Difference between revisions of "Newton's Sums"
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− | <math>\begin{cases}a_n\alpha^n+a_{n-1}\alpha^{n-1}+...+a_0=0\a_n\beta^n+a_{n-1}\beta^{n-1}+...+a_0=0\~~~~~~~~~~~~~~~~~~\vdots\a_n\omega^ | + | <math>\begin{cases}a_n\alpha^n+a_{n-1}\alpha^{n-1}+...+a_0=0\a_n\beta^n+a_{n-1}\beta^{n-1}+...+a_0=0\~~~~~~~~~~~~~~~~~~\vdots\a_n\omega^n+a_{n-1}\omega^{n-1}+...+a_0=0\end{cases}</math> |
Revision as of 22:09, 8 January 2020
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
[hide]Statement
Consider a polynomial of degree ,
Let have roots . Define the following sums:
Newton 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 Sums tell us that:
Solving, first for , and then for the other variables, yields,
Which gives us our desired solutions, and .
Practice
2019 AMC 12A #17