# Difference between revisions of "Newton's Sums"

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<math>\displaystyle a_nS_2 + a_{n-1}S_1 + 2a_{n-2}=0</math> | <math>\displaystyle a_nS_2 + a_{n-1}S_1 + 2a_{n-2}=0</math> | ||

− | <math>\displaystyle a_nS_3 + a_{n-1}S_2 + a_{n-2} | + | <math>\displaystyle a_nS_3 + a_{n-1}S_2 + a_{n-2}S_1 + 3a_{n-3}=0</math> |

<math>\vdots</math> | <math>\vdots</math> |

## Revision as of 16:02, 30 June 2006

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.

## Basic Usage

Consider a polynomial:

Let have roots . Define the following sums:

Newton sums tell us that,

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, -127 and 1.