Question

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

Answer 1

`3/4(x-1) + 1/2(x-1)^2 + 1/4(x+1)`

Explanation:

firstly note that the factors of `(x-1)^2`

are (x-1) and `(x-1)^2`

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

now multiply through by `(x-1)^2(x+1)`

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

now require to find values for A , B and C . Note that if x = 1 , the terms with A and C will be zero and if x = -1 the terms with A and B will be zero. This is the starting point for finding values.

let x = 1 in (1) : 1 = 2B `rArr B = 1/2`

let x = -1 in(1) : 1 = 4C `rArr C = 1/4`

can now choose any value of x , to substitute into equation (1)

let x = 0 in (1) : 0 = -A + B + C

hence A = B + C # = 1/2 + 1/4 = 3/4

`rArr x^2/((x-1)^2(x+1)) = 3/4(x-1) + 1/2(x-1)^2 + 1/4(x+1)`