How do you express (x^2)/((x^4) - (2x^2) - 8) in partial fractions?

Sep 5, 2017

${x}^{2} / \left({x}^{4} - 2 {x}^{2} - 8\right) = \frac{1}{6 \left(x - 2\right)} - \frac{1}{6 \left(x + 2\right)} + \frac{1}{3 \left({x}^{2} + 2\right)}$

Explanation:

If we let $u = {x}^{2}$, then we can rewrite as

$\frac{u}{{u}^{2} - 2 u - 8}$

The denominator can be factored as $\left(u - 4\right) \left(u + 2\right)$. Hence:

$\frac{A}{u - 4} + \frac{B}{u + 2} = \frac{u}{{u}^{2} - 2 u - 8}$

$A \left(u + 2\right) + B \left(u - 4\right) = u$

$A u + 2 A + B u - 4 B = u$

We can now write a system of equations:

$\left\{\begin{matrix}A + B = 1 \\ 2 A - 4 B = 0\end{matrix}\right.$

Simplify the second equation to get:

$\left\{\begin{matrix}A + B = 1 \\ A - 2 B = 0\end{matrix}\right.$

Substitute $A = 1 - B$ into the second equation.

$1 - B - 2 B = 0$

$- 3 B = - 1$

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

This means that $A = \frac{2}{3}$.

Therefore, we can say

$\frac{u}{{u}^{2} - 2 u - 8} = \frac{2}{3 \left(u - 4\right)} + \frac{1}{3 \left(u + 2\right)}$

Reversing our initial substitution:

${x}^{2} / \left({x}^{4} - 2 {x}^{2} - 8\right) = \frac{2}{3 \left({x}^{2} - 4\right)} + \frac{1}{3 \left({x}^{2} + 2\right)}$

But we now see that ${x}^{2} - 4 = \left(x + 2\right) \left(x - 2\right)$, so we can simplify further.

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

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

$A x - 2 A + B x + 2 B = \frac{2}{3}$

Then we have a new system of equations:

$\left\{\begin{matrix}A + B = 0 \\ B - A = \frac{1}{3}\end{matrix}\right.$

From the first equation we deduce $B = - A$ which means that $- 2 A = \frac{1}{3}$ and that $A = - \frac{1}{6}$. Accordingly, $B = \frac{1}{6}$.

Hence:

$\frac{2}{3 \left({x}^{2} - 4\right)} = \frac{1}{6 \left(x - 2\right)} - \frac{1}{6 \left(x + 2\right)}$

Putting everything back together, we get:

${x}^{2} / \left({x}^{4} - 2 {x}^{2} - 8\right) = \frac{1}{6 \left(x - 2\right)} - \frac{1}{6 \left(x + 2\right)} + \frac{1}{3 \left({x}^{2} + 2\right)}$

Hopefully this helps!