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
SUMMARY

The discussion focuses on finding the sum of the geometric series \(1 + x + x^2 + ... + x^{99}\) for the roots of the polynomial \(x^{100} - 3x + 2 = 0\). The sum is derived using the formula for the sum of a geometric series, resulting in \(3\) for all roots except \(x = 1\), where the sum equals \(100\). The derivation involves substituting \(x^{100} = 3x - 2\) into the geometric series formula, confirming the calculations through algebraic manipulation.

PREREQUISITES
  • Understanding of polynomial equations and their roots
  • Knowledge of geometric series and their summation formulas
  • Basic algebraic manipulation skills
  • Familiarity with polynomial identities
NEXT STEPS
  • Study the properties of polynomial roots and their relationships
  • Learn more about geometric series and their applications in mathematics
  • Explore advanced polynomial equations and their solutions
  • Investigate the implications of special cases in polynomial functions
USEFUL FOR

Mathematicians, students studying algebra, and anyone interested in polynomial equations and geometric series.

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
2K
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
5K
Graduate Coefficients of Chebyshev polynomials
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Matrix rank in terms of determinant polynomial
  • · Replies 14 ·
Replies
14
Views
3K
MHB 11.1 Determine if the polynominal.... is the span of (....,....)
  • · Replies 4 ·
Replies
4
Views
1K
MHB Is x+1 a Factor of the Polynomial x^3-5x^2+3x+1?
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Localizing single variable quotient polynomial ring at a prime ideal
  • · Replies 6 ·
Replies
6
Views
1K