How to Find the Sum of Roots of the Polynomial \( x^{100} - 3x + 2 = 0 \)?

  • Context: Undergrad 
  • Thread starter Thread starter Cade
  • Start date Start date
  • Tags Tags
    Polynomial Roots Sum
Click For Summary

Discussion Overview

The discussion revolves around finding the sum of the roots of the polynomial \( x^{100} - 3x + 2 = 0 \) and specifically calculating the sum \( 1 + x + x^2 + ... + x^{99} \) for each possible value of \( x \). The conversation includes approaches to derive this sum and considerations regarding specific roots of the polynomial.

Discussion Character

  • Mathematical reasoning
  • Exploratory
  • Debate/contested

Main Points Raised

  • One participant asks how to approach the problem of finding the sum of roots for the polynomial.
  • Another participant suggests that if \( x \) is a root of the polynomial, the sum can be expressed as \( \frac{x^{100}-1}{x-1} \) and simplifies to 3 under certain conditions.
  • A participant inquires about the derivation of the sum, leading to an explanation involving the geometric series and algebraic manipulation of the polynomial equation.
  • Concerns are raised about the case when \( x = 1 \), suggesting that this root leads to a division by zero in the geometric series formula, which complicates the calculation of the sum.
  • Another participant confirms that for \( x = 1 \), the sum would equal 100, while for other roots, the previously mentioned sum of 3 applies.

Areas of Agreement / Disagreement

Participants express differing views on the implications of \( x = 1 \) for the sum calculation, indicating a lack of consensus on how to handle this case. The discussion remains unresolved regarding the overall implications for the sum of roots.

Contextual Notes

The discussion highlights potential limitations in the approach to the sum, particularly concerning the treatment of specific roots and the conditions under which the geometric series formula is applicable.

Who May Find This Useful

This discussion may be useful for individuals interested in polynomial equations, geometric series, and the nuances of handling specific roots in mathematical problems.

Cade
Messages
90
Reaction score
0
How would I go about approaching this problem?

Given the polynomial:
x^100 - 3x + 2 = 0

Find the sum 1 + x + x^2 + ... + x^99 for each possible value of x.
 
Physics news on Phys.org
Cade said:
How would I go about approaching this problem?

Given the polynomial:
x^100 - 3x + 2 = 0

Find the sum 1 + x + x^2 + ... + x^99 for each possible value of x.


If you meant that x is a root of the polynomial X^{100}-3X+2 , then
1+x+...+x^{99}=\frac{x^{100}-1}{x-1}=\frac{3x-3}{x-1}=3

DonAntonio
 
Interesting, thanks, how did you derive that?
 
Cade said:
Interesting, thanks, how did you derive that?



First equality: sum of a geometric sequence.

Second equality: x^{100}-3x+2=0\Longrightarrow x^{100}=3x-2

Third equality: trivial algebra

DonAntonio
 
Oh, I didn't realize the first part was the sum of a geometric series. Thanks for your help.
 
isn't a trivial solution to the equation equal to 1, then then sum would be greater than 3, This is the solution that makes the geometric sum equation impossible as you are dividing by zero.
 
Last edited:
coolul007 said:
isn't a trivial solution to the equation equal to 1, then then sum would be greater than 3, This is the solution that makes the geometric sum equation impossible as you are dividing by zero.



Indeed. So for \,\,x=1\,\,,\,\,1+1^1+1^2+...+1^{99}=100\,\, , and for all the other roots it is what I wrote before.

Thanx.

DonAntonio
 

Similar threads

Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
3K
Undergrad Equivalent definitons for primitive polynomials
  • · Replies 24 ·
Replies
24
Views
1K
Undergrad Looking for a creative or quick method for finding roots in GF(p^n)
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Finding a polynomial that has solution (root) as the sum of roots
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
  • · Replies 48 ·
2
Replies
48
Views
6K
Graduate Coefficients of Chebyshev polynomials
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Matrix rank in terms of determinant polynomial
  • · Replies 14 ·
Replies
14
Views
4K
MHB 11.1 Determine if the polynominal.... is the span of (....,....)
  • · Replies 4 ·
Replies
4
Views
2K
MHB Is x+1 a Factor of the Polynomial x^3-5x^2+3x+1?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Localizing single variable quotient polynomial ring at a prime ideal
  • · Replies 6 ·
Replies
6
Views
2K