Newton's Sums
Revision as of 22:56, 27 June 2013 by Willalphagamma (talk | contribs) (Changed S to P, since I think it is more widely used, and S conflicts with the notation used in Newton's Inequality page.)
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.
Statement
Consider a polynomial of degree ,
Let have roots . Define the following sums:
Newton sums tell us that,
(Define for .)
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 .