Question

How do you express `(x^2 + 5x - 7 )/( x^2 (x+ 1)^2)` in partial fractions?

Answer 1

The answer is `=-7/x^2+19/x-11/(x+1)^2-19/(x+1)`

Explanation:

`(x^2+5x-7)/(x^2(x+1)^2)=A/x^2+B/x+C/(x+1)^2+D/(x+1)`
`(A(x+1)^2+Bx(x+1)^2+Cx^2+Dx^2(x+1))/(x^2(x+1)^2)`

`x^2+5x-7=A(x+1)^2+Bx(x+1)^2+Cx^2+Dx^2(x+1)`
let `x=0` `=>` `-7=A`
`x=-1` `=>` `-11=C`
coefficentsof `x^2` `=>` `1=A+2B+C+D`
coefficients of `x` `=>` `5=2A+B` `=>` `B=19`
`:.D=-19`
So, `(x^2+5x-7)/(x^2(x+1)^2)=-7/x^2+19/x-11/(x+1)^2-19/(x+1)`