代数恒等式

基础恒等式

\displaystyle (a\pm b)^2=a^2\pm2ab+b^2\\{}\\ (a\pm b)^3=a^3\pm3a^2b+3ab^2\pm b^3\\{}\\ (a\pm b)^4=a^4\pm4a^3b+6a^2b^2\pm 4ab^3+b^4\\{}\\ (a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc\\{}\\ (a+b-c)^2=a^2+b^2+c^2+2ab-2ac-2bc\\{}\\ (a-b-c)^2=a^2+b^2+c^2-2ab-2ac+2bc\\{}\\ (a+b+c)^3=a^3+b^3+c^3+6abc+3(a^2b+ab^2+b^2c+bc^2+c^2a+ca^2)\\{}\\ (a_1+a_2+\cdots a_n)^2=a_1^2+a_2^2+2(a_1a_2+a_1a_3+\cdots a_{n-1}a_n)\\{}\\ a^2-b^2=(a-b)(a+b)\\{}\\ a^3+b^3=(a+b)(a^2-ab+b^2)\\{}\\ a^3-b^3=(a-b)(a^2+ab+b^2)\\{}\\ a^4+b^4=(a^2+b^2)^2-2a^2b^2=(a^2+\sqrt{2}ab+b^2)(a^2-\sqrt{2}ab+b^2)\\{}\\ a^4-b^4=(a^2-b^2)(a^2+b^2)=(a+b)(a-b)(a^2+b^2)\\{}\\ a^5+b^5=(a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)\\{}\\ a^5-b^5=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)\\{}\\ a^{2n+1}-b^{2n+1}=(a-b)(a^{2n}+a^{2n-1}b+a^{2n-2}b^2+\cdots+b^{2n})\\ =(a-b)\left(a^2-2ab\cos\frac{2\pi}{2n+1}+b^2\right)\left(a^2-2ab\cos\frac{4\pi}{2n+1}+b^2\right)\cdots\left(a^2-2ab\cos\frac{2n\pi}{2n+1}+b^2\right)\\{}\\ a^{2n+1}+b^{2n+1}=(a+b)(a^{2n}-a^{2n-1}b+a^{2n-2}b^2-\cdots+b^{2n})\\ =(a+b)\left(a^2+2ab\cos\frac{2\pi}{2n+1}+b^2\right)\left(a^2+2ab\cos\frac{4\pi}{2n+1}+b^2\right)\cdots\left(a^2+2ab\cos\frac{2n\pi}{2n+1}+b^2\right)\\{}\\ a^{2n}-b^{2n}=(a-b)(a+b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\cdots)(a^{n-1}-a{n-2}+a^{n-3}b^2-\cdots)\\ =(a-b)(a+b)\left(a^2-2ab\cos\frac{\pi}{n}+b^2\right)\left(a^2-2ab\cos\frac{2\pi}{n}+b^2\right)\cdots\left(a^2-2ab\cos\frac{(n-1)\pi}{n}+b^2\right)\\{}\\ a^{2n}+b^{2n}=\left(a^2+2ab\cos\frac{\pi}{2n}+b^2\right)\left(a^2+2ab\cos\frac{3\pi}{2n}+b^2\right)\cdots\left(a^2+2ab\cos\frac{(2n-1)\pi}{2n}+b^2\right)