# How do you write the partial fraction decomposition of the rational expression  (9x) / (x^2 + 2x - 8)?

Dec 21, 2015

$\frac{6}{x + 4} + \frac{3}{x - 2}$

#### Explanation:

Factor the denominator.

$\frac{9 x}{\left(x + 4\right) \left(x - 2\right)} = \frac{A}{x + 4} + \frac{B}{x - 2}$

Simplify.

$9 x = A \left(x - 2\right) + B \left(x + 4\right)$

Plug in $- 4$ for $x$:

$9 \left(- 4\right) = A \left(- 4 - 2\right)$
$A = 6$

Plug in $2$ for $x$:

$9 \left(2\right) = B \left(2 + 4\right)$
$B = 3$

Plug these values into the original statement.

$\frac{9 x}{{x}^{2} + 2 x - 8} = \frac{6}{x + 4} + \frac{3}{x - 2}$