# How do you write the partial fraction decomposition of the rational expression  x / (x² - x - 2)?

Oct 22, 2016

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

#### Explanation:

Factor the denominator:

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

Write each factor as a term with an unknown coefficient:

$\frac{x}{{x}^{2} - x - 2} = \frac{A}{x + 1} + \frac{B}{x - 2}$

Multiply both sides by $\left(x + 1\right) \left(x - 2\right)$

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

Make the term containing B become 0 by setting $x = - 1$:

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

$- 1 = A \left(- 3\right)$

$A = \frac{1}{3}$

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

Make the term containing $\frac{1}{3}$ become zero by setting $x = 2$

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

$B = \frac{2}{3}$

Check:

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

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

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

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

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

$\frac{x}{{x}^{2} - x - 2}$ This checks.