How do you express  (x + 2)/((2x+7)(x+1)) in partial fractions?

Nov 12, 2016

$\frac{A}{2 x + 7} + \frac{B}{x + 1} = \frac{x + 2}{\left(2 x + 7\right) \left(x + 1\right)}$

Put on a common denominator.

$\frac{A \left(x + 1\right)}{\left(2 x + 7\right) \left(x + 1\right)} + \frac{B \left(2 x + 7\right)}{\left(2 x + 7\right) \left(x + 1\right)} = \frac{x + 2}{\left(2 x + 7\right) \left(x + 1\right)}$

$A x + A + 2 B x + 7 B = x + 2$

$\left(A + 2 B\right) x + \left(A + 7 B\right) = x + 2$

So, $A + 2 B = 1$ and $A + 7 B = 2$.

Solve.

$A = 1 - 2 B \to 1 - 2 B + 7 B = 2$

$5 B = 1$

$B = \frac{1}{5}$

$\therefore A = 1 - 2 \left(\frac{1}{5}\right) = \frac{3}{5}$

Hence, the partial fraction decomposition is $\frac{3}{5 \left(2 x + 7\right)} + \frac{1}{5 \left(x + 1\right)} = \frac{x + 2}{\left(2 x + 7\right) \left(x + 1\right)}$.

Hopefully this helps!