Is there a quick analytic way to solve this polynomial?

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Discussion Overview

The discussion revolves around finding an analytic solution for a polynomial equation of the form ny = x^(n-1) + x^(n-2) + ... + x + 1, specifically for large values of n (greater than 10,000) and certain conditions on y. The focus is on exploring potential methods for solving this polynomial efficiently, particularly for specific values of y.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests interpreting the polynomial as a sum to facilitate working with it.
  • Another participant expresses a desire for a quick method to find x for specific values of y and n, noting the impracticality of solving large polynomials directly.
  • A participant points out a misunderstanding, clarifying that the original poster is seeking to solve the polynomial rather than perform calculus operations.
  • One participant reiterates the polynomial's form and introduces a formula for the sum, prompting further exploration of this approach.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the existence of a quick analytic solution, with some expressing skepticism about the feasibility of solving high-degree polynomials analytically.

Contextual Notes

The discussion highlights the complexity of solving polynomials of degree 5 and higher, particularly in the context of large n and specific conditions on y. There are unresolved assumptions regarding the nature of solutions and the applicability of proposed methods.

Who May Find This Useful

Individuals interested in polynomial equations, particularly those involving high degrees and specific constraints, as well as those exploring analytic methods in mathematics.

DeanRichards
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ny = x^(n-1) + x^(n-2) + ... + x + 1

for a certain y and n (>10000) with y!=1 and ny > 1.

Is there an analytic way to solve this? Thank you.
 
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yes, you can interpret it as a sum. that way you can work with it easily.
from that point on you can prove any type of given question whatsoever
 

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thank you for your reply, my question was though how to find x for let's say y = 4 and n = 10000 in the shortest possible amount of time. i know that's not feasible for polynomials of this size in general, but i don't know if there are any special kinds and solution strategies, since this one looks quite simple.
i should have added that i am only interested in real solutions which only exist for n*y >= 1.
 
I think poster #2 is confused. The OP was looking to solve the polynomial, not take derivatives or integrals.

AFAIK, there is no simple analytic way to solve polynomials with degree 5 and greater.
 
DeanRichards said:
ny = x^(n-1) + x^(n-2) + ... + x + 1

for a certain y and n (>10000) with y!=1 and ny > 1.

Is there an analytic way to solve this? Thank you.

The sum on the right = (1-xn)/(1-x).

Can you go from there?
 
mathman said:
The sum on the right = (1-xn)/(1-x).

Can you go from there?

incredible. thanks a lot!
 

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