Question

How do you write the partial fraction decomposition of the rational expression `(x-1)/(x^3 +x)`?

Answer 1

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

Explanation:

Let's factorise the denominator `x^3+x=x(x^2+1)`
so the partial fraction decomposition is
`(x-1)/(x^3+x)=(x-1)/(x(x^2+1))=(Ax+B)/(x^2+1)+C/x`
`=(x(Ax+B)+C(x^2+1))/((x)(x^2+1))`
So we have `x-1=x(Ax+B)+C(x^2+1)`
let `x=0` `=>` `-1=C` `=>` `C=-1`
Coefficient of x, `1=B` `=>` `B=1`
coefficients of x^2, `0=A+C` `=>` `A=1`

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