Partial Fraction Decomposition Calculator

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

Factor the denominator: $\frac{\left(u - 1\right) \left(u + 1\right)^{2}}{u^{2} \left(u^{2} + 2 u - 1\right)}=\frac{\left(u - 1\right) \left(u + 1\right)^{2}}{u^{2} \left(u + 1 + \sqrt{2}\right) \left(u - \sqrt{2} + 1\right)}$

The form of the partial fraction decomposition is

$$\frac{u^{3} + u^{2} - u - 1}{u^{2} \left(u + 1 + \sqrt{2}\right) \left(u - \sqrt{2} + 1\right)}=\frac{A}{u}+\frac{B}{u^{2}}+\frac{C}{u + 1 + \sqrt{2}}+\frac{D}{u - \sqrt{2} + 1}$$

Write the right-hand side as a single fraction:

$$\frac{u^{3} + u^{2} - u - 1}{u^{2} \left(u + 1 + \sqrt{2}\right) \left(u - \sqrt{2} + 1\right)}=\frac{u^{2} \left(u + 1 + \sqrt{2}\right) D + u^{2} \left(u - \sqrt{2} + 1\right) C + u \left(u + 1 + \sqrt{2}\right) \left(u - \sqrt{2} + 1\right) A + \left(u + 1 + \sqrt{2}\right) \left(u - \sqrt{2} + 1\right) B}{u^{2} \left(u + 1 + \sqrt{2}\right) \left(u - \sqrt{2} + 1\right)}$$

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

$$u^{3} + u^{2} - u - 1=u^{2} \left(u + 1 + \sqrt{2}\right) D + u^{2} \left(u - \sqrt{2} + 1\right) C + u \left(u + 1 + \sqrt{2}\right) \left(u - \sqrt{2} + 1\right) A + \left(u + 1 + \sqrt{2}\right) \left(u - \sqrt{2} + 1\right) B$$

Expand the right-hand side:

$$u^{3} + u^{2} - u - 1=u^{3} A + u^{3} C + u^{3} D + 2 u^{2} A + u^{2} B - \sqrt{2} u^{2} C + u^{2} C + u^{2} D + \sqrt{2} u^{2} D - u A + 2 u B - B$$

Collect up the like terms:

$$u^{3} + u^{2} - u - 1=u^{3} \left(A + C + D\right) + u^{2} \left(2 A + B - \sqrt{2} C + C + D + \sqrt{2} D\right) + u \left(- A + 2 B\right) - B$$

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

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

Solving it (for steps, see system of equations calculator), we get that $A=3$, $B=1$, $C=-1 + \sqrt{2}$, $D=- \sqrt{2} - 1$

Therefore,

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

Answer: $\frac{\left(u - 1\right) \left(u + 1\right)^{2}}{u^{2} \left(u^{2} + 2 u - 1\right)}=\frac{3}{u}+\frac{1}{u^{2}}+\frac{-1 + \sqrt{2}}{u + 1 + \sqrt{2}}+\frac{- \sqrt{2} - 1}{u - \sqrt{2} + 1}$