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

Dec 14, 2015

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

#### Explanation:

Factor the denominator as a sum of cubes.

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

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

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

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

Determine the following system:
$\left\{\begin{matrix}A + B = 0 \\ - A + B + C = 0 \\ A + C = 2\end{matrix}\right.$

Solve to find that $\left\{\begin{matrix}A = \frac{2}{3} \\ B = - \frac{2}{3} \\ C = \frac{4}{3}\end{matrix}\right.$

Therefore,

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