n乗根の因数分解
\[z^{n}-1=\prod_{k=1}^{n}\left(z-e^{\frac{2\pi}{n}ki}\right)\]
n乗同士の和と差の因数分解
\[a^{2n+1}\pm b^{2n+1}=\left(a\pm b\right)\left(\sum_{k=0}^{2n}\left(\mp1\right)^{k}a^{2n-k}b^{k}\right)\]
ソフィー・ジェルマンの恒等式
\[a^{4}+4b^{4}=\left(a^{2}+2ab+2b^{2}\right)\left(a^{2}-2ab+2b^{2}\right)\]
オイラーの4平方恒等式
\[\left(a_{0}^{\;2}+a_{1}^{\;2}+a_{2}^{\;2}+a_{3}^{\;2}\right)\left(b_{0}^{\;2}+b_{1}^{\;2}+b_{2}^{\;2}+b_{3}^{\;2}\right)=\left(a_{0}b_{0}-a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}\right)^{2}+\left(a_{0}b_{1}+a_{1}b_{0}+a_{2}b_{3}-a_{3}b_{2}\right)^{2}+\left(a_{0}b_{2}-a_{1}b_{3}+a_{2}b_{0}+a_{3}b_{1}\right)^{2}+\left(a_{0}b_{3}+a_{1}b_{2}-a_{2}b_{1}+a_{3}b_{0}\right)^{2}\]
ブラーマグプタ2平方恒等式
\[\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)=\left(ac\pm bd\right)^{2}+\left(ad\mp bc\right)^{2}\]
ビネ・コーシーとラグランジュの恒等式
\[\left(\sum_{i=1}^{n}a_{i}c_{i}\right)\left(\sum_{j=1}^{n}b_{j}d_{j}\right)-\left(\sum_{i=1}^{n}a_{i}d_{i}\right)\left(\sum_{j=1}^{n}b_{j}c_{j}\right)=\sum_{1\leq i<j\leq n}\left(a_{i}b_{j}-a_{j}b_{i}\right)\left(c_{i}d_{j}-c_{j}d_{i}\right)\]
交代式の因数分解
\[\text{交代式}=\text{差積}\times\text{対称式}\]
差積の定義と性質
\[\Delta\left(x_{1},\cdots,x_{n}\right):=\prod_{1\leq i<j\leq n}\left(x_{i}-x_{j}\right)\]
複二次式の定義と因数分解
\[a_{4}x^{4}+a_{2}x^{2}+a_{0}=\frac{1}{4a_{4}}\left(2a_{4}x^{2}+a_{2}+\sqrt{a_{2}^{\;2}-4a_{4}a_{0}}\right)\left(2a_{4}x^{2}+a_{2}-\sqrt{a_{2}^{\;2}-4a_{4}a_{0}}\right)\]
3次式の実数の範囲で因数分解
\[a^{3}\pm b^{3}=\left(a\pm b\right)\left(a^{2}\mp ab+b^{2}\right)\]
2次式の実数の範囲で因数分解
\[a^{2}\pm2ab+b^{2}=\left(a\pm b\right)^{2}\]