Definition:Depressed Polynomial

Definition
Let $f \left({x}\right)$ be a polynomial function over a field $k$:


 * $f \left({x}\right) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_1 x + a_0$

If $a_{n-1} = 0_k$, then we call $f$ a depressed polynomial.

It has been suggested that a polynomial with further zero terms might be referred to as "downright despondent", though this convention has yet to gain widespread usage by the community.

Tschirnhaus Substitution
When looking for solutions $f \left({x}\right) = 0$, we can make the linear substitution $x = y - \frac {a_{n-1}} n$.

Letting $O(y^j)$, $j \in \Z$ formally denote any finite sum of terms of degree at most $j$, we find that

This shows that the search for roots of $f$ can be reduced to the case when $f$ is depressed.

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

It is known as a Tschirnhaus Transformation.