# How do you write the partial fraction decomposition of the rational expression  1/((x^4)-6(x^3))?

Dec 14, 2015

$\frac{1}{216 \left(x - 6\right)} - \frac{1}{216 x} - \frac{1}{36 {x}^{2}} - \frac{1}{6 {x}^{3}}$

#### Explanation:

Factor the denominator.

$\frac{1}{{x}^{3} \left(x - 6\right)} = \frac{A}{x} + \frac{B}{x} ^ 2 + \frac{C}{x} ^ 3 + \frac{D}{x - 6}$

$1 = A \left({x}^{2}\right) \left(x - 6\right) + B \left(x\right) \left(x - 6\right) + C \left(x - 6\right) + D \left({x}^{3}\right)$

$1 = A {x}^{3} - 6 A {x}^{2} + B {x}^{2} - 6 B x + C x - 6 C + D {x}^{3}$

$1 = {x}^{3} \left(A + D\right) + {x}^{2} \left(- 6 A + B\right) + x \left(- 6 B + C\right) + 1 \left(- 6 C\right)$

Use this to write the following system:

$\left\{\begin{matrix}A + D = 0 \\ - 6 A + B = 0 \\ - 6 B + C = 0 \\ - 6 C = 1\end{matrix}\right.$

Solve to see that:

$\left\{\begin{matrix}A = - \frac{1}{216} \\ B = - \frac{1}{36} \\ C = - \frac{1}{6} \\ D = \frac{1}{216}\end{matrix}\right.$

The fraction decomposes into:

$\frac{1}{216 \left(x - 6\right)} - \frac{1}{216 x} - \frac{1}{36 {x}^{2}} - \frac{1}{6 {x}^{3}}$