# How do you express  f(x) = (3x + 2) / [(x - 1)(x + 4)] in partial fractions?

Feb 7, 2016

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

#### Explanation:

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

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

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

$\implies \left\{\begin{matrix}A + B = 3 \\ 4 A - B = 2\end{matrix}\right.$

$\implies \left\{\begin{matrix}A = 1 \\ B = 2\end{matrix}\right.$

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