# Newton's Sums

**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 sum:

Newton's sums tell us that,

(Define for .)

We also can write:

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,

Note: This technically only proves the statements for the cases where . For the cases where , an argument based on analyzing individual monomials in the expansion can be used (see http://web.stanford.edu/~marykw/classes/CS250_W19/Netwons_Identities.pdf, for example.)

## Example

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

Newton's Sums tell us that:

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

Which gives us our desired solutions, and .