identities other than $(a-b)^2=(a+b)^2-4ab$?

Question

We know that for any integer (or real in general) the following equation $$ (a-b)^2=(a+b)^2-4ab $$ holds for all $a,b$.

I am looking for some other identities which involves $a+b$ and $ab$ and derive $a-b$ or similar relations.

Answer 1

Here are some example of $(a+b)^n-(a-b)^n$:

\begin{align*} (a+b)^3-(a-b)^3 & = 2b(3a^2 + b^2), \\ (a+b)^4-(a-b)^4 & = 8ab(a^2+b^2),\\ (a+b)^5-(a-b)^5 & = 2b(5a^4 + 10a^2b^2 + b^4),\\ (a+b)^6-(a-b)^6 & =4ab(3a^2 + b^2)(a^2 + 3b^2) \end{align*}