# How do you express 1/[(1+x)(1-2x)] in partial fractions?

Mar 9, 2016

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

#### Explanation:

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

Simplifying RHS, it is equal to

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

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

i.e. $B - 2 A = 0$ and $A + B = 1$

From first we get $B = 2 A$ and putting this in second we get

$A + 2 A = 1$ or $3 A = 1$ or $A = \frac{1}{3}$. As $B = 2 A = 2 \times \frac{1}{3} = \frac{2}{3}$, we have

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