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

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

#### Explanation:

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

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

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

Equate the numerators

$x = A x + A + B x - B$

$x = A x + B x + A - B$

$1 \cdot x + 0 = \left(A + B\right) x + \left(A - B\right)$

so that

$A + B = 1$ and $A - B = 0$

solving for A and B

$A = \frac{1}{2}$ and $B = \frac{1}{2}$

so therefore

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

have a nice day ! from the Philippines..