Help with solving a polynomial

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    Polynomial
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Discussion Overview

The discussion revolves around solving the polynomial equation x^10 + a*x + 1 = 0, specifically finding all real numbers a such that there exists a real solution r for which 1/r is also a solution. The scope includes mathematical reasoning and problem-solving techniques related to polynomial equations.

Discussion Character

  • Mathematical reasoning
  • Homework-related
  • Exploratory

Main Points Raised

  • One participant seeks help in finding the values of a by equating the polynomial for r and 1/r, noting that their initial attempts were unsuccessful.
  • Another participant suggests rearranging the equation to express a in terms of x, proposing that the right-hand sides of the equations for r and 1/r should be equal.
  • A later reply indicates that equating the two forms leads to the conclusion that r must be ±1, resulting in two equations for a: a = ±2.
  • Further contributions clarify that for r = 1, the polynomial simplifies correctly, while for r = -1, the conditions also appear to be satisfied, although there is uncertainty about the validity of r = -1 as a solution.

Areas of Agreement / Disagreement

Participants express differing views on the validity of r = -1 as a solution, with some asserting it satisfies the conditions while others question its correctness. The discussion remains unresolved regarding the implications of these findings.

Contextual Notes

There are limitations in the assumptions made about the solutions and the dependence on the specific values of a. The discussion does not fully resolve the mathematical steps or the implications of the derived equations.

Who May Find This Useful

Readers interested in polynomial equations, mathematical problem-solving techniques, or those seeking assistance with similar homework problems may find this discussion relevant.

galois427
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I need some help on how to solve this question. It asks me to find all real numbers a with the property that the polynomial equation x^10 + a*x +1 = 0 has a real solution r such that 1/r is also a solution. I tried plugging in r and 1/r and equating the 2 equations, but that got me nowhere.
 
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rearrange the equation as,
a = (-x^10-1)/x
put x=r and call it eqn 1
then put x=1/r and call it eqn 2
shouldn't RHS of both 1 and 2 be same?

-- AI
 
Well, could you show what you got when you plugged in r and 1/r?
 
TenaliRaman said:
rearrange the equation as,
a = (-x^10-1)/x
put x=r and call it eqn 1
then put x=1/r and call it eqn 2
shouldn't RHS of both 1 and 2 be same?

-- AI

i euqated that equation and found out that r= + or - 1 so a = + or - 2.
well, that yielded 2 equations. x^10 - 2x + 1 and x^10 + 2x +1. r = 1 is a zeo, but r=-1 is not. what am i doing wrong?
 
Do you know the answer?
 
for r = 1, a=-2
so x^10 - 2x + 1 = 0
put r = 1 , it is zero ... put 1/r = 1 .. again it is zero.

for r=-1, a = 2
so x^10 + 2x + 1 = 0
put r = -1 , it is zero ... put 1/r = -1 .. again it is zero.

so our conditions are satisfied...

-- AI
 

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