Tschirnhaus Transformation yields Depressed Polynomial

Theorem
Let $$P_n \left({x}\right) = 0$$ be a polynomial equation of order $$n$$:
 * $$a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0$$

Then the substitution: $$y = x + \frac {a_{n-1}} {n a_n}$$

converts $$P_n$$ into a depressed polynomial:
 * $$b_n y^n + b_{n-1} y^{n-1} + \cdots + b_1 y + b_0 = 0$$

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

Such a substitution is called a Tschirnhaus transformation.

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

By the Binomial Theorem:
 * $$a_n x^n = a_n \left({y^n - \frac n {a_n} y^{n-1} + P'_{n-2} \left({y}\right)}\right)$$

where $$P'_{n-2} \left({y}\right)$$ is a polynomial in $$y$$ of order $$n-2$$.

Now we note that:
 * $$a_n x^{n-1} = a_n y^{n-1} - P''_{n-2} \left({y}\right)$$

where $$P''_{n-2} \left({y}\right)$$ is another polynomial in $$y$$ of order $$n-2$$.

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

Hence the result.