# Math Help - Partial fraction decomposition

1. ## Partial fraction decomposition

How do you solve: 1 / x^2(x^2+2x+2)

I put it in the form:
A/x + B/x^2 + (Cx+D)/(x^2+2x+2) which can be solved by completing the square

2. Originally Posted by fanban
How do you solve: 1 / x^2(x^2+2x+2)

I put it in the form:
A/x + B/x^2 + (Cx+D)/(x^2+2x+2) which can be solved by completing the square

$\frac{1}{x^2(x^2+2x+2)} = \frac{A}{x} + \frac{B}{x^2} + \frac{Cx+D}{x^2+2x+2}$

$1 = Ax(x^2+2x+2) + B(x^2+2x+2) + (Cx+D)x^2$

$1 = (A+C)x^3 + (2A+B+D)x^2 + (2A+2B)x + 2B$

equating coefficients ...

$A+C = 0$

$2A+B+D = 0$

$2A+2B = 0$

$2B = 1$

$B = \frac{1}{2}$

$A = -\frac{1}{2}$

$C = \frac{1}{2}$

$D = \frac{1}{2}$

$\frac{1}{x^2(x^2+2x+2)} = -\frac{1}{2}\left(\frac{1}{x} - \frac{1}{x^2} - \frac{x+1}{x^2+2x+2}\right)$