Viète's Formulas/Examples/Monic Polynomial

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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 $a_N$:

\((1):\quad\) \(\displaystyle a_N\) \(=\) \(\displaystyle 1\)
\((2):\quad\) \(\displaystyle a_i\) \(=\) \(\displaystyle b_i\) 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 $(1)$

$\blacksquare$