Question

Resolve into a partial fraction for the following:

`(x^2 - 6x + 2)/(x^2 (x + 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 - 6x + 2)/(x^2 (x + 2)) = "A"/x + "B"/x^2 + "C"/(x + 2)`   ....(2)

Multiply both sides by x² (x + 2) we get

`(x^2 - 6x + 2)/(x^2 (x + 2)) xx x^2(x + 2) = "A"/x x^2 (x + 2) + "B"/x^2 (x + 2) + "C"/(x + 2) x^2(x + 2)`

x² – 6x + 2 = Ax(x + 2) + B(x + 2) + C(x²) ……… (2)

Put x = 0 in (2) we get

0 – 0 + 2 = 0 + B(0 + 2) + 0

2 = B(2)

B = 1

Put x = -2 in (2) we get

(-2)² – 6(-2) + 2 = 0 + 0 + C(-2)²

4 + 12 + 2 = C(4)

18 = 4C

C = `9/2`

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

1 = A + C

1 = A + `9/2`

A = `1 - 9/2 = (2 - 9)/2 = (-7)/2`

Using A = `(-7)/2`, B = 1, C = `9/2` in (1) we get,

`(x^2 - 6x + 2)/(x^2 (x + 2)) = (-7)/(2x) + 1/x^2 + 9/(2(x + 2))`