Question

How do you write the partial fraction decomposition of the rational expression `x/((x-1)(x^2+4)`?

Answer 1

The answer is `=(1/5)/(x-1)+(-1/5x+4/5)/(x^2+4)`

Explanation:

Let's write the decomposition
`x/((x-1)(x^2+4))=A/(x-1)+(Bx+C)/(x^2+4)`

`=(A(x^2+4)+(Bx+C)(x-1))/((x-1)(x^2+4))`

`:. x=A(x^2+4)+(Bx+C)(x-1)`

Let `x=1` `=>` `1=5A` `=>` `A=1/5`
Coefficients of `x^2` `=>` `0=A+B` `=>` `B=-1/5`
And `0=4A-C` `=>` `C=4/5`

So, `x/((x-1)(x^2+4))=(1/5)/(x-1)+(-1/5x+4/5)/(x^2+4)`