Viète's Formulas/Examples/Monic Polynomial
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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 $a_N$:
\(\text {(1)}: \quad\) | \(\ds a_N\) | \(=\) | \(\ds 1\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds a_i\) | \(=\) | \(\ds b_i\) | 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:
\(\ds b_k\) | \(=\) | \(\ds a_k\) | from $(2)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds a_{N-j}\) | change of index $k = N-j$ | |||||||||||
\(\ds \) | \(=\) | \(\ds a_N \paren {-1}^j \map {e_j} U\) | Viète's Formulas | |||||||||||
\(\ds \) | \(=\) | \(\ds a_N \paren {-1}^{N - k} \map {e_{N - k} } U\) | change of index $j = N - k$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{N - k} \map {e_{N - k} } U\) | $a_N = 1$ by $(1)$ |
$\blacksquare$