Can You Find Real Roots in an Integer Polynomial Using Calculus?

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Discussion Overview

The discussion centers around determining whether an integer coefficient single variable polynomial has at least one real root. It explores various conditions related to the degree of the polynomial and the nature of its coefficients, touching on both theoretical and practical aspects of the problem.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants suggest that if the polynomial has an odd degree and real coefficients, it must have at least one real root.
  • Others mention that for quadratic polynomials, the condition for having real roots is that the discriminant must be non-negative.
  • A participant notes that for higher even degree polynomials, determining the existence of real roots is more complex and may involve methods such as Sturm sequences.
  • One participant expresses the need to work out the general case, indicating that while it may only require elementary calculus, it is still considered difficult.

Areas of Agreement / Disagreement

There is some agreement on the conditions for odd degree and quadratic polynomials having real roots, but the discussion remains unresolved regarding higher even degree polynomials and the general case.

Contextual Notes

The discussion does not fully resolve the complexities involved in determining real roots for higher even degree polynomials and lacks specific definitions or assumptions that may affect the conclusions drawn.

Dragonfall
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I'll try my luck here:

How do you determine whether an integer coefficient single variable polynomial has at least one real root?
 
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if the order is odd and coefficients are real?
 
Dragonfall said:
I'll try my luck here:

How do you determine whether an integer coefficient single variable polynomial has at least one real root?
Integer coefficients don't make any difference. If the coefficients are real, then for odd degree, there has to be (as trambolin notes) at least one real root. For quadratic, the discriminant has to be non-negative. For higher even degree it is more complicated. Try looking at Sturm sequences.
 
Of course if the degree is odd or is 2 then it's trivial. I have to work out the general case, which "only requires elementary calculus, but is difficult."
 

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