Question

Resolve into partial fraction for the following:

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

Answer 1

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

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

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

Put x = 1 in (2) we get

1 – 2 = A(1 – 1)² + B(1 + 2) (1 – 1) + C(1 + 2)

-1 = 0 + 0 + 3C

C = `- 1/3`

Put x = -2 in (2) we get

-2 – 2 = A(-2 – 1)² + B(-2 + 2) (-2 – 1) + C(-2 + 2)

-4 = A(-3)² + 0 + 0

-4 = 9A

A = `(-4)/9`

From (2) we have,

0x² + x – 2 = A(x – 1)² + B(x + 2) (x – 1) + C(x + 2)

Equating coefficients of x² on both sides we get
0 = A + B

0 = `(-4)/9 + "B"`   ...`(∵ "A" = (-4)/9)`

B = `4/9`

Using, `"A" = (-4)/9`, B = `4/9`, C = `- 1/3` in (1) we get

`(x - 2)/((x + 2)(x - 1)^2) = ((-4)/9)/(x + 2) + (4/9)/(x - 1) + ((-1)/3)/(x - 1)^2`

`= 4/(9(x - 1)) - 4/(9(x + 2)) - 1/(3(x - 1))`