The first step in solving a cubic equation is to apply a change of variables to reduce an equation of the form

*x*³ + *bx*² + *cx* + *d* = 0

to one of the form

*y*³ + *py* + *q* = 0.

This process can be carried further through Tschirnhausen transformations, a generalization of an idea going back to Ehrenfried Walther von Tschirnhaus in 1683.

For a polynomial of degree *n* > 4, a Tschirnhausen transformations is a rational change of variables

*y* = *g*(*x*) / *h*(*x*)

turning the equation

*x*^{n} + *a*_{n−1} *x*^{n−1} + *a*_{n−2} *x*^{n−2} + … + *a*_{0} = 0

into

*y*^{n} + *b*_{n−4} *y*^{n−4} + *b*_{n−5} *y*^{n−5} + … + *b*_{0} = 0

where the denominator *h*(*x*) of the transformation is not zero at any root of the original equation.

I believe the details of how to construct the transformations are in An essay on the resolution of equations by G. B. Jerrad.