Generic solution of a specific polynomial equation

Asked Nov 12 '21 at 16:19

Active Nov 12 '21 at 16:19

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I am trying to find out if there is a generic solution(s) for x in terms of n for the following polynomial equation.

$$ x^{n} = \sum_{p=0}^{n-1}x^p $$

I tried to do google search for similar form but nothing came out of the search. Wondering if there is such a form?

asked Nov 12 '21 at 16:19

sm535

1

@Moo They would be equal at the $n$ roots of the polynomial $x^n - x^{n-1} - \dots - x - 1$. – Leander Tilsted Kristensen Nov 12 '21 at 16:43
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There is no closed form, see 1, 2 found by approach0.xyz. – dxiv Nov 12 '21 at 18:39
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Thanks @dxiv I also found this gem: https://math.stackexchange.com/questions/3207962/can-the-roots-to-the-polynomial-x6-x5-x4-x3-x2-x-1-be-found-i?rq=1. – sm535 Nov 13 '21 at 06:04

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