# How do you express 2/(x^3-x^2) in partial fractions?

Mar 23, 2016

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

#### Explanation:

The denominator splits into factors as: ${x}^{3} - {x}^{2} = {x}^{2} \left(x - 1\right)$

So we solve for a partial fraction decomposition of the form:

$\frac{2}{{x}^{3} - {x}^{2}}$

$= \frac{A}{x - 1} + \frac{B}{x} + \frac{C}{x} ^ 2$

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

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

Equating coefficients, we get:

$\left\{\begin{matrix}A + B = 0 \\ C - B = 0 \\ - C = 2\end{matrix}\right.$

Hence we find:

$\left\{\begin{matrix}C = - 2 \\ B = - 2 \\ A = 2\end{matrix}\right.$

So:

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