# How do you write the partial fraction decomposition of the rational expression (x^2)/(x+1)^3?

Dec 14, 2015

$\frac{1}{x + 1} - \frac{2}{x + 1} ^ 2 + \frac{1}{x + 1} ^ 3$

#### Explanation:

${x}^{2} / {\left(x + 1\right)}^{3} = \frac{A}{x + 1} + \frac{B}{x + 1} ^ 2 + \frac{C}{x + 1} ^ 3$

${x}^{2} = A {\left(x + 1\right)}^{2} + B \left(x + 1\right) + C$

${x}^{2} = A {x}^{2} + 2 A x + A + B x + B + C$

${x}^{2} = {x}^{2} \left(A\right) + x \left(2 A + B\right) + 1 \left(A + B + C\right)$

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

Solve to see that $\left\{\begin{matrix}A = 1 \\ B = - 2 \\ C = 1\end{matrix}\right.$

Therefore,

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