Partial Fraction Decomposition Calculator

Your input: perform the partial fraction decomposition of $\frac{u^{2}}{\left(u - 1\right)^{2} \left(u + 1\right)^{2}}$

The form of the partial fraction decomposition is

$$\frac{u^{2}}{\left(u - 1\right)^{2} \left(u + 1\right)^{2}}=\frac{A}{u + 1}+\frac{B}{\left(u + 1\right)^{2}}+\frac{C}{u - 1}+\frac{D}{\left(u - 1\right)^{2}}$$

Write the right-hand side as a single fraction:

$$\frac{u^{2}}{\left(u - 1\right)^{2} \left(u + 1\right)^{2}}=\frac{\left(u - 1\right)^{2} \left(u + 1\right) A + \left(u - 1\right)^{2} B + \left(u - 1\right) \left(u + 1\right)^{2} C + \left(u + 1\right)^{2} D}{\left(u - 1\right)^{2} \left(u + 1\right)^{2}}$$

The denominators are equal, so we require the equality of the numerators:

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

Expand the right-hand side:

$$u^{2}=u^{3} A + u^{3} C - u^{2} A + u^{2} B + u^{2} C + u^{2} D - u A - 2 u B - u C + 2 u D + A + B - C + D$$

Collect up the like terms:

$$u^{2}=u^{3} \left(A + C\right) + u^{2} \left(- A + B + C + D\right) + u \left(- A - 2 B - C + 2 D\right) + A + B - C + D$$

The coefficients near the like terms should be equal, so the following system is obtained:

$$\begin{cases} A + C = 0\\- A + B + C + D = 1\\- A - 2 B - C + 2 D = 0\\A + B - C + D = 0 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $A=- \frac{1}{4}$, $B=\frac{1}{4}$, $C=\frac{1}{4}$, $D=\frac{1}{4}$

Therefore,

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

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