Viète's Formulas/Examples/Monic Polynomial

Example of Use of Viète's Formulas
Let:


 * $\ds \map P x = x^N + \sum_{k \mathop = 0}^{N - 1} b_k x^k$

be a monic polynomial of degree $N$.

Let $U$ be the set of $N$ roots of equation $\map P x = 0$.

Then:


 * $b_k = \paren {-1}^{N - k} \map {e_{N - k} } U, \quad 0 \le k \le N - 1$

where $\map {e_m} U$ denotes an elementary symmetric function.

Proof
Let:


 * $U = \set {x_1, \ldots, x_N}$

Translate Viète's Formulas from notation $a_0$ to $a_N$:

Let $N-k = j$ define a change of index.

Then $k = N-j$.

Apply the change of index: