Question

How do you write the partial fraction decomposition of the rational expression `(x-5)/(x-2)^2`?

Answer 1

`(x-5)/(x-2)^2=1/(x-2)-3/(x-2)^2`

Explanation:

`(x-5)/(x-2)^2=A/(x-2)+B/(x-2)^2`

`x-5=A(x-2)+B`

`x-5=Ax-2A+B`

`x-5=x(A)+1(-2A+B)`

Thus, `{(A=1),(-2A+B=-5):}`

So, `{(A=1),(B=-3):}`

Plug back in:

`(x-5)/(x-2)^2=1/(x-2)-3/(x-2)^2`