Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 1

The difference of two cubes identity gives the real polynomial factorizationExpanding the right side cancels the two mixed terms of each degree. The quadratic factor satisfies $a^2+ab+b^2=(a+b/2{)}^2+3b^2/4$.

For real cube roots, set $u=a^{1/3}$ and $v=b^{1/3}$, so $(ab{)}^{1/3}=uv$. The sum of two cubes identity gives