# How do you write the partial fraction decomposition of the rational expression 1/((x+7)(x^2+9))?

Dec 14, 2015

$\frac{1}{\left(x + 7\right) \left({x}^{2} + 9\right)} = \frac{7 - x}{58 \left({x}^{2} + 9\right)} + \frac{1}{58 \left(x + 7\right)}$

#### Explanation:

$\frac{1}{\left(x + 7\right) \left({x}^{2} + 9\right)} = \frac{A}{x + 7} + \frac{B x + C}{{x}^{2} + 9}$

$1 = A \left({x}^{2} + 9\right) + \left(B x + C\right) \left(x + 7\right)$

IF $x = - 7$:

$1 = 58 A$
$A = \frac{1}{58}$

IF $x = 0$:

$1 = \frac{9}{58} + 7 C$
$C = \frac{7}{58}$

IF $x = 7$:

$1 = 1 + 14 \left(7 B + \frac{7}{58}\right)$
$B = - \frac{1}{58}$

Plug in these values:

$\frac{1}{\left(x + 7\right) \left({x}^{2} + 9\right)} = \frac{7 - x}{58 \left({x}^{2} + 9\right)} + \frac{1}{58 \left(x + 7\right)}$