How do you write the partial fraction decomposition of the rational expression 3x2+12x−20x4−8x2+16?

1 Answer
Narad T.
Dec 3, 2016

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

Explanation:

To factorise the denominator,

We use (a−b)2=a2−2ab+b2

and a2−b2=(a+b)(a−b)

So,

x4−8x2+16=(x2−4)2=(x+2)2(x−2)2

Therefore,

3x2+12x−20x4−8x2+16=3x2+12x−20(x+2)2(x−2)2

We can now do the decomposition in partial fractions

3x2+12x−20x4−8x2+16=A(x+2)2+Bx+2+C(x−2)2+Dx−2

=A(x−2)2+B(x−2)2(x+2)+C(x+2)2+D(x+2)2(x−2)

Therefore,

(3x2+12x−20)=A(x−2)2+B(x−2)2(x+2)+C(x+2)2+D(x+2)2(x−2)

Let x=2, ⇒, 16=16C, ⇒, C=1

Let x=−2, ⇒, −32=16A, ⇒, A=−2

Let x=0, ⇒, −20=4A+8B+4C−8D

B−D=−2

Coefficients of x3

0=B+D

So, B=−1 and D=1

So,

3x2+12x−20x4−8x2+16=−2(x+2)2−1x+2+1(x−2)2+1x−2

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