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.
Basic Usage
Consider a polynomial of degree
,
![$\displaystyle <math>\displaystyle P(x)$](http://latex.artofproblemsolving.com/b/0/b/b0b4edfc2f4698ec44c7a4df2fed27ac5940d3bc.png)
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, and
.