Tschirnhausen Transformation (original) (raw)

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A transformation of a polynomial equation f(x)=0 which is of the form y=g(x)/h(x) where and 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 (1683) 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+px+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 .