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:
![$\displaystyle P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$](http://latex.artofproblemsolving.com/0/4/8/048f3d9f71b9de5006ba1da753b7631c32eff33b.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, -127 and 1.