Category:Viète's Formulas

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This category contains pages concerning Viète's Formulas:

Let $P_n$ be a polynomial of degree $n$ with real or complex coefficients:

\(\ds \map {P_n} x\) \(=\) \(\ds \sum_{i \mathop = 0}^n a_i x^i\)
\(\ds \) \(=\) \(\ds a_n x^n + a_{n - 1} x^{n - 1} + \dotsb + a_1 x + a_0\)

where $a_n \ne 0$.

Let $z_1, \ldots, z_n$ be the roots of $P_n$ (be they real or complex), not assumed distinct.


\(\ds \forall k \in \set {0, 1, \ldots, n}: \, \) \(\ds \paren {-1}^k \dfrac {a_{n - k} } {a_n}\) \(=\) \(\ds \map {e_k} {\set {z_1, \ldots, z_n} }\)
\(\ds \) \(=\) \(\ds \sum_{\substack {1 \mathop \le i_1 \mathop < \dotsb \mathop < i_k \mathop \le n \\ 1 \mathop \le k \mathop \le n} } z_{i_1} \dotsm z_{i_k}\)

where $\map {e_k} {\set {z_1, \ldots, z_n} }$ denotes the elementary symmetric function of degree $k$ on $\set {z_1, \ldots, z_n}$.

Source of Name

This entry was named for François Viète.


This category has only the following subcategory.

Pages in category "Viète's Formulas"

The following 3 pages are in this category, out of 3 total.