How do you express (x+1)/(x^2 + 6x) in partial fractions?

Mar 8, 2016

$\frac{x + 1}{x \left(x + 6\right)} = \frac{\frac{1}{6}}{x} + \frac{\frac{5}{6}}{x + 6} = \frac{1}{6 x} + \frac{5}{6 \left(x + 6\right)}$

Explanation:

$\frac{x + 1}{x \left(x + 6\right)} = \frac{A}{x} + \frac{B}{x + 6}$
$\left(x + 1\right) = A \left(x + 6\right) + B x$
$x + 1 = A x + 6 A + B x$
$1 = A + B , 6 A = 1$
$A = \frac{1}{6} , B = 1 - \frac{1}{6} = \frac{5}{6}$
$\frac{x + 1}{x \left(x + 6\right)} = \frac{\frac{1}{6}}{x} + \frac{\frac{5}{6}}{x + 6} = \frac{1}{6 x} + \frac{5}{6 \left(x + 6\right)}$