z =
8) 2 x
= ‘ =1 x + 04
1) 84 y
= ‘ 4; y + =7
7) 0
1w= ‘ =wz + 1 z =
4) x
= ‘=
7 xz +
0
=7 z =
9) 2 x
: ‘ =1 x
1 y8 + 04 y4
:) 1 x
1 ‘ 0= x
= y= + 2 y1
2) 41 x0; ‘ 41 x
7 y
8 + 04 y4
0;) 04 x
: ‘ 1; x
1 y2 + =7 y
0:
00) 2e1l4 ‘ 1:e=l8 pq= + 41 p=q1
0=) 0
2w= ‘ 4wz + :0 z =
08) 04
2
x1 ‘=;
8
x
= y8 +=7
1
y
4
M.
;) 1 x
= ‘ 2 y
=
0) p= ‘ q=
=) y
= ‘ 04
8) w= ‘ 84
1) 2 x= ‘ y
=
7) 12
`= ‘ m=
4) 41=7
e1 ‘ 2
9) =7 z = ‘ 41
:) 04 y
1
‘ z
=
2) 1 x
1 ‘ 2
0;) 2 x
= ‘ 04 y
=
00) =7 x
= ‘ 0;; y
=
0=) 01 x
= ‘ =7 y
=
08) =704
e
1 ‘ 2l
=
H. ;) `= + :` + 07
0) x= + 09 x + 9=
=) y=
+ 9 y + 0= 8) z = + 2 z + 01
1) `= + :` + 04
7) m= + 0=m + 84
4) o 1 + 0;o = + 04
9) 2w4 + 88w8 + 8;
:) `= ‘ 0;` + =1
2) 1h= ‘ 0;h + 4
0;) =7 x= ‘ 4; x + 8=
00) 2 y= ‘ 17 y + 74
0=) 1 x= ‘ 04 x + 07
08) 0;; x= ‘ 4; x + 2
01) :0 z 1 ‘ 011 z = + 4;
07) 2 x1 y= ‘ 0: x= y + :
Gcgrhbhbf 80) (1w + z )8 5
(1w)8 + 8(1w)=( z ) + 8(1w)( z )= + ( z )8 5
41w8 + 8(04w=)( z ) + 8(1w)( z =) + z 8 5
41w8 + 1:w= z + 0=wz = + z 8
=) (8 x= + = y
=)8 5
(8 x=)8 + 8(8 x=)=(= y=) + 8(8 x=)(= y=)= + (= y=)8 5
=9 x4 + 8(2 x
1)(= y
=) + 8(8 x
=)(1 y
1) + : y
4 5
=9 x 4 + 71 x 1 y
= + 84 x = y1 + : y4
Gcgrhbhbf 10) (=h ‘ a )8 5
(=h)8 ‘ 8(=h)=(a ) + 8(=h)(a )= ‘ (a )8 5
:h8 ‘ 8(1h=)(a ) + 8(=h)(a =) ‘ a 8 5
:h8 ‘ 0=h=a + 4ha = ‘ a 8
=) (1 y8 ‘ =z)8 5
(1 y
8)8 ‘ 8(1 y8)=(= z ) + 8(1 y8)(= z )= + (= z )8 5
41 y
2 ‘ 8(04 y
4)(= z ) + 8(1 y8)(1 z =) ‘ : z 8 5
41 y
2 ‘ 24 y4 z + 1: y
8 z = ‘ : z 8
Ags`rrfnng lugv`s j`mbnba`ags
`. x=
+ mx + `x + `m 5 x=
+ x(` + m) + `mm. ( x + `)( x + m)
h. [rfauhtf ag afs mblfebfs hfl ul târeblf hfeól ysbilfs biu`ngs.
=2;BIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
http://slidepdf.com/reader/full/libro-polochic-matematica-1er-sem 303/326
Xge`l` 08¡@ tr`m`c`r!2; =
17 807 8
7 7
0
89: =
0:2 848 8
=0 8
9 9
0
=:4 =
018 0008 08
0
Gcgrhbhbf 00) `m=(‘ 8 z = ‘ `=m=hx= + ` + 0)
=) 4 x=(‘8 x7+ = x8 + = x ‘ 0)
8) y=(‘7w= y + 0; x= + 7= xyz 8)
1) 0= x=
(8 x1
‘ 7 x8
+ 1 x ‘ 4)7) 8 y(‘8`m8 + 1my + = yz )
4) =e=l(‘:` + 2ml ‘ 0;ez = + 00e)
9) ‘4(`8m= z ‘ =`= z 8 ‘ =`m8h ‘ 1mh)
:) 7e=(0;`=el1 ‘ m=el + 8ez ‘ 7l8r )
Gcgrhbhbf = 0) (8`= ‘ 7 x)(=` + mx)
=) (8 x= + 4m)(= y ‘ 0)
8) (= y= + m)(9`= y+1mx=)
1) (1 xy= ‘ 8 z )(‘ m= + 1`)
7) (8`= ‘ 8m)(8 x= + y=)
4) (=e ‘ l)( =l
8
y + 2e
=
x ‘ 7m) 9) (‘8l8 + ez )(=e= y + 7m)
:) ( x + = y)(00`8 x ‘ 2 y= z )
2) (8e= ‘ `z )(8l8 ‘ :`e)
0;) (my ‘ =)(‘7m8r + 2 x)
00) (7 x= + 4 s)(8rx ‘ 7s8 y + 1 z )
0=) (‘ z = + = y)(1 xy= ‘ 7 z + 8w)
Gcgrhbhbf 8 0) (` ‘ m)=
=) ( x + 9)=
8) (` ‘ 7)=
1) ( x ‘ 8)=
7) (7`= ‘ 8m)=
4) (`8 + = x)=
9) (7 x= + 1)=
:) ( x ‘ 0;)=
2) ( x8 + 7)=
0;) (2 z ‘ 0;)=
00) (8`
1 y4 ‘ 0; x
7)=
0=) (` ‘ 0=e= x=)=
Gcgrhbhbf 1 0) ( x8 + =)( x8 ‘ =)
=) (m + 0)(m ‘ 0)
8) (1 + e=)(1 ‘ e=) 1) (= + x)(= ‘ x)
7) (7`7 y= + 9m4 x8)(7`7 y= ‘ 9m4 x8)
4) (1 x= y1 + 0;)(1 x= y1 ‘ 0;)
9) (4 x + `8m=)(4 x ‘ `8m=)
:) (2m= x + 4`h)(2m= x ‘ 4`h)
2) (9 x= y + =`=m)(9 x= y ‘ =`=m)
0;) (`
4 +
x8
7 )(`
4 ‘
x8
7 )00) (0
= + 8`)(0
= ‘ 8`)
0=)
(0 +
`
7 )(0 ‘
`
7 )Ags`rrfnng lugv`s j`mbnba`ags[gråegtrfs3@ 5 x + x + 7 + 7 5 = x + 0;
M 5 x + x + y + y 5 = x + = y H 5 7 + 7 + 7 + 7 5 =;
A 5 y + y + 7 + 7 5 = y + 0;
Àrg`s3@ 5 7 x
M 5 xy
H 5 7 • 7 5 =7A 5 7 y
=20E`tgeàtbh` ∝ Hn`vgs
8/20/2019 Libro Polochic Matemática 1er. Sem
http://slidepdf.com/reader/full/libro-polochic-matematica-1er-sem 304/326
Xge`l` 01¡@ tr`m`c`r!X 5
(0 + 0;;)0;;=
5 0; ;;;=
5 7;7;
Gcgrhbhbf 0;) (x ‘ 1) (x + 0)
0) ( x + =)( x + 0)
=) ( x ‘ 4)( x ‘ 7)
8) ( x ‘ 8)( x + 0)
1) ( x + =)( x + =)
7) ( x ‘ =0)( x ‘ 1)
4) ( x ‘ 7 y)( x ‘ 8 y)
9) ( x + 4 y)( x ‘ = y)
:) ( x ‘ 7 y)( x + y)
Gcgrhbhbf =;) = x
= + 9 x + 8 5
1 x
= + =(9 x) + 4 5
(= x)= + 9(= x) + 4 5
(= x + 4)(= x + 0) 5
( = x + 4
= )( = x + 0
0 ) 5 ( x + 8)(= x + 0)
0) 4 x= + 00 x + 8 5
84 x
= + 4(00 x) + 0: 5
(4 x)= + 00(4 x) + 0: 5
(4 x + 2)(4 x + =) 5
(
4 x + 2
8
)(
4 x + =
= ) 5 (= x + 8)(8 x + 0)
=) 4 x
= + 09 x + 7 5
84 x
= + 4(09 x) + 8; 5
(4 x)= + 09(4 x) + 8; 5
(4 x + 07)(4 x + =) 5
( 4 x + 07
8 )( 4 x + =
= ) 5 (= x + 7)(8 x + 0)
8) 0= x
= ‘ 08 x + 8 5
011 x
= ‘ 0=( x) + 84 5
(0= x)= ‘ (0= x) + 84 5
(0= x ‘ 2)(0= x ‘ 1) 5
(0= x ‘ 2
8 )( 0= x ‘ 1
1 ) 5 (1 x ‘ 8)(8 x ‘ 0)
Gcgrhbhbf 8;) w
8 + x
8 5
(w + x)_(w)
= ‘ (w • x) + ( x)=U 5 (w + x )(w
= ‘ wx + x =)
0) y 8 + z 8 5
( y + z )_( y) = ‘ ( y • z ) + ( z )=U 5 ( y + z)( y
= ‘ yz + z=)
=) x 8 + : 5
( x + =)_( x) = ‘ ( x • =) + (=)=U 5 ( x + =)( x = ‘ = x + 1)
8) =9 + j 8 5
(8 + j)_(8) = ‘ (8 • j) + (j)=U 5 (8 + j)(2 ‘ 8j + j=)
1) : + :w 8 5
(= + =w)_(=) =
‘ (= • =w) + (=w)=
U 5 (= + =w)(1 ‘ 1w + 1w=
)7) x 8 + 41 5
( x + 1)_( x) = ‘ ( x • 1) + (1)=U 5 ( x + 1)( x = ‘ 1 x + 04)
Gcgrhbhbf 1;) x8 ‘ 0 5
( x ‘ 0)_( x)= + ( x • 0) + (0)=U 5 ( x ‘ 0)( x = + x + 0)
0) w8 ‘ x8 5
(w ‘ x)_(w)= + (w • x) + ( x)=U 5 (w ‘ x )(w= + wx + x =)
=) y8 ‘ : z 8 5
( x ‘ = z )_( y)= + ( y • = z ) + (= z )=U 5 ( y ‘ = z )( y= + = yz + 1 z =)
8) : x8 ‘ 41 5 (= x ‘ 1)_(= x)= + (= x • 1) + (1)=U 5 (= x ‘ 1)(1 x = + : x + 04)
1) =9 ‘ y8 5
(8 ‘ y)_(8)= + (8 • y) + ( y)=U 5 (8 ‘ y)(2 + 8 y + y=)
7) 0=7 ‘ :w8 5
(7 ‘ =w)_(7)= + (7 • =w) + (=w)=U 5 (7 ‘ =w)(=7 + 0;w + 1w=)
Ags`rrfnng lugv`s j`mbnba`ags0) Àrg` 5 x= ‘ 04 x + 1: 5 ( x ‘ 1)( x ‘ 0=)
Gn ftrf n`af agn rghtàliunf ebag x ‘ 0=
=) @ tft n 5=1 x
= + 1: x
= 5 9= x
=
@ tft n 5 : x(2 x)
Gn ftrf n`af agn rghtàliunf ebag 2 x
8) @ 5 7 x= + 07 xy + 4 y= 5 ( x + = y) (7 x + 8 y)
Gn ftrf n`af agn røemf ebag 7 x + 8 y
=2=BIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
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Xge`l` 07¡@ tr`m`c`r!0) 7
0= 8) :
7
=) 0;
2 1) 7
0:
Gcgrhbhbf 0 0) x= ‘ 7 x + 4
=`x ‘ 4` 5
( x ‘ 8)( x ‘ =)
=`( x ‘ 8) 5 ( x ‘ 8)( x ‘ =)
=`( x ‘ 8) 5 x ‘ =
=`
=) z = ‘ 7 z + 4
z = ‘ 9 z + 0= 5 ( z ‘ 8)( z ‘ =)
( z ‘ 1)( z ‘ 8) 5 ( z ‘ 8)( z ‘ =)
( z ‘ 1)( z ‘ 8) 5 z ‘ =
z ‘ 1
8) :`8 + =9
1`= + 0=` + 25 (=` + 8)(1`= ‘ 4` + 2)
(=` + 8)= 5
(=` + 8)(1`=
‘ 4` + 2)(=` + 8)= 5 1`
=
‘ 4` + 2=` + 8
1) `= ‘ 1`m + 1m=
`8 ‘ :m8 5 (` ‘ =m)=
(` ‘ =m)( = + =`m + 1m=)5
( ‘ =m)=
( ‘ =m)( = + =`m + 1m=) 5 ` ‘ =m
`= + =`m + 1m=
7) y: ‘ =7 y
= y8 + : y= ‘ 0; y 5 y( y9 ‘ =7)
= y( y= + 1 y ‘ 7) 5
y( y9 ‘ =7)
= y( y
= + 1 y ‘ 7) 5 y9 ‘ =7
=( y= + 1 y ‘ 7) 5 y9 ‘ =7
= y= + : y ‘ 0;
4) x8
‘ 1 x=
‘ 0= x x8 ‘ 8 x= ‘ 0; x
5 x ( x + =)( x ‘ 4) x ( x + =)( x ‘ 7)
5 x( x + =)( x ‘ 4) x( x + =)( x ‘ 7)
5 x ‘ 4 x ‘ 7
9) w= ‘ 8w
w= + 8w 5
w(w ‘ 8)
w(w + 8) 5 w(w ‘ 8)
w(w + 8) 5 w ‘ 8
w + 8
:) 4 x= + 7 x ‘ 4
07 x= ‘ 9 x ‘ = 5 (8 x ‘ =)(= x + 8)
(7 x + 0)(8 x ‘ =) 5
(8 x ‘ =)(= x + 8)
(7 x + 0)(8 x ‘ =) 5 = x + 8
7 x + 0
2) x= + x ‘ =
x8 ‘ x= ‘ x + 0 5 ( x + =) ( x ‘ 0)
( x8 ‘ x=) ‘ ( x ‘ 0) 5 ( x + =) ( x ‘ 0)
x=( x ‘ 0) ‘ ( x ‘ 0) 5
( x + =)( x ‘ 0)
( x= ‘ 0)( x ‘ 0) 5 ( x + =)( x ‘ 0)
( x ‘ 0)( x + 0)( x ‘ 0) 5 ( x + =) ( x ‘ 0)
( x + 0)( x ‘ 0)=
( x + =) ( x ‘ 0)
( x + 0)( x ‘ 0)= 5 x + =
( x + 0)( x ‘ 0)
0;) = x= + 9 x + 8
= x= ‘ 9 x ‘ 1 5 (= x + 0)( x + 8)
(= x + 0)( x ‘ 1)5
(= x + 0)( x + 8)
(= x + 0)( x ‘ 1)5 x + 8
x ‘ 1
Gcgrhbhbf = 0) x + 8
x ‘ =•
x= + = x ‘ :
x= ‘ 2 5 ( x + 8)( x + 1)( x ‘ =)
( x ‘ =)( x ‘ 8)( x + 8) 5
( x + 8)( x + 1)( x ‘ =)( x ‘ =)( x ‘ 8)( x + 8)
5 x + 1 x ‘ 8
=) x= ‘ = x + 0
x= ‘ 2 •
x= + x ‘ 4
x= ‘ 0 5 ( x ‘ 0)=( x + 8)( x ‘ =)
( x + 8)( x ‘ 8)( x + 0)( x ‘ 0)5
( x ‘ 0)=( x + 8)( x ‘ =)
( x + 8)( x ‘ 8)( x + 0)( x ‘ 0)5 ( x ‘ 0)( x ‘ =)
( x ‘ 8)( x + 0)
8) e + l
el ‘ l= •
l=
e= ‘ l= 5
l=(e + l)
l(e ‘ l)(e ‘ l)(e + l)
l=(e + l)
l(e ‘ l)(e ‘ l)(e + l) 5 l
(e ‘ l)=
1) x= ‘ = x
x= ‘ 7 x + 4 •
x= + 1 x + 1
x= ‘ 1 5 x( x ‘ =)( x + =)=
( x ‘ =)( x ‘ 8)( x ‘ =)( x + =)5
x( x ‘ =)( x + =)=
( x ‘ =)( x ‘ 8)( x ‘ =)( x + =) 5 x ( x + =)
( x ‘ 8)( x ‘ =)
7) xy ‘ = y=
x= + xy •
x= + = xy + y=
x= ‘ = xy 5 y( x ‘ = y)( x + y)=
= x( x + y)( x ‘ = y)5
y( x ‘ = y)( x + y)=
= x( x + y)( x ‘ = y)5 y( x + y)
= x
4) 2 ‘ 4 x + x=
x= ‘ 2 •
x= ‘ 7 x + 4
8 x= ‘ 2 x 5 ( x ‘ 8)=( x ‘ =)( x ‘ 8)
8 x( x ‘ 8)( x + 8)( x ‘ 8) 5
( x ‘ 8)=
( x ‘ =)( x ‘ 8)8 x( x ‘ 8)( x + 8)( x ‘ 8) 5 ( x ‘ =)( x ‘ 8)
8 x ( x + 8)
9) x= ‘ 1 xy + 1 y=
x= ‘ = xy •
x=
x= ‘ 1 y
= 5
x =( x ‘ = y)=
x ( x ‘ = y)( x ‘ = y)( x + = y)5
x=( x ‘ = y)=
x( x ‘ = y)=( x + = y)5 x
x + = y
:) =` ‘ =
=`= ‘ 7; •
`= ‘ 1` ‘ 7
8` + 8 5 =( ‘ 0)(` + 0)( ‘ 7)
4( ‘ 7)( + 7)( + 0)5
=( ‘ 0)( + 0)(` ‘ 7)
4( ‘ 7)( + 7)( + 0)5 ` ‘ 0
8(` + 7)
2) x= ‘ 2 x= ‘ 1
• x= ‘ 9 x + 0; x= ‘ : x + 07
5 ( x ‘ 8)( x + 8)( x ‘ =)( x ‘ 7)( x ‘ =)( x + =)( x ‘ 8)( x ‘ 7)
5
( x ‘ 8)( x + 8)( x ‘ =)( x ‘ 7)
( x ‘ =)( x + =)( x ‘ 8)( x ‘ 7)5 x + 8
x + =
0;) = x= ‘ 8 x ‘ =
4 x + 8 •
8 x + 4
x= ‘ 1 5 8(= x + 0)( x ‘ =)( x + =)
8(= x + 0)( x ‘ =)( x + =) 5
8(= x + 0)( x ‘ =)( x + =)
8(= x + 0)( x ‘ =)( x + =)5 0
=28E`tgeàtbh` ∝ Hn`vgs
8/20/2019 Libro Polochic Matemática 1er. Sem
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Gcgrhbhbf 80) x8 ‘ x
= x= + 4 x ù 7 x= ‘ 7 x
= x + 4 5 x8 ‘ x
= x= + 4 x • = x + 4
7 x= ‘ 7 x 5
= x ( x ‘ 0)( x + 0)( x + 8)0; x =( x + 8)( x ‘ 0)
5 = x( x ‘ 0)( x + 0)( x + 8)0; x=( x + 8)( x ‘ 0)
5
x + 0
7 x
=) : x= + =4 x + 07
04 x= ‘ 2 ù 4 x= + 08 x ‘ 7
2 x= ‘ 05
: x= + =4 x + 07
04 x= ‘ 2 • 2 x= ‘ 0
4 x= + 08 x ‘ 75
(1 x + 8)(= x + 7)(8 x ‘ 0)(8 x + 0)
(1 x ‘ 8)(1 x + 8)(8 x ‘ 0)(= x + 7) 5
(1 x + 8)(= x + 7)(8 x ‘ 0)(8 x + 0)
(1 x ‘ 8)(1 x + 8)(8 x ‘ 0)(= x + 7) 5 8 x + 0
1 x ‘ 8
8)=; x= ‘ 8; x
07 x8 + 07 x= ù1 x ‘ 4
x + 0 5
=; x= ‘ 8; x
07 x8 + 07 x= • x + 0
1 x ‘ 4 5
0; x (= x ‘ 8)( x + 0)
8; x =( x + 0)(= x ‘ 8) 5 0; x(= x ‘ 8)( x + 0)
8; x=( x + 0)(= x ‘ 8) 5 0
8 x
1)`x= + 7
1`= ‘ 0 ù
`8 x= + 7`=
=` ‘ 0 5
`x= + 7
1`= ‘ 0 •
=` ‘ 0
`8 x= + 7`= 5
(`x = + 7)(=` ‘ 0)
`=(= ‘ 0)(=` + 0)(`x = + 7) 5 (`x= + 7)(=` ‘ 0)
`=(= ‘ 0)(= + 0)(`x= + 7) 5
0
`=(=` + 0)
7)`= ‘ 4 + 7
`= ‘ 07 + 74 ù
`= + = ‘ 87
`= ‘ 7 ‘ =1 5
`= ‘ 4 + 7
`= ‘ 07 + 74 •
`= ‘ 7 ‘ =1
`= + = ‘ 87 5
( ‘ 7)( ‘ 0)( ‘ :)( + 8)
( ‘ :)( ‘ 9)( + 9)(` ‘ 7) 5
( ‘ 7)( ‘ 0)( ‘ :)( + 8)
( ‘ :)( ‘ 9)( + 9)(` ‘ 7) 5
(` ‘ 0)(` + 8)
(` ‘ 9)( + 9)
4)`= ‘ 4`
`8 + 8`= ù`= + 8` ‘ 71
`= + 2` 5
`= ‘ 4`
`8 + 8`= • `= + 2`
`= + 8` ‘ 71 5
`=(` ‘ 4)(` + 2)
`=(` + 8)(` + 2)(` ‘ 4) 5 `=(` ‘ 4)( + 2)
`=(` + 8)(` + 2)(` ‘ 4) 5
0` + 8
Ags`rrfnng lugv`s j`mbnba`agsN` rga`hhbøl pugag v`rb`r, n` bag` prblhbp n agmg e`ltglgrsg3
N` `sbstgltg gstà ebltbglaf pfrqug n`s pàibl`s 022 y =;;hfrrgspflagl ul` ebse` jfc`.
Xge`l` 04¡@ tr`m`c`r!0) 4
= 5 8 1) 8 + =
45 7
4
=) 4
9 7) 0; ‘ 8 + 9
45 01
4 5 9
8
8) ‘
=
2 4) 04 ‘ 2 ‘ 0;
=1 5 ‘
8
=1 5 ‘
0
:
Gcgrhbhbf 00)
x
x + = +
4 x
x + = 5
x + 4 x
x + =5
9 x
x + =
=)7 x= ‘ 1
: x8 +
x= + 1
: x8 5
7 x= ‘ 1 + x= + 1
: x8 5
4 x=
: x85
8
1 x
8)4 x=
4 x ‘ 9 ‘
9 x
4 x ‘ 9 5
4 x= ‘ 9 x
4 x ‘9 5
x(4 x ‘ 9)
4 x ‘ 9 5 x
1) x= + 8 x
x= ‘ x ‘ 4 ‘
x= ‘ 4
x= ‘ x ‘ 4 5
x= + 8 x ‘ x= + 4
x= ‘ x ‘ 4 5
8 x + 4
x= ‘ x ‘ 45
8( x + =)
( x ‘ 8)( x + =)5
8
x ‘ 8
Gcgrhbhbf =
0)8 x + 0
7 x +
x ‘ =
0; x 5
(8 x + 0)(=) + ( x ‘ =)(0)
0; x5
4 x + = + x ‘ =
0; x5
9 x
0; x 5
9
0;
=) x + 8
8 x
‘7 x ‘ 0
7 x=
5( x + 8)(7 x) ‘ (7 x ‘ 0)(8)
07 x=
5
7 x= + 07 x ‘ 07 x + 8
07 x= 5
7 x = + 8
07 x =
Gcgrhbhbf 8
0)
x + 0
0
0 + x
0
5
x + 0
0
0 + x
0
50( x + 0)
0(0 + x)5
0( x + 0)
0( x + 0)5 0
=) x ‘
1
x
x + 1 +1
x
5
x
0 ‘
1
x x
0
+1
0
+1
x
5
x= ‘ 1
x
x= + 1 x + 1
x
5
x( x= ‘ 1)
x( x= + 1 x + 1)5
( x + =)( x ‘ =)
( x + =)( x + =) 5
x ‘ =
x + =
8) x +
00
x +
4
x=
8 + 1
x ‘
1
x=
5
x
0 +
00
x +
4
x=
8
0 +
1
x ‘
1
x=
5
x8 + 00 x + 4
x=
8 x= + 1 x ‘ 1
x=
5
x=( x8 + 00 x + 4)
x=(8 x= + 1 x ‘ 1) 5
x 8 + 00 x + 4
8 x = + 1 x ‘ 1
=21BIG] ∝ [fnfhjbh
8/20/2019 Libro Polochic Matemática 1er. Sem
http://slidepdf.com/reader/full/libro-polochic-matematica-1er-sem 307/326
Xge`l` 09
1)
`= ‘ m=
`
` + m
`8
5
`=( =) ‘ `=m=
`8
` + m
`8
5
`1 ‘ `=m=
`8
` + m
`8
5
`8( 1 ‘ `=m=)
`8( + m) 5
`=( = ‘ m=)
( + m) 5
`=( + m)( ‘ m)
( + m)5 `=(` ‘ m)
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0;`h + =7mh
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]gp`sf p`r` n` prugm` ag ir`au`lafs agn EBLGAWH
Gstbe`a` (f) gstuab`ltg agn irupf [fnfhjbh3
Ng s`nua`efs hfl h`rbðf y gspgr`efs qug gstâ gstuab`laf hfl gltusb`sef gstg prbegr ðfag m`hjbnngr`tf.
Gn Eblbstgrbf ag Gauh`hbøl rg`nbz` h`a` `ðf prugm`s ag e`tgeàtbh` y ag hfeprglsbøl nghtfr`` tfafs nfs gstuab`ltgs qug gstàl hurs`laf gn óntbef ðf agn hbhnf màsbhf f gn óntbef ðf ag
abvgrsbh`af. Gn prfpøsbtf ag gst`s prugm`s gs fmtglgr bldfre`hbøl qug pgrebt` egcfr`r n`h`nba`a gauh`tbv` ag lugstrf p`ås.
Hflsghugltgs hfl gst` egcfr` gauh`tbv`, jgefs blhnubaf, gl gstg nbmrf, ul` sghhbøl nn`e`a`Q`nngr e`tgeàtbhf qug hfltbglg afs prugm`s, p`r` qug sg v`y` prgp`r`laf pfhf ` pfhf.
¶Puâ hfltglbafs sg rgp`s`l gl gstg t`nngr6
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[rgsglt` 88 åtges qug `m`rh`l nfs hfltglbafs ag n`s sge`l`s 0 ` n` 9 y tge`s ag prfpfrhbf-l`nba`a, sghuglhb`s lueârbh`s, fpgr`hbflgs hfl nfs lóegrfs r`hbfl`ngs, sbilfs ag hfep`r`-hbøl, e.h.e., E.H.A., pfrhglt`cg y egaba`s ag vfnuegl.
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[rgsglt` 8= åtges qug `m`rh`l nfs hfltglbafs ag n`s sge`l`s 2 ` n` 04 y tge`s ag ghu`hbf-lgs ag prbegr ir`af, ngliu`cg nigmr`bhf, rgin` ag trgs sbepng y pgråegtrf ag hu`arbnàtgrfs.
¶Høef sg gv`nu`ràl gstfs hfltglbafs6
• Gl n` prbegr` prugm` p`rhb`n sg gv`nu`ràl nfs hfltglbafs ag n` prugm` @.
• Gl n` prugm` l`n sg blhnuygl nfs hfltglbafs ag n` prugm` M.
Nfs tge`s qug lf `p`rghgl gl gstg nbmrf nfs glhfltr`rà gl gn e`tgrb`n ag e`tgeàtbh` ag nfs
irupfs Pubrbiuà, Wt`tnàl y S`hungu f t`embâl pugag blvgstbi`r gl bltgrlgt.
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Blstruhhbflgs3
0. Wsg gst`s pàibl`s sfnf p`r` nggr n`s prgiult`s y glulhb`afs. Lf n`s sumr`yg,lb j`i` `lft`hbflgs f e`rh`s gl gnn`s.
=. N` jfc` ag rgspugst`s sg glhugltr` n l`n ag h a` prugm . ]ghørtgn .
8. Ng` h`a` prgiult` f glulhb`af y n s pfsbmngs rgspugst`s u fphbflgs qugn` rgspflagl.
1. Xgnghhbflg n` rgspugst` hfrrght` y rgnnglg gn hårhunf hfrrgspflabgltg gl n`jfc` ag rgspugst`s. Xfn`egltg ul` fphbøl rgspflag hfrrght`egltg ` n`prgiult`.
7. Lf `rruiug lb j`i` tr`zfs gl ftrfs nui`rgs ag n` jfc` ag rgspugst`s.
4. [`r` rgspflagr gst` prugm` agmgrà utbnbz`r n`pbhgrf lgirf.
9. Wtbnbhg jfc s `p rtg p`r` rg`nbz`r gn prfhgabebgltf ag n`s fpgr`hbflgs.
:. Eba` gn tbgepf qug t`ra` gl rgsfnvgr n` prugm`. [r`htbqug j`st` qug hflsbi`j`hgrnf gl gn eglfr tbgepf pfsbmng.
Iuågsg hfl gn gcgepnf agn rghu`arf.
;) ¶Huàn gs gn rgsunt`af ag fpgr`r ‘0;= + 1;;6
`. =2:m. 1=;h. 8:;a. 10;=
Jfc` ag rgspugst`s
;. `. m. h. a.
0. `. m. h. a.
=. `. m. h. a.
Ag gst` dfre`sg agmg rgnngl`rgn hårhunf.
Gl su jfc` ag rgspugst`sagmgrà rgnngl`r gn hårhunfag n` rgspugst` hfrrght`.
@jfr` rgvbsg su jfc` ag rgspugst`s, n` fphbøl ; y` gstà e`rh`a`.
Óntbe`s rghfegla`hbflgs3
• Xb lf s`mg ul` rgspugst`, lf sg agtgli`, sbi` `agn`ltg.
• Gstà prfjbmbaf gn usf ag h`nhun`afr`.
Blstruhhbflgs3 E`rqug gl n` jfc` ag rgspugst`s gn rgsunt`af hfrrghtf ag n`fpgr`hbøl.
8/20/2019 Libro Polochic Matemática 1er. Sem
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0) ¶Puâ lfemrg rghbmg gn hflcultf ag nfs lóegrfs gltgrfs pfsbtbvfs ag hgrf j`st` bllbtf6
`. l`tur`ngs m. r`hbfl`ngs h. lgi`tbvfs a. brr`hbfl`ngs
=) ¶Huàn gs gn gngegltf lgutrf gl n` euntbpnbh`hbøl6
`. ; m. ‘0 h.0
` a. 0
8) N` gxprgsbøl (7 • 8) • = 5 7 • (8 • =) ¶quâ prfpbga`a gcgepnbh`6
`. abstrbmutbv` m. hfleut`tbv` h. `sfhb`tbv` a. hn`usur`tbv`
1) ¶Huàn ag nfs sbiubgltgs gs ul lóegrf prbef6
`. 9 m. 0; h. ;.=7 a.7
1
7) ¶Puâ v`nfr agmg tglgr ` p`r` qug 0;; + ` 5 0;;6
`. 0;; m. 0; h. 0 a. ;
4) ¶Huàn gs gn v`nfr `msfnutf ag ‘076
`. |‘07| 5 ; m. |‘07| 5 0 h. |‘07| 5 07 a. |‘07| 5 ‘07
9) ¶Huàn gs gn blvgrsf euntbpnbh`tbvf ag 0;; 6
`. ‘ 0;; m.0;;
0 h. 0
0;; a. ‘ 0;;
0
:) ¶Huàn ag nfs lóegrfs sbiubgltgs gs brr`hbfl`n6
`. =.=8=8=8=… m. 8.01072=4… h. 7.4444444… a. :.8888888…
2) ¶Huàn ag nfs glulhb`afs lf rgprgsglt` ul` prfpfrhbøl6
`.1
7 5
7
1 m.
=
1 5
1
: h.
7
8 5
07
2 a.
0
8 5
8
2
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`. 0; m. 04 h. 8= a. 41
00) ¶Huàn gs n` pftglhb` ag (=9)= gxprgs`a` gl m`sg 86
`. 84 m. 87 h. 81 a. 88
0=) ¶Huàn gs gn lóegrf eàs pgqugðf ag n` sgrbg ag lóegrfs rg`ngs 8.9, ‘:.4, ‘1.9, 2.8, ‘0.:6
`. 8.9 m. ;.0 h. ‘1.9 a. ‘:.4
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08) Gl n` sgrbg3 1, 9, 00, 04, ¶huàn gs gn lóegrf sbiubgltg6
`. 02 m. =; h. =0 a. ==
01) Gl n` sgrbg3 0;, 0:, 81, 44, ¶huàn gs gn lóegrf sbiubgltg6
`. 0=: m. 0=2 h. 08; a. 080
07) ¶Huàn gs n` rgspugst` hfrrght` ag n` fpgr`hbøl =; ù 1 + 8 • = ‘ 06
`. 0; m. 07 h. =; a.=;
08
04) ¶Puâ lóegrf gs gn e.h.e. ag 0;, 07, =;6
`. 7 m. 8; h. 4; a. 0=;
09) ¶Huàn gs gn E.H.A. ag 0=, 0:, =16
`. 0 m. = h. 8 a. 4
0:) ¶Puâ agshfepfsbhbøl gl d`htfrgs prbefs gs hfrrght`6
`. =1 5 1 • 4 m. =1 5 : • 8 h. = • 8 • 1 a. =1 5 = • = • = • 8
02) ¶Huàn ag n s sbiubgltgs `re`hbflgs gs vgra`agr`6
`. 8 • _ 07 ‘ = • (4 ‘ = • =) U 5 ‘8 h. 8 • _ 07 ‘ = • (4 ‘ = • =) U 5 ‘82
m. 8 • _ 07 ‘ = • (4 ‘ = • =) U 5 88 a. 8 • _ 07 ‘ = • (4 ‘ = • =) U 5 08
=;) ¶Huàn ag n s gxprgsbflgs sbiubgltgs gs hfrrght`6
`. 8 < ‘8 m. 8 > ‘8 h. ‘8 < 8 a. ‘8 < ‘8
=0) ¶Huàn gs n` rgspugst` hfrrght` ag gngv`r ( 0;;
1 ); 6
`. =7 m. 0; h. 0 a. ;
==) ¶Huàn ag n s sbiubgltgs fpgr`hbflgs gs hfrrght 6
`. 1= • 18 5 044 m. 1= • 18 5 047 h. 1= • 18 5 17 a. 1= • 18 5 14
=8) ¶Puâ gxprgsbøl ag n`s sbiubgltgs gs gqubv`ngltg ` 2= ù 8=6
`. (2 ù 8)= ‘ = m. (2 ù 8)= ù = h. (2 ù 8)= a (2 ù 8)= • =
=1) ¶Puâ rgsunt`af sg fmtbglg `n sue`r8
: +
0
16
`.1
: m.
7
: h.
1
8= a.
1
0=
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=7) ¶Huàn gs gn rgsunt`af agn prfauhtf7
= •
1
8 •
=
7 6
`.8
1 m.
00
0; h.
:7
8; a.
1
8
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1
> =
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7
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9
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1 ≠
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1 ag i`nøl ag pbltur` `zun,
0
= ag i`nøl ag pbltur` mn`lh` y
=
8 ag i`nøl ag pbltur` rfc`. ¶Huàltfs i`nflgs ag pbltur` sg utbnbz`rfl gl tft`n6
`. 8 i`nflgs `. 7 i`nflgs h.0:
7 ag i`nflgs a.
=8
0= ag i`nflgs
=2) Mglc`eål tbglg trgs rfnnfs ag n`emrg hfl ul n`rif ag 07, =; y =7 egtrfs h`a` ulf. Xb agsg` abvbabrtfaf gn `n`emrg gl pbgz`s biu`ngs, ag n` e`yfr nflibtua pfsbmng p`r` lf agspgrabhb`r `n`emrg, ¶quân`rif agmg tglgr h`a` pbgz`6
`. 0 e m. 8 e h. 7 e a. 07 e
8;) ]gibl` vbsbt` ` su jgre`lf [garf h`a` 4 aå`s y ` su jgre`l` ]fs`rbf h`a` 1, sb jfy vbsbtø ` `emfs,¶huàltfs aå`s agmgl p`s`r p`r` qug vbsbtg ` nfs afs gn ebsef aå`6
`. 1 a. 4 h. 0= a. 0:
80) Afs pbltfrgs t`ra`l 2 aå`s gl pblt`r ul` h`s`. Xb hfltr`t`l ul tgrhgr pbltfr, ¶huàltfs aå`s t`ra`ràlgl pblt`r n` h`s` nfs trgs cultfs6
`. 8 aå`s m. 1 aå`s h. 7 aå`s a. 4 aå`s
8=) Gl ul hårhunf ag gstuabf ag BIG] nfs gstuab`ltgs ag màsbhf y m`hjbnngr`tf sue`l 0=;, ag nfs hu`ngs gn 9; % sfl eucgrgs y gn rgstf, jfemrgs. Ag nfs jfemrgs, gn =7 % gstuab` m`hjbnngr`tf. ¶Huàltfs jfemrgsgstuab`l m`hjbnngr`tf6
`. : m. 2 h. 0; a. 07
88) Gl ul rghbpbgltg j`y 7;;; he8 ag ul` sfnuhbøl. Xb 0 nbtrf gs biu`n ` 0;;; he8, ¶huàltfs nbtrfs tglgefsag n` sfnuhbøl6
`. 7 nbtrfs m. 8 nbtrfs h. 1 nbtrfs a. 4 nbtrfs
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8/20/2019 Libro Polochic Matemática 1er. Sem
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0) ¶Puâ rgsunt`af sg fmtbglg `n fpgr`r 7(= + 8 7 )6
`. =7 7 m. 9 + : 7 h. 0; + 07 7 a. 0; + 8 7
=) ¶Huàn gs n` hflst`ltg lueârbh` gl n` gxprgsbøl 7 xy86
`. x m. y h. 7 a. 8
8) ¶Puâ prfpbga`a eugstr` n` gxprgsbøl `(h + a ) 5 `h + `a 6
`. hgrr`aur` m. abstrbmutbv` h. hfleut`tbv` a. gngegltf `msfrmgltg
1) ¶Puâ pfnblfebf rgprgsglt` gn pgråegtrf ag ul rghtàliunf qug ebag x + 0 ag `lhjf y = x ag n`rif6
`. 1 x m. 8 x + 0 h. 4 x + = a. ( x + 0) + (= x)
7) ¶Puâ gxprgsbøl gcgepnbh` hfrrght`egltg n` euntbpnbh`hbøl ag pftglhb`s ag biu`n m`sg6
`. `e • `l 5 `e + l h. `e • `l 5 `e • l
m. `e • `l 5 `e ‘ l a. `e • `l 5 `e ù l
4) ¶Huàn ag n`s `re`hbflgs sbiubgltgs gs vgra`agr`6
`. 1 i • 1 i • 1 i 5 0=i8 h. 1 i • 1 i • 1 i 5 0= i
m. 1 i • 1 i • 1 i 5 41 i 8 a. 1 i • 1 i • 1 i 5 1 i 8
9) ¶Høef sg gxprgs` 7 x= + 8 gl gn ngliu`cg usu`n6
`. Gn afmng ag ul lóegrf x eàs trgsm. Gn quåltupnf ag ul lóegrf x `n hu`ar`af eàs trgsh. Wl lóegrf x `n hu`ar`af eàs trgsa. Hblhf pfr x eàs trgs
:) ¶Huàn gs gn rgsunt`af hfrrghtf ag (: x= y8)(‘8 x4 y= z 8)6
`. ‘=1 x1 yz 8 m. ‘=1 x: y7 z 8 h. ‘00 x: y7 z 8 a. =1 x0= y4 z 8
2) ¶Huàltfs târeblfs dfre`l gn pfnblfebf =
8 z + 8 z 7 + 1 z = ‘ : y + 4 y=6
`. 7 m. 0; h. 00 a. 01
0;) ¶Høef sg gxprgs` gl ngliu`cg e`tgeàtbhf "Gn pgråegtrf ([) ag ul rghtàliunf gs gn afmng prfauhtfag n` sue` agn `lhjf (Y) y gn n`rif (N)"6
`. [ 5 =Y + N m. [ 5 Y + =N h. [ 5 =(Y + N) a. [ 5 =(Y • N)
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00) ¶Huàn gs gn d`htfr hfeól gl n` gxprgsbøl = x8 ‘ 1 x= + : x6
`. = m. x h. x= a. = x
0=) ¶Huàn ag n`s gxprgsbflgs sbiubgltgs gs ul trblfebf hu`ar`af pgrdghtf6
`. x= ‘ 04 m. x= ‘ 7 z + 01 h. x= ‘ : x + 41 a. x= + 0; x + =7
08) ¶Huàn gs gn rgsunt`af ag sbepnbh`r ‘8 x ‘ = + 8 x= y ‘ 1 + 7 x ‘ 4 x=6
`. 8 x= y ‘ : x= ‘ 4 m. 8 x= y ‘ 4 x= + = x ‘ 4 h. 8 x= y ‘ 4 x= + : x + 4 a. ‘9 x= y
01) ¶Puâ rgsunt`af sg fmtbglg ag (1 x= + 8 x + 0) + (9 x= ‘ 1 x + 4)6
`. 09 x m. 09 x= h. 0; x + 9 a. 00 x= ‘ x + 9
07) ¶Huàn gs n` abdgrglhb` ag (:`m= ‘ 4mh + 1) ‘ (=`m= ‘ 2mh + 0;)6
`. 4`m= + 8mh + 4 m. 4`m= + 8mh ‘ 4 h. 4`m= ‘ 8mh + 4 a. 4`m= ‘ 8mh ‘ 4
04) ¶Huàn gs gn rgsunt`af hfrrghtf ag (8 x4 y1)(‘1 x8 y= z 1)6
`. 0= x8 y= z 1 m. ‘0= x2 y4 z 1 h. ‘0= x= y= z 1 a. ‘9 x2 y4 z 1
09) ¶Huàn gs gn rgsunt`af hfrrghtf ag euntbpnbh`r ( x + 2)( x ‘ =)6
`. x= ‘ 0: m. = x + 9 h. x= + 9 x ‘ 0: a. x= ‘ 9 x + 0:
0:) ¶Puâ mblfebf gstà gxp`labaf hfrrght`egltg6`. (j ‘ o )= 5 j= ‘ o h. (q ‘ =)= 5 q= ‘ 1q + 1
m. ( x + 7)= 5 x= + =7 a. (m ‘ 1)= 5 m= + :m ‘ 04
02) ¶Huàn gs gn rgsunt`af ag abvbabr (07 x1 + 0= x8 ‘ 2 x=) ù 8 x6
`. 7 x8 + 1 x= ‘ 8 x m. 7 x1 + 1 x8 ‘ 8 x= h. 07 x7 + 1 x1 ‘ 8 x8 a. 07 x8 + 07 x= ‘ 07 x
=;) ¶Huàngs sfl nfs d`htfrgs ag x= ‘ 0= x + =96
`. ( x + :)( x ‘ 1) m. ( x ‘ 4)( x + 4) h. ( x ‘ 2)( x ‘ 8) a. ( x + 1)( x + 8)
=0) ¶Puâ rgsunt`af sg fmtbglg `n v`nu`r gn pfnblfebf = x= + 07 sb x 5 86
`. 88 m. =4 h. =0 a. 09
==) ¶Puâ v`nfr sg fmtbglg ag =`h • =`m sb ` 5 =, m 5 2 y h 5 0
86
`. 1 m. : h. =1 a. 0;
8
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=8) ¶Puâ rgsunt`af sg fmtbglg `n sbepnbh`r0;w8 z 4
7w= z 6
`. 7wz 7 m. =wz 7 h. =w8/= z 4/0 a. 7w8/= z 4/0
=1) ¶Huàn gs n` rgspugst` hfrrght` ag x= + 4 x + :
x + =6
`. x + = m. x + 1 h. x ‘ = a. x ‘ 1
=7) ¶Huàn ag n s sbiubgltgs fpgr`hbflgs gs hfrrght 6
`. ( y ‘ 1)= 5 ‘04 y= h. ( y ‘ 1)= 5 y= ‘ 04
m. ( y ‘ 1)= 5 y= ‘ : y + 04 a. ( y ‘ 1)= 5 y= + 04
=4) ¶Huàn gs gn v nfr ag x gl n` ghu`hbøl = x ‘ 8 5 786
`. = m. 4 h. 0; a. =:
=9) ¶Huàn gs gn v nfr ag x gl n` ghu`hbøl 7 x + 4 5 0; x + 76
`. 0 m. = h. 0
= a. 0
7
=:) Gl agtgrebl`af aå`, n` tgepgr`tur` gl Pugtz`ntgl`lif ` n`s 83;; ag n` e`ð`l` gr` ag ‘1 »H. Nugifn` tgepgr`tur` `uegltø 0 »H h`a` jfr`, ¶huàn gr` n` tgepgr`tur` ` n`s 0;3;; ag n` e`ð`l` gsg aå`6
`. ; »H m. 8 »H h. 9 »H a. ‘=
=2) Wl` ltgl` ag 0= egtrfs ag `ntf prfyght` ul` sfemr` ag : egtrfs. ¶Huàn sgrà n` `ntur` ag ul gabhbf
qug ` n` ebse` jfr` prfyght` ul` sfemr` ag 4 egtrfs6`. 2 egtrfs m. 0; egtrfs h. 07 egtrfs a. =; egtrfs
8;) Qfafs nfs aå`s, X`lar` h`ebl` 0
= oe pfr n` e`ð`l` y 8
1 ag oe pfr n` t`rag. ¶Huàltfs obnøegtrfs
h`ebl` h`a` aå`6
`. 8
= oe m. 1
4 oe h. 7
1 oe a. 8
: oe
80) Gl ul e`p` h`a` 1 he rgprgsglt`l =0; oe ag abst`lhb` rg`n. Xb afs hbua`ags gstàl sgp`r`a`s pfr 0; he gl gn e`p`, ¶quâ abst`lhb` rg`n j`y gltrg gnn`s6
`. 1=; oe m. 7=7 oe h. :4; oe a. 0;7; oe
8=) Wl h`eblf hfeul n gs hflstrubaf pfr 4; tr`m`c`afrgs gl 7; aå`s. ¶Huàltfs tr`m`c`afrgs sg jumbgr`llghgsbt`af p`r` hflstrubr gn ebsef h`eblf gl =; aå`s6
`. 0;; tr`m`c`afrgs m. 00; tr`m`c`afrgs h. 0=; tr`m`c`afrgs a. 07; tr`m`c`afrgs
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