# Difference between revisions of "Newton's Sums"

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− | <math>\begin{cases}a_n\alpha^n+a_{n-1}\alpha^{n-1}+...+a_0=0\\a_n\beta^n+a_{n-1}\beta^{n-1}+...+a_0=0\\~~~~~~~~~~~~~~~~~~\vdots\\a_n\omega^ | + | <math>\begin{cases}a_n\alpha^n+a_{n-1}\alpha^{n-1}+...+a_0=0\\a_n\beta^n+a_{n-1}\beta^{n-1}+...+a_0=0\\~~~~~~~~~~~~~~~~~~\vdots\\a_n\omega^n+a_{n-1}\omega^{n-1}+...+a_0=0\end{cases}</math> |

## Revision as of 23:09, 8 January 2020

**Newton sums** give us a clever and efficient way of finding the sums of roots of a polynomial raised to a power. They can also be used to derive several factoring identities.

## Contents

## Statement

Consider a polynomial of degree ,

Let have roots . Define the following sums:

Newton sums tell us that,

(Define for .)

We also can write:

etc., where denotes the -th elementary symmetric sum.

## Proof

Let be the roots of a given polynomial . Then, we have that

Thus,

Multiplying each equation by , respectively,

Sum,

Therefore,

## Example

For a more concrete example, consider the polynomial . Let the roots of be and . Find and .

Newton Sums tell us that:

Solving, first for , and then for the other variables, yields,

Which gives us our desired solutions, and .

## Practice

2019 AMC 12A #17