Question

How do you write the partial fraction decomposition of the rational expression `(9x^2 + 1)/(x^2(x − 2)^2)`?

Answer 1

`(9x^2+1)/(x^2(x-2)^2) = 1/(4x)+1/(4x^2)-1/(4(x-2))+37/(4(x-2)^2)`

Explanation:

Given that we have squared linear factors in the denominator, the decomposition will take a form like:

`(9x^2+1)/(x^2(x-2)^2) = A/x+B/x^2+C/(x-2)+D/(x-2)^2`

Multiplying both sides by `x^2(x-2)^2` this becomes:

`9x^2+1 = Ax(x-2)^2+B(x-2)^2+Cx^2(x-2)+Dx^2`

Putting `x=0` we get:

`1 = 4B" "` so `" "B = 1/4`

Putting `x=2` we get:

`37 = 4D" "` so `" "D=37/4`

Looking at the coefficient of `x`, we have:

`0 = 4A-4B = 4A-1" "` and hence `A=1/4`

Looking at the coefficient of `x^3`, we have:

`0 = A+C" "` and hence `C=-A=-1/4`

So:

`(9x^2+1)/(x^2(x-2)^2) = 1/(4x)+1/(4x^2)-1/(4(x-2))+37/(4(x-2)^2)`