Roots of polynomial equations ( Substitution )

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Discussion Overview

The discussion revolves around the reduction of the polynomial equation u^4 + 5u^3 + 6u^2 + 5u + 1 = 0 to the form v^2 + 5v + 4 = 0 using the substitution v = u + 1/u. Participants explore methods and implications of this substitution in the context of polynomial equations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant asks how to perform the reduction using the substitution v = u + 1/u.
  • Another participant suggests dividing through by u^2, noting that u must not equal zero, and prompts others to observe the implications of this step.
  • A third participant reiterates the initial question and suggests either following a previous suggestion or substituting v into the equation v^2 + 5v + 4 = 0 to derive the original polynomial.
  • One participant introduces a different polynomial equation, x^6 - 6x^5 + 14x^4 - 18x^3 + 14x^2 - 6x + 1 = 0, suggesting that the same substitution method could be applied.
  • Another participant presents a similar polynomial, x^6 - 9x^5 + 29x^4 - 42x^3 + 29x^2 - 9x + 1 = 0, indicating a preference for this equation for exploration.

Areas of Agreement / Disagreement

Participants are engaged in a collaborative exploration of the substitution method, but there is no consensus on a single approach or resolution to the problem. Multiple methods and examples are presented without agreement on a definitive solution.

Contextual Notes

Some participants mention the necessity of the condition u ≠ 0 for certain steps, and the implications of dividing by u^2 are not fully resolved. The discussion includes various polynomial forms that may require different considerations.

Erfan1
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How do I reduce u^4 + 5u^3 + 6u^2 + 5u + 1 = 0 to v^2 + 5v + 4 = 0 by using v = u + 1/u ?
 
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Erfan said:
How do I reduce u^4 + 5u^3 + 6u^2 + 5u + 1 = 0 to v^2 + 5v + 4 = 0 by using v = u + 1/u ?

Since $$u \ne 0$$ you may divide through by $$u^2$$.

What do you notice now?
 
Erfan said:
How do I reduce u^4 + 5u^3 + 6u^2 + 5u + 1 = 0 to v^2 + 5v + 4 = 0 by using v = u + 1/u ?
You can do as M R suggested or you can put $v=u+1/u$ in $v^2+5v+4=0$. You should get $u^4 + 5u^3 + 6u^2 + 5u + 1 = 0$.
 
You can use the same idea to solve $$x^6-6x^5+14x^4-18x^3+14x^2-6x+1=0$$, which I made specially for you. :)
 
And a slightly nicer one $$x^6-9x^5+29x^4-42x^3+29x^2-9x+1=0$$.
 

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