# How do you express (3x)/(x^2+x-2) in partial fractions?

Nov 24, 2016

the partial fraction decomposition is $\frac{2}{x + 2} + \frac{1}{x - 1}$

#### Explanation:

We can factor the denominator as $\left(x + 2\right) \left(x - 1\right)$.

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

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

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

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

We can now write a system of equations.

$\left\{\begin{matrix}A + B = 3 \\ 2 B - A = 0\end{matrix}\right.$

Solving:

$B = 3 - A$

$2 \left(3 - A\right) - A = 0$

$6 - 2 A - A = 0$

$- 3 A = - 6$

$A = 2$

$A + B = 3$

$2 + B = 3$

$B = 1$

Hence, the partial fraction decomposition is $\frac{2}{x + 2} + \frac{1}{x - 1}$.

Hopefully this helps!