Important Identities

An equation that holds for every value in its domain is called an identity (as opposed to a conditional equation, true only for some values). The following identities, valid for all $x,a,b\in R$ (and $x^n\pm a^n$/$x^n+b^n$ for $n\in N$), are the standard factoring toolkit:

1. $(x+a)^3=(x+a)^2(x+a)=x^3+3x^2a+3xa^2+a^3=x^3+3xa(x+a)+a^3$.
2. $(x-b)^3=x^3-3x^2b+3xb^2-b^3=x^3-3xb(x-b)+b^3$ (set $a=-b$ in identity 1).
3. $x^3+a^3=(x+a)(x^2-xa+a^2)$.
4. $x^3-b^3=(x-b)(x^2+xb+b^2)$ (set $a=-b$ in identity 3).
5. $x^n-a^n=(x-a)(x^{n-1}+x^{n-2}a+\cdots+x^{n-k-1}a^k+\cdots+a^{n-1})$, for $n\in N$.
6. $x^n+b^n=(x+b)(x^{n-1}-x^{n-2}b+\cdots+x^{n-k-1}(-b)^k+\cdots+(-b)^{n-1})$, for $n\in N$. …