# Viète's Formulas/Examples/Monic Polynomial

## Example of Use of Viète's Formulas

Let:

 $\displaystyle \map P x$ $=$ $\displaystyle x^N + \displaystyle \sum_{k \mathop = 0}^{N - 1} b_k x^k$ Monic polynomial of degree $N$.

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

Then:

 $\displaystyle b_k$ $=$ $\displaystyle \paren {-1}^{N - k} \, \map {e_{N - k} } U, \quad 0 \le k \le N - 1$ Definition of Elementary Symmetric Function $\map {e_m} U$

## Proof

Let:

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

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

 $\text {(1)}: \quad$ $\displaystyle a_N$ $=$ $\displaystyle 1$ $\text {(2)}: \quad$ $\displaystyle a_i$ $=$ $\displaystyle b_i$ for $0 \le i \le N-1$

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

Then $k = N-j$.

Apply the change of index:

 $\displaystyle b_k$ $=$ $\displaystyle a_k$ from $(2)$ $\displaystyle$ $=$ $\displaystyle a_{N-j}$ change of index $k = N-j$ $\displaystyle$ $=$ $\displaystyle a_N \paren {-1}^j \map {e_j} U$ Viète's Formulas $\displaystyle$ $=$ $\displaystyle a_N \paren {-1}^{N - k} \map {e_{N - k} } U$ change of index $j = N - k$ $\displaystyle$ $=$ $\displaystyle \paren {-1}^{N - k} \map {e_{N - k} } U$ $a_N = 1$ by $(1)$

$\blacksquare$