# How do you write the partial fraction decomposition of the rational expression  (4x+23)/(x^2-x-6)?

Dec 25, 2015

$\frac{4 x + 23}{{x}^{2} - x - 6} = \frac{7}{x - 3} - \frac{3}{x + 2}$

#### Explanation:

Factor the denominator and split apart the expression.

$\frac{4 x + 23}{\left(x - 3\right) \left(x + 2\right)} = \frac{A}{x - 3} + \frac{B}{x + 2}$

Find a common denominator of $\left(x - 3\right) \left(x + 2\right)$.

$4 x + 23 = A \left(x + 2\right) + B \left(x - 3\right)$

Plug in $- 2$ for $x$:

$4 \left(- 2\right) + 23 = B \left(- 2 - 3\right)$
$B = - 3$

Plug in $3$ for $x$:

$4 \left(3\right) + 23 = A \left(3 + 2\right)$
$A = 7$

Plug this into the original expression:

$\frac{4 x + 23}{{x}^{2} - x - 6} = \frac{7}{x - 3} - \frac{3}{x + 2}$