How do you write the partial fraction decomposition of the rational expression 6(x2−25)2?

1 Answer
George C.
Dec 30, 2015

6(x2−25)2

=−3250(x−5)+350(x−5)2+3250(x+5)+350(x+5)2

Explanation:

Note that:

(x2−25)2=((x−5)(x+5))2

So need to solve:

6(x2−25)2=Ax−5+B(x−5)2+Cx+5+D(x+5)2

=Ax+(B−5A)(x−5)2+Cx+(D+5C)(x+5)2

=(Ax+(B−5A))(x+5)2+(Cx+(D+5C))(x−5)2(x2−25)2

=(Ax+(B−5A))(x2+10x+25)+(Cx+(D+5C))(x2−10x+25)(x2−25)2

=(A+C)x3+(B+D+5A−5C)x2+(10B−10D−25A−25C)x+25(B+D−5A+5C)(x2−25)2

Hence:

A+C=0

B+D+5A−5C=0

10B−10D−25A−25C=0

25(B+D−5A+5C)=6

From the first of these C=−A, so the rest become:

B+D+10A=0

10(B−D)=0

25(B+D−10A)=6

From the second of these D=B, so the rest become:

2B+10A=0

25(2B−10A)=6

From the first of these B=−5A, so the second becomes:

25(−20A)=6

Hence A=−3250, B=350, C=3250, D=350

So:

6(x2−25)2

=−3250(x−5)+350(x−5)2+3250(x+5)+350(x+5)2

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