Partial Fraction Decomposition Calculator

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

Simplify the expression: $$$\frac{t}{\left(1 - t\right)^{2} \left(t + 1\right) \left(- t^{2} - 2 t + 1\right)}=\frac{- t}{\left(t - 1\right)^{2} \left(t + 1\right) \left(t^{2} + 2 t - 1\right)}$$$

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

The form of the partial fraction decomposition is

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

Write the right-hand side as a single fraction:

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

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

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

Expand the right-hand side:

$$- t=t^{4} A + t^{4} C + t^{4} D + t^{4} E + 2 t^{3} A + t^{3} B - \sqrt{2} t^{3} D + \sqrt{2} t^{3} E - 2 t^{2} A + 3 t^{2} B - 4 t^{2} C - 2 t^{2} D + \sqrt{2} t^{2} D - 2 t^{2} E - \sqrt{2} t^{2} E - 2 t A + t B + 4 t C + \sqrt{2} t D - \sqrt{2} t E + A - B - C - \sqrt{2} D + D + E + \sqrt{2} E$$

Collect up the like terms:

$$- t=t^{4} \left(A + C + D + E\right) + t^{3} \left(2 A + B - \sqrt{2} D + \sqrt{2} E\right) + t^{2} \left(- 2 A + 3 B - 4 C - 2 D + \sqrt{2} D - 2 E - \sqrt{2} E\right) + t \left(- 2 A + B + 4 C + \sqrt{2} D - \sqrt{2} E\right) + A - B - C - \sqrt{2} D + D + E + \sqrt{2} E$$

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

$$\begin{cases} A + C + D + E = 0\\2 A + B - \sqrt{2} D + \sqrt{2} E = 0\\- 2 A + 3 B - 4 C - 2 D + \sqrt{2} D - 2 E - \sqrt{2} E = 0\\- 2 A + B + 4 C + \sqrt{2} D - \sqrt{2} E = -1\\A - B - C - \sqrt{2} D + D + E + \sqrt{2} E = 0 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=\frac{3}{8}$$$, $$$B=- \frac{1}{4}$$$, $$$C=- \frac{1}{8}$$$, $$$D=- \frac{1}{8} + \frac{\sqrt{2}}{8}$$$, $$$E=- \frac{\sqrt{2}}{8} - \frac{1}{8}$$$

Therefore,

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

Answer: $$$\frac{t}{\left(1 - t\right)^{2} \left(t + 1\right) \left(- t^{2} - 2 t + 1\right)}=\frac{\frac{3}{8}}{t - 1}+\frac{- \frac{1}{4}}{\left(t - 1\right)^{2}}+\frac{- \frac{1}{8}}{t + 1}+\frac{- \frac{1}{8} + \frac{\sqrt{2}}{8}}{t + 1 + \sqrt{2}}+\frac{- \frac{\sqrt{2}}{8} - \frac{1}{8}}{t - \sqrt{2} + 1}$$$