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Calcolatore di decomposizione in fratti semplici

Trova la scomposizione in fratti semplici passo dopo passo

Questo calcolatore online calcolerà la decomposizione in fratti semplici della funzione razionale, con i passaggi mostrati.

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Solution

Your input: perform the partial fraction decomposition of $$$\frac{x^{2}}{1 - x^{4}}$$$

Simplify the expression: $$$\frac{x^{2}}{1 - x^{4}}=\frac{- x^{2}}{x^{4} - 1}$$$

Factor the denominator: $$$\frac{- x^{2}}{x^{4} - 1}=\frac{- x^{2}}{\left(x - 1\right) \left(x + 1\right) \left(x^{2} + 1\right)}$$$

The form of the partial fraction decomposition is

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

Write the right-hand side as a single fraction:

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

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

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

Expand the right-hand side:

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

Collect up the like terms:

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

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

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

Solving it (for steps, see system of equations calculator), we get that $$$A=\frac{1}{4}$$$, $$$B=0$$$, $$$C=- \frac{1}{2}$$$, $$$D=- \frac{1}{4}$$$

Therefore,

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

Answer: $$$\frac{x^{2}}{1 - x^{4}}=\frac{\frac{1}{4}}{x + 1}+\frac{- \frac{1}{2}}{x^{2} + 1}+\frac{- \frac{1}{4}}{x - 1}$$$