# How do you write the partial fraction decomposition of the rational expression  (2x-2)/((x-5)(x-3))?

Dec 14, 2015

$\frac{4}{x - 5} - \frac{2}{x - 3}$

#### Explanation:

$\frac{2 x - 2}{\left(x - 5\right) \left(x - 3\right)} = \frac{A}{x - 5} + \frac{B}{x - 3}$

$2 x - 2 = A \left(x - 3\right) + B \left(x - 5\right)$

IF $x = 3$:

$2 \left(3\right) - 2 = B \left(3 - 5\right)$
$B = - 2$

IF $x = 5$:

$2 \left(5\right) - 2 = A \left(5 - 3\right)$
$A = 4$

Thus,

$\frac{2 x - 2}{\left(x - 5\right) \left(x - 3\right)} = \frac{4}{x - 5} - \frac{2}{x - 3}$