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))#