# How do you express ( 2x)/(1-x^3) in partial fractions?

Mar 14, 2016

2/(3(1-x)) +(2(x-1))/(3(1+x+x^2)

#### Explanation:

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

On simplifying the Right Hand Side and comparing the coefficients
on both sides it would be

$2 x = A + A x + A {x}^{2} + B x - B {x}^{2} + C - C x$, so that,
A-B=0, A+C=0 and A+B-C=2

Add the first two equation to get B-C= 2A.
Plug in this value in the next equation to get 3A=2, that is $A = \frac{2}{3}$ and hence B= $\frac{2}{3}$
and C= $- \frac{2}{3}$.

The required partial fractions would be
2/(3(1-x)) +(2(x-1))/(3(1+x+x^2)