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

Feb 9, 2016

$\frac{3}{4} \left(x - 1\right) + \frac{1}{2} {\left(x - 1\right)}^{2} + \frac{1}{4} \left(x + 1\right)$

#### Explanation:

firstly note that the factors of ${\left(x - 1\right)}^{2}$

are (x-1) and ${\left(x - 1\right)}^{2}$

hence ${x}^{2} / \left({\left(x - 1\right)}^{2} \left(x + 1\right)\right) = \frac{A}{x - 1} + \frac{B}{x - 1} ^ 2 + \frac{C}{x + 1}$

now multiply through by ${\left(x - 1\right)}^{2} \left(x + 1\right)$

${x}^{2} = A \left(x - 1\right) \left(x + 1\right) + B \left(x + 1\right) + C {\left(x - 1\right)}^{2} \ldots \ldots \left(1\right)$

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 $\Rightarrow B = \frac{1}{2}$

let x = -1 in(1) : 1 = 4C $\Rightarrow C = \frac{1}{4}$

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

$\Rightarrow {x}^{2} / \left({\left(x - 1\right)}^{2} \left(x + 1\right)\right) = \frac{3}{4} \left(x - 1\right) + \frac{1}{2} {\left(x - 1\right)}^{2} + \frac{1}{4} \left(x + 1\right)$