Question

Resolve into a partial fraction for the following:

`(x + 2)/((x - 1)(x + 3)^2)`

Answer 1

Here the denominator has three factors. So given fraction can be expressed as a sum of three simple fractions.

Let `(x + 2)/((x - 1)(x + 3)^2) = "A"/(x - 1) + "B"/(x + 3) + "C"/(x + 3)^2`   ....(1)

Multiply both sides by (x – 1) (x + 3)² we get

`(x + 2)/((x - 1)(x + 3)^2) (x – 1) (x + 3)^2 = "A"/(x - 1) (x – 1) (x + 3)^2
+ "B"/(x + 3) (x – 1) (x + 3)^2 + "C"/(x + 3)^2 (x – 1) (x + 3)^2`

x + 2 = A(x + 3)² + B(x – 1) (x + 3) + C(x – 1) ……. (2)

Put x = 1 in (2) we get

1 + 2 = A(1 + 3)² + 0 + 0

3 = A(4)²

A = `3/16`

Put x = -3 in (2) we get

-3 + 2 = 0 + 0 + C(-3 – 1)

-1 = C(-4)

C = `1/4`

Comparing coefficient of x² on both sides of (2) we get,

0 = A + B

0 = `3/16 + "B"`

B = `- 3/16`

Using A = `3/16`, B = `- 3/16`, C = `1/4` in (1) we get,

`(x + 2)/((x - 1)(x + 3)^2) = 13/(16(x - 1)) - (4/3)/(16(x + 3)) + 1/(4(x + 3)^2)`