# How do you write the partial fraction decomposition of the rational expression 1/(x^3-5x^2)?

Dec 14, 2015

$\frac{1}{{x}^{3} - 5 {x}^{2}} = \frac{1}{5 x} - \frac{1}{5 {x}^{2}} - \frac{1}{5 \left(x - 5\right)}$

#### Explanation:

Factor the denominator.

$\frac{1}{{x}^{2} \left(x - 5\right)} = \frac{A}{x} + \frac{B}{x} ^ 2 + \frac{C}{x - 5}$

$1 = A x \left(x - 5\right) + B \left(x - 5\right) + C {x}^{2}$

$1 = A {x}^{2} - 5 A x + B x - 5 B + C {x}^{2}$

$1 = {x}^{2} \left(A + C\right) + x \left(- 5 A - 5 B\right) + 1 \left(- 5 B\right)$

Thus, $\left\{\begin{matrix}A + C = 0 \\ - 5 A - 5 B = 0 \\ - 5 B = 1\end{matrix}\right.$

Solve to find that $\left\{\begin{matrix}A = \frac{1}{5} \\ B = - \frac{1}{5} \\ C = - \frac{1}{5}\end{matrix}\right.$

Plug these back in:

$\frac{1}{{x}^{2} \left(x - 5\right)} = \frac{1}{5 x} - \frac{1}{5 {x}^{2}} - \frac{1}{5 \left(x - 5\right)}$