Question

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

Answer 1

`(x+1)/(x^2(x-1))=-2/x-1/x^2+2/(x-1)`

Explanation:

The decomposition that should be set up from this problem is

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

Note that since `x^2` was in the denominator, both `x` and `x^2` are included in the denominators of the decomposition.

From here, we can get a common denominator of `x^2(x-1)` in each term.

`(x+1)/(x^2(x-1))=(Ax(x-1))/(x^2(x-1))+(B(x-1))/(x^2(x-1))+(Cx^2)/(x^2(x-1))`

The denominators are equal, so they can be removed, giving us the equation

`x+1=Ax(x-1)+B(x-1)+Cx^2`

Set `x=1` so both the `A` and `B` terms will equal `0`.

`1+1=A(1)(0)+B(0)+C(1)`

`ul(C=2`

Set `x=0` so both the `A` and `C` terms will equal `0`.

`0+1=A(0)(-1)+B(-1)+C(0)`

`ul(B=-1`

Now that we know the values of `B` and `C`, we can arbitrarily set `x=2` and substitute in the known values of `B` and `C` to solve for `A`.

`2+1=A(2)(1)+(-1)(1)+2(4)`

`3=2A+7`

`ul(A=-2`

This leaves us with the decomposition of

`(x+1)/(x^2(x-1))=-2/x-1/x^2+2/(x-1)`