部分分数分解計算機
部分分数分解を段階的に求める
このオンライン計算機は、有理関数の部分分数分解を、手順を示しながら求めます。
Solution
Your input: perform the partial fraction decomposition of $$$\frac{1}{7 - x^{2}}$$$
Factor the denominator: $$$\frac{1}{7 - x^{2}}=\frac{1}{- \left(x - \sqrt{7}\right) \left(x + \sqrt{7}\right)}=- \frac{1}{x^{2} - 7}$$$
The form of the partial fraction decomposition is
$$\frac{-1}{\left(x - \sqrt{7}\right) \left(x + \sqrt{7}\right)}=\frac{A}{x + \sqrt{7}}+\frac{B}{x - \sqrt{7}}$$
Write the right-hand side as a single fraction:
$$\frac{-1}{\left(x - \sqrt{7}\right) \left(x + \sqrt{7}\right)}=\frac{\left(x - \sqrt{7}\right) A + \left(x + \sqrt{7}\right) B}{\left(x - \sqrt{7}\right) \left(x + \sqrt{7}\right)}$$
The denominators are equal, so we require the equality of the numerators:
$$-1=\left(x - \sqrt{7}\right) A + \left(x + \sqrt{7}\right) B$$
Expand the right-hand side:
$$-1=x A + x B - \sqrt{7} A + \sqrt{7} B$$
Collect up the like terms:
$$-1=x \left(A + B\right) - \sqrt{7} A + \sqrt{7} B$$
The coefficients near the like terms should be equal, so the following system is obtained:
$$\begin{cases} A + B = 0\\- \sqrt{7} A + \sqrt{7} B = -1 \end{cases}$$
Solving it (for steps, see system of equations calculator), we get that $$$A=\frac{\sqrt{7}}{14}$$$, $$$B=- \frac{\sqrt{7}}{14}$$$
Therefore,
$$\frac{-1}{\left(x - \sqrt{7}\right) \left(x + \sqrt{7}\right)}=\frac{\frac{\sqrt{7}}{14}}{x + \sqrt{7}}+\frac{- \frac{\sqrt{7}}{14}}{x - \sqrt{7}}$$
Answer: $$$\frac{1}{7 - x^{2}}=\frac{\frac{\sqrt{7}}{14}}{x + \sqrt{7}}+\frac{- \frac{\sqrt{7}}{14}}{x - \sqrt{7}}$$$