Question

Decompose the fraction into partial fractions. (7x^2+2x+3)/x^2(x^2+1)

Answer 1

The fraction (7x²  + 2x + 3)/x²(x² + 1)

(7x²  + 2x + 3)/x²(x² + 1) = (A/x) + (B/x²) + [(Cx + D)/(x² + 1)] ----> (i)

(7x²  + 2x + 3)/x²(x² + 1) = {[Ax(x² + 1)] + [B(x² + 1)] + [x²(Cx + D)]}/x²(x² + 1)

7x²  + 2x + 3 = [Ax(x² + 1)] + [B(x² + 1)] + [x²(Cx + D)]

7x² + 2x + 3 = Ax³ + Ax + Bx²+ B + Cx³ + Dx²

Equate x³ coefficients.

A + C = 0 ---> (1)

Equate x² coefficients.

B + D = 7 ---> (2)

Equate x coefficients.

A  = 2

Equate constants.

B = 3

Substitute A in equation (1).

2 + C = 0

C = - 2

Substitute B value in equation (2).

3 + D = 7

D = 7 - 3

D = 4

Substitute A, B , C and D in equation (i).

Partial decomposition of (7x²  + 2x + 3)/x²(x² + 1) = (2/x) + (3/x²) + [(- 2x + 4)/(x² + 1)].