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

Feb 16, 2016

$\frac{4}{x - 1} - \frac{3}{x}$

#### Explanation:

first step is to factor the denominator

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

Since these factors are linear,the numerators will be constants, say A and B.

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

multiply through by x(x-1)

${x}^{3} + 3 = A \left(x - 1\right) + B x \text{..............................................(1)}$

The aim now is to find the values of A and B. Note tat if x = 1 the term with A will be zero and if x = 0 the term with B will be zero.
This is the starting point for finding A and B.

let x = 1 in (1) : 4 = B

let x = 0 in (1) :$3 = - A \Rightarrow A = - 3$

$\Rightarrow \frac{{x}^{3} + 2}{{x}^{2} - x} = \frac{4}{x - 1} - \frac{3}{x}$