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Rechner zur Partialbruchzerlegung

Finde die Partialbruchzerlegung Schritt für Schritt

Dieser Online-Rechner bestimmt die Partialbruchzerlegung einer rationalen Funktion und zeigt dabei die Rechenschritte an.

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Solution

Your input: perform the partial fraction decomposition of $$$\frac{1}{x^{9} \left(x - 1\right)}$$$

The form of the partial fraction decomposition is

$$\frac{1}{x^{9} \left(x - 1\right)}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x^{3}}+\frac{D}{x^{4}}+\frac{E}{x^{5}}+\frac{F}{x^{6}}+\frac{G}{x^{7}}+\frac{H}{x^{8}}+\frac{J}{x^{9}}+\frac{K}{x - 1}$$

Write the right-hand side as a single fraction:

$$\frac{1}{x^{9} \left(x - 1\right)}=\frac{x^{9} K + x^{8} \left(x - 1\right) A + x^{7} \left(x - 1\right) B + x^{6} \left(x - 1\right) C + x^{5} \left(x - 1\right) D + x^{4} \left(x - 1\right) E + x^{3} \left(x - 1\right) F + x^{2} \left(x - 1\right) G + x \left(x - 1\right) H + \left(x - 1\right) J}{x^{9} \left(x - 1\right)}$$

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

$$1=x^{9} K + x^{8} \left(x - 1\right) A + x^{7} \left(x - 1\right) B + x^{6} \left(x - 1\right) C + x^{5} \left(x - 1\right) D + x^{4} \left(x - 1\right) E + x^{3} \left(x - 1\right) F + x^{2} \left(x - 1\right) G + x \left(x - 1\right) H + \left(x - 1\right) J$$

Expand the right-hand side:

$$1=x^{9} A + x^{9} K - x^{8} A + x^{8} B - x^{7} B + x^{7} C - x^{6} C + x^{6} D - x^{5} D + x^{5} E - x^{4} E + x^{4} F - x^{3} F + x^{3} G - x^{2} G + x^{2} H - x H + x J - J$$

Collect up the like terms:

$$1=x^{9} \left(A + K\right) + x^{8} \left(- A + B\right) + x^{7} \left(- B + C\right) + x^{6} \left(- C + D\right) + x^{5} \left(- D + E\right) + x^{4} \left(- E + F\right) + x^{3} \left(- F + G\right) + x^{2} \left(- G + H\right) + x \left(- H + J\right) - J$$

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

$$\begin{cases} A + K = 0\\- A + B = 0\\- B + C = 0\\- C + D = 0\\- D + E = 0\\- E + F = 0\\- F + G = 0\\- G + H = 0\\- H + J = 0\\- J = 1 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=-1$$$, $$$B=-1$$$, $$$C=-1$$$, $$$D=-1$$$, $$$E=-1$$$, $$$F=-1$$$, $$$G=-1$$$, $$$H=-1$$$, $$$J=-1$$$, $$$K=1$$$

Therefore,

$$\frac{1}{x^{9} \left(x - 1\right)}=\frac{-1}{x}+\frac{-1}{x^{2}}+\frac{-1}{x^{3}}+\frac{-1}{x^{4}}+\frac{-1}{x^{5}}+\frac{-1}{x^{6}}+\frac{-1}{x^{7}}+\frac{-1}{x^{8}}+\frac{-1}{x^{9}}+\frac{1}{x - 1}$$

Answer: $$$\frac{1}{x^{9} \left(x - 1\right)}=\frac{-1}{x}+\frac{-1}{x^{2}}+\frac{-1}{x^{3}}+\frac{-1}{x^{4}}+\frac{-1}{x^{5}}+\frac{-1}{x^{6}}+\frac{-1}{x^{7}}+\frac{-1}{x^{8}}+\frac{-1}{x^{9}}+\frac{1}{x - 1}$$$