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
SUMMARY

The discussion focuses on the polynomial equation $x^{2012}-7x+6=0$ and seeks to determine the possible values of the sum $1+a+a^2+\cdots+a^{2011}$, where $a$ is a root of the polynomial. Participants confirm the correctness of the proposed solutions, indicating a collaborative effort to validate the findings. The analysis of the polynomial's roots and their implications on the geometric series sum is central to the conversation.

PREREQUISITES
  • Understanding of polynomial equations, specifically degree 2012 polynomials.
  • Knowledge of geometric series and their summation formulas.
  • Familiarity with root-finding techniques for polynomials.
  • Basic algebraic manipulation skills.
NEXT STEPS
  • Explore the properties of polynomial roots using the Rational Root Theorem.
  • Study the derivation and application of the geometric series formula.
  • Investigate numerical methods for approximating roots of high-degree polynomials.
  • Learn about the implications of Vieta's formulas in polynomial equations.
USEFUL FOR

Mathematicians, students studying algebra, and anyone interested in polynomial equations and their properties.

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
1K
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