Question

Resolve the following rational expressions into partial fractions

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

Answer 1

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

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

7 + x = A(1 + x²) + Bx(1 + x) + C(1 + x)   ......1)

Put x = – 1, in equation (1)

7 – 1 = A(1 + (– 1)²) + B(– 1)(1 – 1) + C(1 – 1)

6 = A(1 + 1) + 0 + 0

A = `6/3` = 3

⇒ A = 3

Put x = 0, in equation (1)

7 + 0 = A(1 + 0²) + B × 0(1 + 0) + C(1 + 0)

7 = A + 0 + C

7 = 3 + C

⇒ C = 4

Equating the coefficient of x² in equation (I) we have

0 = A + B

0 = 3 + B

⇒ B = – 3

∴ The required partial fraction is

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