# How do you write the partial fraction decomposition of the rational expression 5 / (x^2 + x - 6)?

Dec 21, 2015

Factorize and solve

#### Explanation:

First step is to factorize the denominator.

$D e n \left(\frac{5}{{x}^{2} + x - 6}\right) = {x}^{2} + x - 6$
$= {x}^{2} + 3 x - 2 x - 6$
$= x \left(x + 3\right) - 2 \left(x + 3\right)$
$= \left(x + 3\right) \left(x - 2\right)$

Now the partial fractions are
$\frac{5}{{x}^{2} + x - 6} = \frac{A}{x + 3} + \frac{B}{x - 2}$

Solve for A & B.

Multiply both sides by denominator
$A \left(x - 2\right) + B \left(x + 3\right) = 5$

If $x = 2$ we will get B

$5 B = 5$
$B = 1$

If $x = - 3$ we will get A
$- 5 A = 5$
$A = - 1$

PFs are
$\frac{5}{{x}^{2} + x - 6} = \frac{1}{x - 2} - \frac{1}{x + 3}$