How do you write the partial fraction decomposition of the rational expression x3−x2+1x4−x3?

1 Answer
Tom
Dec 23, 2015

x3−x2+1x4−x3=−1x3−1x2+1x−1

Explanation:

x3−x2+1x4−x3=x3−x2x4−x3+1x4−x3

=x−1x2−x+1x3(x−1)

=x−1x(x−1)+1x3(x−1)

=1x+1x3(x−1)

now focus on 1x3(x−1)

1x3(x−1)=Ax3+Bx2+Cx+Dx−1

Multiply both side by x3(x−1)

1=A(x−1)+Bx(x−1)+Cx2(x−1)+Dx3

1=Ax−A+Bx2−Bx+Cx3−Cx2+Dx3

1=x3(C+D)+x2(B−C)+x(A−B)−A

C+D=0
B−C=0
A−B=0
A=−1

Just by looking we have

A=−1
B=−1
C=−1
D=1

So

x3−x2+1x4−x3=−1x3−1x2+1x−1

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