8/13/2019 3 6 Partial Fractions

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^orteoc Iromtejks

=.9

Ektrjhumtejk

Et es jitdk bdcpiuc tj `rdoa hjwk o mjgpcemotdh ocfd`roem iromtejk ektj o sug ji segpcdr iromtejks. Ijr

dxogpcd et mok `d sbjwk tbot 8x+ 6

x0 + =x+ 0bos tbd sogd vocud os

?

x+ 0+

=

x+ ?ijr oky vocud jix.

Vd soy tbot

8x+ 6

x0 + =x+ 0 es ehdktemoccy dquoc tj

?

x+ 0+

=

x+ ?

okh tbot tbd porteoc iromtejks ji

8x+ 6

x0 + =x+ 0 ord

?

x+ 0 okh

=

x+ ? .

Xbd o`ecety tj dxprdss o iromtejk os ets porteoc iromtejks es portemucorcy usdiuc ek tbd stuhy ji Copcomdtroksijrgs, \-troksijrgs, Mjktrjc Xbdjry okh Ektdfrotejk. Ek tbes ]dmtejk wd dxpcoek bjw porteociromtejks ord ijukh.

^rdrdquesetds

@dijrd stortekf tbes ]dmtejk yju sbjuch . . .

`d iogeceor wetb ohhetejk, su`tromtejk,guctepcemotejk okh hevesejk ji ocfd`roemiromtejks

Cdorkekf Jutmjgds

Jk mjgpcdtejk yju sbjuch `d o`cd tj . . .

hestekfuesb `dtwddk prjpdr okh egprjpdriromtejks

dxprdss ok ocfd`roem iromtejk os tbd sug ji ets

porteoc iromtejks

9> BDCG (0>>4) 7 O(>) +@(0 + =)

= 7 @

sj tbot tbd mjkstokt @ gust dquoc =. Xbd mjkstokt O mok `d ijukh detbdr `y su`stetutekf sjgdjtbdr vocud ijr x jr octdrkotevdcy `y dquotekf mjdmedkts.

J`sdrvd tbot, `y rdorrokfekf tbd refbt-bokh sehd, Dquotejk (*) mok `d wrettdk os

6x+ ?> 7 (O+ 0@)x+ (O+ =@)

Mjgporekf tbd mjdmedkts ji x jk `jtb sehds wd sdd tbot 6 7 O+ 0@. Vd ocrdohy akjw@ 7 =

okh sj

6 7 O+ 0(=)

7 O+ 9

irjg wbemb O7 ?. Vd mok tbdrdijrd wretd

6x+ ?>

0x0 + 4x+ =7

?

0x+ =+

=

x+ ?

Vd bovd summddhdh ek dxprdssekf tbd fevdk iromtejk os tbd sug ji ets porteoc iromtejks. Xbd rdsuctmok ocwoys `d mbdmadh `y ohhekf tbd iromtejks jk tbd refbt.

BDCG (0>>4)+

M

x ?

Ierst guctepcy `jtb sehds `y (x0 +x+ ?>)(x ?))

Dvocuotd M`y cdttekf x7 ?)+

?

=(x ?)

92 BDCG (0>>4)>4) es o mjkstokt, O soy,

o pjcykjgeoc ji hdfrdd ? bos tbd ijrg Ox+@,

o pjcykjgeoc ji hdfrdd 0 bos tbd ijrg Ox0 +@x+M,

okh sj jk.

Ei, ijr dxogpcd, tbd egprjpdr iromtejk es sumb tbot tbd kugdrotjr bos hdfrdd 4 okh tbd hdkjgekotjrbos hdfrdd =, tbdk k h7 0, okh wd kddh tj ohh o pjcykjgeoc ji tbd ijrg Ox0 +@x+M.

Ady ^jekt ?6

Ei o iromtejk es egprjpdr ok ohhetejkoc tdrg es ekmcuhdh toaekf tbd ijrg ji o pjcykjgeoc ji hdfrddk h, wbdrd k es tbd hdfrdd ji tbd kugdrotjr okh h es tbd hdfrdd ji tbd hdkjgekotjr.

6> BDCG (0>>4)x0

08x 05(x+ 0)0(x =)

, (i) 8x4

+ 2x8

+ 0=x=

+ 06x0

+ 04x+ 5(x0 +x+ ?)(0x+ ?)

Okswdrs

(o) ? + ?

x+ 0 (`) = +

0

x =, (m) ? + x+

?

x+ ? (h) 0x+ = +

?

x+ 0,

(d) ?

(x+ 0)0+

?

x+ 0+

?

x =+ =x0 +x+ 0, (i) 0x0 +x+ 6 +

?

0x+ ?+

?

x0 +x+ ?

BDCG (0>>4)