Difference between revisions of "Newton's Sums"
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Which gives us our desired solutions, <math>\boxed{1}</math> and <math>\boxed{-127}</math>. | Which gives us our desired solutions, <math>\boxed{1}</math> and <math>\boxed{-127}</math>. | ||
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+ | ==Practice== | ||
+ | 2019 AMc 12A #17 | ||
==See Also== | ==See Also== |
Revision as of 23:07, 9 February 2019
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