# Elementary Symmetric Function/Examples/Monic Polynomial

Jump to navigation
Jump to search

It has been suggested that this page or section be merged into Viète's Formulas.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mergeto}}` from the code. |

## Example of Elementary Symmetric Function: Monic Polynomial

Let $\set {x_1, x_2, \ldots, x_n}$ be a set of real or complex values, not required to be unique.

The expansion of the monic polynomial in variable $x$ with roots $\set {x_1, x_2, \ldots, x_n}$ has coefficients which are parity operators times an elementary symmetric function:

- $\ds \prod_{j \mathop = 1}^n \paren {x - x_j} = x^n - \map {e_1} {\set {x_1, \ldots, x_n} } x^{n - 1} + \map {e_2} {\set {x_1, \ldots, x_n} } x^{n - 2} - \dotsb + \paren {-1}^n \map {e_n} {\set {x_1, \ldots, x_n} }$

## Proof

This is an exposition of Viète's Formulas.

$\blacksquare$