# How do you express 1/(x^3-1) in partial fractions?

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

#### Explanation:

the factors of $\left({x}^{3} - 1\right)$ are $\left(x - 1\right)$ and $\left({x}^{2} + x + 1\right)$

Set the equations using variables A, B, C so that

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

solve for the variables so that
$A = \frac{1}{3}$
$B = - \frac{1}{3}$
$C = - \frac{2}{3}$

God bless you ...