n乗同士の和と差の因数分解

n乗同士の和と差の因数分解

\(a,b\in\mathbb{R}\;,\;n\in\mathbb{N}_{0}\)とする

(1)

\[ a^{n}-b^{n}=\left(a-b\right)\left(\sum_{k=0}^{n-1}a^{n-1-k}b^{k}\right) \]

(2)

\[ a^{2n+1}\pm b^{2n+1}=\left(a\pm b\right)\left(\sum_{k=0}^{2n}\left(\mp1\right)^{k}a^{2n-k}b^{k}\right) \]

(1)

\begin{align*} a^{n}-b^{n} & =a^{n}+\left(\sum_{k=1}^{n-1}a^{n-k}b^{k}\right)-\left(\sum_{k=1}^{n-1}a^{n-k}b^{k}\right)-b^{n}\\ & =\left(\sum_{k=0}^{n-1}a^{n-k}b^{k}\right)-\left(\sum_{k=1}^{n}a^{n-k}b^{k}\right)\\ & =a\left(\sum_{k=0}^{n-1}a^{n-1-k}b^{k}\right)-b\left(\sum_{k=1}^{n}a^{n-k}b^{k-1}\right)\\ & =a\left(\sum_{k=0}^{n-1}a^{n-1-k}b^{k}\right)-b\left(\sum_{k=0}^{n-1}a^{n-1-k}b^{k}\right)\\ & =\left(a-b\right)\left(\sum_{k=0}^{n-1}a^{n-1-k}b^{k}\right) \end{align*}

(2)

\begin{align*} a^{2n+1}\pm b^{2n+1} & =a^{2n+1}+\left(\sum_{k=1}^{2n}\left(\mp1\right)^{k}a^{2n+1-k}b^{k}\right)-\left(\sum_{k=1}^{2n}\left(\mp1\right)^{k}a^{2n+1-k}b^{k}\right)\pm b^{2n+1}\\ & =\left(\sum_{k=0}^{2n}\left(\mp1\right)^{k}a^{2n+1-k}b^{k}\right)-\left(\sum_{k=1}^{2n+1}\left(\mp1\right)^{k}a^{2n+1-k}b^{k}\right)\\ & =a\left(\sum_{k=0}^{2n}\left(\mp1\right)^{k}a^{2n-k}b^{k}\right)-b\left(\sum_{k=1}^{2n+1}\left(\mp1\right)^{k}a^{2n+1-k}b^{k-1}\right)\\ & =a\left(\sum_{k=0}^{2n}\left(\mp1\right)^{k}a^{2n-k}b^{k}\right)-b\left(\mp\sum_{k=0}^{2n+1}\left(\mp1\right)^{k}a^{2n-k}b^{k}\right)\\ & =\left(a\pm b\right)\left(\sum_{k=0}^{2n}\left(\mp1\right)^{k}a^{2n-k}b^{k}\right) \end{align*}

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n乗同士の和と差の因数分解

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