By a linear change of variable, a cubic polynomial over ${\mathbb{C}}$ can be given the form $x^{3}+3bx+c$. To find the zeros of this cubic in the form of surds in $b$ and $c$, make the substitution $x=y^{{1/3}}+z^{{1/3}},$ thus replacing one unknown with two, and then write down identities which are suggested by the resulting equation in two unknowns. Specifically, we get

This will be true if

which in turn requires

The pair of equations (2) and (4) is a quadratic system in $y$ and $z$, readily solved. But notice that (3) puts a restriction on a certain choice of cube roots.