Tschirnhaus Transformation yields Depressed Polynomial

Theorem
Let $\map f x$ be a polynomial of order $n$:
 * $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 $

Then the Tschirnhaus transformation: $y = x + \dfrac {a_{n-1}} {n a_n}$

converts $f$ into a depressed polynomial:
 * $b_n y^n + b_{n-1} y^{n-1} + \cdots + b_1 y + b_0$

where $b_{n-1} = 0$.

Proof
Substituting $y = x + \dfrac {a_{n-1}} {n a_n}$ gives us $x = y - \dfrac {a_{n-1}} {n a_n}$.

By the Binomial Theorem:
 * $a_n x^n = a_n \paren{ y^n - \dfrac {a_{n-1}} {a_n} y^{n-1} + \map { f'_{n-2} } y }$

where $\map { f'_{n-2} } y$ is a polynomial in $y$ of order $n-2$.

Now we note that:
 * $a_{n-1} x^{n-1} = a_{n-1} y^{n-1} - \map { f''_{n-2} } y$

where $\map { f''_{n-2} } y$ is another polynomial in $y$ of order $n-2$.

The terms in $y^{n-1}$ cancel out.

Hence the result.