Question

Resolve the following rational expressions into partial fractions

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

Answer 1

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

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

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

Put x = 1 in equation (1)

⇒ 1 = A(1 – 1)² + B(1 – 1) + C

1 = 0 + 0 + C

⇒ C = 1

In equation (1), equating the coefficient of x² on both sides

0 = A ⇒ A = 0

Put x = 0 in equation (1) ⇒ 0 = A(0 – 1)² + B(0 – 1) + C

⇒ 0 = A – B + C

0 = 0 – B + 1

⇒ B = 1

∴ The required partial fraction is `x/(x - 1)^3 = 1/(x - 1)^2 + 1/(x - 1)^3`