Eliminating polynomial terms

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

xⁿ + a_n-1 x^n-1 + a_n-2 x^n-2 + ... + a₀ = 0

into

yⁿ + b_n-4 y^n-4 + b_n-5 y^n-5 + ... + b₀ = 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.

• Higher-order Newton-like methods for root-finding
• The fundamental theorem of algebra
• Discriminant of a cubic