# Elementary symmetric sum

An **elementary symmetric sum** is a type of summation.

## Contents

## Definition

The -th **elementary symmetric sum** of a set of numbers is the sum of all products of of those numbers (). For example, if , and our set of numbers is , then:

1st Symmetric Sum =

2nd Symmetric Sum =

3rd Symmetric Sum =

4th Symmetric Sum =

## Notation

The first elementary symmetric sum of is often written . The th can be written

## Uses

Any symmetric sum can be written as a polynomial of the elementary symmetric sum functions. For example, . This is often used to solve systems of equations involving sums of powers, combined with Vieta's formulas.

Elementary symmetric sums show up in Vieta's formulas. In a monic polynomial of degree , the coefficient of the term is , and the coefficient of the term is , where the symmetric sums are taken over the roots of the polynomial.