Question

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

Answer 1

`1/(3(x-2))-1/(3(x+2))-1/(3(x^2+2))`

Explanation:

Factor the denominator.

`(x^2+4)/((x+2)(x-2)(x^2+2))=A/(x+2)+B/(x-2)+(Cx+D)/(x^2+2)`

`x^2+4=A(x-2)(x^2+2)+B(x+2)(x^2+2)+(Cx+D)(x^2-4)`

If `x=-2`:

`8=-24A`
`A=-1/3`

If `x=2`:

`8=24B`
`B=1/3`

If `x=0`:

`4=-4A+4B-4D`
`1=-A+B-D`

Plug in `A` and `B` to see that `D=-1/3`.

If `x=1`:

`5=-3A+9B-3C-3D`

Plug in all the known values to find that `C=0`.

Plug these all back in to see that the expression decomposes into

`1/(3(x-2))-1/(3(x+2))-1/(3(x^2+2))`