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
\(a,b\in\mathbb{R}\;,\;n\in\mathbb{N}_{0}\)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*}ページ情報
タイトル | n乗同士の和と差の因数分解 |
URL | https://www.nomuramath.com/ddv6jkbk/ |
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ブラーマグプタ2平方恒等式
\[
\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)=\left(ac\pm bd\right)^{2}+\left(ad\mp bc\right)^{2}
\]
差積の定義と性質
\[
\Delta\left(x_{1},\cdots,x_{n}\right):=\prod_{1\leq i<j\leq n}\left(x_{i}-x_{j}\right)
\]
2次式の実数の範囲で因数分解
\[
a^{2}\pm2ab+b^{2}=\left(a\pm b\right)^{2}
\]
交代式の因数分解
\[
\text{交代式}=\text{差積}\times\text{対称式}
\]