How do you write the partial fraction decomposition of the rational expression #(x^2 + 5x - 7 )/( x^2 (x+ 1)^2)#?

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Answer 1 · 2016-10-23T19:01:40

The decomposition is

#(x^2+5x-7)/(x^2(x+1)^2)=-7/x^2+19/x-19/(x+1)-11/(x+1)^2#

Let A,B,C,D be constants
The fraction decomposition of the rational expression

#(x^2+5x-7)/(x^2(x+1)^2)=A/x^2+B/x+C/(x+1)+D/(x+1)^2#

#=(A(x+1)^2+Bx(x+1)^2+Cx^2(x+1)+Dx^2)/(x^2(x+1)^2)#

So #x^2+5x-7=A(x+1)^2+Bx(x+1)^2+Cx^2(x+1)+Dx^2#

let #x=0#, then #-7=A#

let #x=-1# then #-11=D#

Comparing coefficients of #x#

#5=2A+B# so #B=5-2A=5+14=19#
Comparing the coefficients of #x^2#
#1=A+2B+C+D# so #C=1-A-2B-D=1+7-38+11=-19#

so the final result is
#(x^2+5x-7)/(x^2(x+1)^2)=-7/x^2+19/x-19/(x+1)-11/(x+1)^2#