Category:Viète's Formulas

This category contains pages concerning Viète's Formulas:

$\ds \map {P_n} x$

$=$

$\ds \sum_{i \mathop = 0}^n a_i x^i$

$\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.

Then:

$\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.

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