A transformation of a polynomial equation f(x) = 0 which is of the form y = g(x)/h(x) where g and h are polynomials and h(x) does not vanish at a root of f(x) = 0. The cubic equation is a special case of such a transformation. Tschirnhaus showed that a polynomial of degree n>2 can be reduced to a form in which the x^(n - 1) and x^(n - 2) terms have 0 coefficients. In 1786, E. S. Bring showed that a general quintic equation can be reduced to the form x^5 + p x + q = 0. In 1834, G. B. Jerrard showed that a Tschirnhaus transformation can be used to eliminate the x^(n - 1), x^(n - 2), and x^(n - 3) terms for a general polynomial equation of degree n>3.

Bring quintic form | cubic equation

Ehrenfried Walter von Tschirnhaus