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

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Answer 1 · 2015-12-14T05:32:35

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

Simplify: $x/(x^3-x)$

Factor the denominator.

$x/(x(x+1)(x-1))=A/x+B/(x+1)+C/(x-1)$

$x=A(x^2-1)+B(x^2-x)+C(x^2+x)$

If $x=0$ :

$0=-A$
$A=0$

If $x=1$ :

$1=2C$
$C=1/2$

If $x=-1$ :

$-1=2B$
$B=-1/2$

Therefore, the expression is equal to

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

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