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

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