Question

Resolve the following rational expressions into partial fractions

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

Answer 1

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

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

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

(x – 1)² = A(x² + 1) + Bx² + Cx  ......(1)

Put x = 0 in equation (1)

(0 – 1)² = A(0² + 1) + B × 0 + C × 0

1 = A + 0 + 0

⇒ A = 1

Equating the coefficient of x² on both sides

1 = A + B

1 = 1 + B

⇒ B = 0

Put x = 1 in equation (1)

(1 – 1)² = A(1² + 1) + B × 1² + C × 1

0 = 2A + B + C

0 = 2 × 1 + 0 + C

⇒ C = – 2

∴ The required partial fraction is

`(x - 1)^2/(x^3 + x) = 1/x + (0*x - 2)/(x^2 + 1)`

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