Question

How do you express `(x-1)/(x^3 +x^2)` in partial fractions?

Answer 1

The answer is `(x-1)/(x^3+x^2)=2/x-1/x^2-2/(x+1)`

Explanation:

Some factorisation to start with
`(x-1)/(x^3+x^2)=(x-1)/((x^2)(x+1))`
`=A/x+B/x^2+C/(x+1)`
`=(Ax(x+1)+B(x+1)+Cx^2)/((x^2)(x+1))`

So now we can solve for `A, B,and C`
`x-1=Ax(x+1)+B(x+1)+Cx^2`
let x=-1, `-2=C` `=>` `C=-2`
Coefficients of `x^2`, `0=A+C` `=>` `A=2`
Coefficients of `x`, `1=A+B` `=>` `B=-1`

And finally, we have
`(x-1)/(x^3+x^2)=2/x-1/x^2-2/(x+1)`