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

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Answer 1 · 2016-03-14T18:11:59

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

Write $(2x)/(1-x^3) = A/(1-x) + (Bx+C)/(1+x+x^2)$

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

$2x= A+Ax +Ax^2 +Bx-Bx^2+C-Cx$ , 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 $A=2/3$ and hence B= $2/3$
and C= $-2/3$ .

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

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