What Are the Possible Values of the Sum from a 2012-Degree Polynomial Solution?

  • Context: MHB 
  • Thread starter Thread starter magneto1
  • Start date Start date
Click For Summary

Discussion Overview

The discussion centers around finding the possible values of the sum \(1 + a + a^2 + \cdots + a^{2011}\) where \(a\) is a solution to the polynomial equation \(x^{2012} - 7x + 6 = 0\). The scope includes mathematical reasoning and exploration of polynomial roots.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant poses the problem of determining the sum for a solution \(a\) of the polynomial equation.
  • Another participant indicates they have a solution but does not provide details in the post.
  • Subsequent replies express approval of the proposed solution, suggesting it may be correct, but do not elaborate on the reasoning or calculations involved.

Areas of Agreement / Disagreement

The discussion does not reach a consensus, as the details of the proposed solutions are not fully articulated, leaving the correctness of the claims unverified.

Contextual Notes

Limitations include the lack of detailed mathematical steps or assumptions that underpin the proposed solutions, as well as the absence of a complete exploration of the polynomial's roots.

Who May Find This Useful

Readers interested in polynomial equations, mathematical problem-solving, and the exploration of roots in higher-degree polynomials may find this discussion relevant.

magneto1
Messages
100
Reaction score
0
Let $x=a$ be a solution of the equation $x^{2012}-7x+6=0$. Find all the possible values for: $1+a+a^2+\cdots+a^{2011}$.
 
Mathematics news on Phys.org
magneto said:
Let $x=a$ be a solution of the equation $x^{2012}-7x+6=0$. Find all the possible values for: $1+a+a^2+\cdots+a^{2011}$.

My solution:

We're told $x=a$ is a solution of the equation $x^{2012}-7x+6=0$, therefore we have $a^{2012}-7a+6=0$.

It can be rewritten as

$a^{2012}-1-7a+6+1=0$

$a^{2012}-1-7a+7=0$

$(a^{2012}-1)-7(a-1)=0$

$(a-1)(a^{2011}+a^{2010}+\cdots+a+1)-7(a-1)=0$

$(a-1)(a^{2011}+a^{2010}+\cdots+a+1-7)=0$

So the value of the expression $1+a+a^2+\cdots+a^{2011}$ could be either 2012 or 7.
 
anemone said:
My solution:...

Quite clever! (Nod)
 
anemone said:
My solution:

We're told $x=a$ is a solution of the equation $x^{2012}-7x+6=0$, therefore we have $a^{2012}-7a+6=0$.

It can be rewritten as

$a^{2012}-1-7a+6+1=0$

$a^{2012}-1-7a+7=0$

$(a^{2012}-1)-7(a-1)=0$

$(a-1)(a^{2011}+a^{2010}+\cdots+a+1)-7(a-1)=0$

$(a-1)(a^{2011}+a^{2010}+\cdots+a+1-7)=0$

So the value of the expression $1+a+a^2+\cdots+a^{2011}$ could be either 2012 or 7.

Nicely done that's correct.
 

Similar threads

MHB Evaluate the constant in polynomial function
  • · Replies 7 ·
Replies
7
Views
2K
High School About a definition: What is the number of terms of a polynomial P(x)?
  • · Replies 48 ·
2
Replies
48
Views
4K
MHB Solving a Third-Degree Polynomial with Real Coefficients
  • · Replies 1 ·
Replies
1
Views
2K
MHB Solving $(x+2011)(x+2012)(x+2013)(x+2014)+1=0$
  • · Replies 5 ·
Replies
5
Views
2K
MHB [ASK] polynomial f(x) divided by (x - 1)
  • · Replies 2 ·
Replies
2
Views
2K
MHB Understanding Cubic Equation Formula for Polynomials of Degree Three
  • · Replies 9 ·
Replies
9
Views
3K
High School General explicit solution to polynomial interpolation
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad A doubt about the multiplicity of polynomials in two variables
  • · Replies 8 ·
Replies
8
Views
3K
High School What are the applications of roots of a polynomial?
  • · Replies 5 ·
Replies
5
Views
3K
MHB Solving 8 Roots: Can 3 Quadratic Polynomials Fulfill $f(g(h(x)))=0$?
  • · Replies 1 ·
Replies
1
Views
1K