Difference between revisions of "Newton's Sums"
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2019 AMC 12A
2019 AMC 12A 17
Revision as of 13:59, 31 July 2020
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.
Let be the roots of a given polynomial . Then, we have that
Multiplying each equation by , respectively,
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 .