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
,
![$P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$](http://latex.artofproblemsolving.com/d/a/4/da42fa5fa45147429584e2685f8dc2e22247c497.png)
Let have roots
. Define the sum:
Newton's sums tell us that,
(Define for
.)
We also can write:
etc., 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,
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
.