Find the sum of the thirteenth powers of the roots of x^(13)+x−2>=0.

Alduccii2 2022-07-22 Answered
Find the sum of the thirteenth powers of the roots of x 13 + x 2 0
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Answers (1)

Reinfarktq6
Answered 2022-07-23 Author has 18 answers
Any root r i of x 13 + x 2 = 0 satisfies r i 13 + r i 2 = 0 , or r i 13 = 2 r i . A polynomial of degree 13 has 13 roots (counting repititons). Sum them up:
i = 1 13 r i 13 = 26 i = 1 13 r i .
Also observe that the x n k th coefficient of a polynomial is the kth symmetric polynomials in the roots, with
ceoff of  x 12 = r 1 + r 2 + + r 13 = 0.
(to convince yourself of the latter fact, expand a smaller example: ( x r 1 ) ( x r 2 ) ( x r 3 ) , and observe the coefficient of x 2 .)

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