# Definition:Tschirnhaus Transformation

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## Definition

Let $\map f x$ be a polynomial over a field $k$:

$\map f x = a_n x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + \cdots + a_1 x + a_0$

Then the Tschirnhaus transformation is the linear substitution $x = y - \dfrac {a_{n - 1} } {n a_n}$.

The Tschirnhaus transformation produces a resulting polynomial $\map {f'} y$ which is depressed, as shown on Tschirnhaus Transformation yields Depressed Polynomial.

This technique is used in the derivation of Cardano's Formula for the roots of the general cubic.

## Also known as

The Tschirnhaus transformation is also called the Tschirnhaus substitution.

## Also see

• Results about Tschirnhaus transformations can be found here.

## Source of Name

This entry was named for Ehrenfried Walther von Tschirnhaus.