Difference of two cubes ($a^3-b^3=(a-b)(a^2+ab+b^2)$)

The polynomial identityfollows by multiplying out the right side. For real $a,b$, the quadratic factor is $a^2+ab+b^2=(a+b/2{)}^2+3b^2/4$, so it is nonnegative and vanishes only when $a=b=0$. The identity is a special case of the finite geometric series factorisation $a^n-b^n=(a-b){\sum }_{j=0}^{n-1}{a}^{n-1-j}{b}^j$.