Section 5.3 Partial Fractions - Dublin, Virginia · 2020. 9. 11. · Section 5.3 Partial Fractions Friday, April 19, 2019 8:54 A M Paral_fra ...

,. i -+ ')

~~ {~+Cl.)

(e~ (Sf-«.) t'I

5.3 Partial Fraction Decomposition

Sl-'l .-~A, A-2 /J;,

----- + ---,~ .. . -a. ----s f-0- (Sto. t ls ,.. .. >" A. S-tr ~ ~

t;,. +as +-b A,s~A~ A3 S -1--Ay ~'2-J.tt~ib ~(,;,,_,+as +b) 2

Section 5.3 Partial Fractions Friday, April 19, 2019 8:54 AM

Par al_fra…

5.3 Partial Fraction Decomposition

Partial Frac ion consist of decomposing a rational function into simpler component fracti

Exam:~: 5;3.1. :nornint~ m_; et"ct of di ctinct ,r· factorn 8 x2 + 6x+5 --- ::: -- + f'{ fS }(X-1-1)

(~r4i)(Xf.l) Xt-) Xi- f

3 X + 1- = A- ( x ~ 1) + (3( X .,_. S )

3,~ -i '=- A-Ji t-8;( + 4 +-S-!s ......._______...., ~ -~----(~-:-A--t-B) '.3~ 4+-1

t = A-r~ -::l ~ A-'-f : L(~ 2-

3~1-'4 ~-4-, ~e - ' l~~)(Xtl) = V +f) (x +-fl

~l-St-1-1~ =-'-(~(-J.-1)2- + ?:,{ >l-l){J£~1) +-l~ (.l('-f-l) <E: X-:.\ s- i+r~-=- c.(l x-=--3 H'l\t-i'/+-13 -:.A(q,)

i ::::. I./(! ~ C( "= I I 4 Q.=C_ 'i-= II

... •~-='-l+i(-1)(3)+ C, 3-=-~~

- \ -:: s -

Example 5.3.3. Den minator fntain irreducible quadratic factor , none of which is rep(

0A-2x2 +x -8 _ ~)( +-~--8 A- Bxre.. v(x~J-'(I\ "',tl- - ____ + A. ~ x3 +4x ...__ -- --- 'T

xlx 2 +«4) - X ~'l-'I

l x,. ~x-i ;\{x2+<() +- (~.x"-<!) )(

X-=O -~:<-(ti -2= ,4

2 o ._ ~tz. 1-l ( 2. 'I. 2 I- f -~ -: -2 X' - V

Ex'- / _, sL .,_ 'I <54- sa.

r.,,-+rd t-r6.v(.r~s 52 + l/ A- (3 c.. ~

-------- ::: f"rz +- - .,_ ---- .,_ --- s «'·•)

si. ls -1){ sr1) s s~ s-, sf-/ rH,)

D

e, ,,_ +ti -= 4 s c ~ -,Jc, -1-1) ~ Brr-, v .s"'') 1- e ( s-,..,) i2. .,. I) s''-r s-,;

CS = 0 '-I -: - ~ -:::::') \s ... ..... '-/

S-::. , s-~ , c ,_-, C. : >/z

S =- -1 )- -2~ ='? b ::.-'r'"/2 z. 6-

5-= 1 t--=- A- (2)( 1)(3) -'( ( ,}(~)-,. .f (~)l)(j-½ (14}{ 1)

3 =- (,A -12. F~D -10

~ ~ b A + 8 JI

;1_-I s -L/ -i- )/z. t-

l. ~~ S-1 S ~ I row 3 r o w tf row 'I _1

r f t-' f ~-, f -f J (

~.fl

Ln" f.n( ~l~<-e 1YM\S kY-hl1 of.

5"0 s

B + CSt-1:) (Sfl)z. S2 ~4S .f..13

- ~ 50 s-: A cs~1)£s2flfS'-1'?) ~(~2+l(s1-r3)+-@s1-~JCs-1-1/-

~~ _, -SO::. B (I-'( 4-13) -',0-: l0{3 - ~ = l3 ,..._ ______

i)ui\l~ i\lt ols [ l ':) 0 ;'A._[(~ H) ( '2. S -1-lf .,_ ( ~ 7. f&{S t-1 :s)J - ', (--;-; ,-. l/) 4-

+ (c_ c; ~ Q) l ( S' f I)~ {SH)i.(e)

s = - ' r o -= A ( , ") - 0 r 2 c -,) H/) ) 0 = lo A -- 1° Io 2

c,o =to A- -1-, ~ A y;,, ~ "" . - , ., - /I

t:,v - ·- -

'1 ~ A~ ----.. ~

45: -~ !,O:= b( q-i+1~)~ (-2CY\))fl)(-1)+ C. - c..1 : + Lf e - 2 ~ 1, e - 1.t = ;-C - '2 l:)

')-=. c, 5" O ~ ' [ '( ~ I '3 l - ~ ( ~) + 'D ( 2.l I ) +- c_ j O ::: l o 2 - 2 o +- 2 t) +e_ s-0 ~ «z +-zf).,.e

,- '32-:::r+-zt> -~l = -b-1--z.,h

-4= '>r -z l) -2,-=ZJ)

-1- - ~l = t i-L D <E- - 13 = 1) ..... - 3,-= 'c_

-b=

r

-------- ---------

-+(; 5 • I 3 - -(.5-1-1 )2.

11. attn n 1, 2, ... 12. nt ' iawl

(G~w

'J' ( _ c )2 J w2 ._.._. ,s ,.,3

s ~ 1--'IS t 13 - S 2 .,_ 'IS .,. '-I ? 13 - 4 -

" (S+2) I - ---~- + ..._ ___ _ - t 5 #- t.) 2 ~ 1 i.

_, ) ' - i 2 ;; ~, C e 7 I

-lb

=' ,e Ceb (3 t)

(<Jt.u ~ I/

foe...., .VI 3 0(-:. - ~

<:v :. .3 roc..o .. , 2.