Question

How do you write the partial fraction decomposition of the rational expression `(2x - 3) / (x - 1)^2`?

Answer 1

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

Explanation:

Note that `(x-1)^2`

has factors (x-1) and `(x-1)^2`

`rArr (2x-3)/(x-1)^2 = A/(x-1) + B/((x-1)^2` .................(*)

Multiply both sides by `(x-1)^2`

so 2x -3 = A(x-1) + B

To find A and B choose values for x and substitute into (*)

Note: choose x =1 and the term with A becomes 0.

x = 1 : substitute into )*) : -1 = 0 + B → B= - 1

Choose any value for x , say x = 0

x = 0 : substitute into (*) : - 3 = - A+B = -A - 1

so - A -1 = -3 → A = 2