# How do you write the partial fraction decomposition of the rational expression  (3x + 2) / [(x - 1)(x + 4)]?

Jan 2, 2016

$\frac{3 x + 2}{\left(x - 1\right) \left(x + 4\right)} = \frac{1}{x - 1} + \frac{2}{x + 4}$

#### Explanation:

Since the denominator already in the factor form we can rewrite it as

$\frac{3 x + 2}{\left(x - 1\right) \left(x + 4\right)} = \frac{A}{x - 1} + \frac{B}{x + 4}$

Multiply both sides by the LCD to get

$3 x + 2 = A \left(x + 4\right) + B \left(x - 1\right)$

Multiple the expression to get

$3 x + 2 = A x + 4 A + B x - B$

Set up the system of equation like this

$x : \text{ " " } A + B = 3$
${x}^{0} \text{ " " } 4 A - B = 2$

Solve system by elimination method

$A + B = 3$

$4 A - B = 2$

$5 A \text{ " = 5}$

$\implies A = 1$

Solve for B

$\left(1\right) + B = 3 \implies B = 2$

Partial fraction of $\frac{3 x + 2}{\left(x - 1\right) \left(x + 4\right)} = \frac{1}{x - 1} + \frac{2}{x + 4}$