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

Jul 18, 2018

The answer is $= \frac{\frac{3}{5}}{x - 1} + \frac{\frac{17}{5}}{x + 4}$

#### Explanation:

Perform the decomposition into partial fractions

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

$= \frac{A}{x - 1} + \frac{B}{x + 4}$

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

The denominators are the same, compare the numerators

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

Let $x = 1$, $\implies$, $3 = 5 A$, $\implies$, $A = \frac{3}{5}$

Let $x = - 4$, $\implies$, $- 17 = - 5 B$, $\implies$, $B = \frac{17}{5}$

Therefore,

$\frac{4 x - 1}{{x}^{2} + 3 x - 4} = \frac{\frac{3}{5}}{x - 1} + \frac{\frac{17}{5}}{x + 4}$