How do you express x2−2x−1(x−1)2(x2+1) in partial fractions?

1 Answer
Narad T.
Nov 15, 2017

The answer is =−1(x−1)2+1x−1−(x−1)x2+1

Explanation:

Perform the decomposition into partial fractions

x2−2x−1(x−1)2(x2+1)=A(x−1)2+Bx−1+Cx+Dx2+1

=A(x2+1)+B(x−1)(x2+1)+(Cx+D)(x−1)2(x−1)2(x2+1)

The denominators are the same, compare the numerators

(x2−2x−1)=A(x2+1)+B(x−1)(x2+1)+(Cx+D)(x−1)2

Let x=1

−2=2A, A=−1

Coefficients of x2

1=A−B−2C+D

Coefficients of x

−2=B+C−2D

and

−1=A−B+D, ⇒, −B+D=0, B=D

1=−1−B−2C+B, ⇒, 2C=−2, ⇒, C=−1

−2=B−1−2B, ⇒, B=1

⇒, D=1

Finally,

x2−2x−1(x−1)2(x2+1)=−1(x−1)2+1x−1−(x−1)x2+1

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