Question on the logic of rational roots theorem

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Discussion Overview

The discussion centers on the rational roots theorem and its implications for the polynomial function f(x) = x^3 + x^2 - 4x - 7. Participants explore the logic behind the theorem, particularly in relation to the existence of rational and irrational roots, and the confusion surrounding the concept of conjugates.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the rational roots theorem, noting that while the theorem indicates no rational roots exist for the given polynomial, a graph suggests there is a root, leading to a contradiction in their reasoning.
  • Another participant points out that if a root is irrational, it does not imply that its conjugate must also be a root, clarifying the distinction between irrational and complex roots.
  • Several participants emphasize that the rational roots theorem does not involve complex conjugation and that the absence of rational roots is confirmed by evaluating specific values of the polynomial.
  • A participant acknowledges their misunderstanding of the terms "irrational" and "complex," indicating a mix-up in the concepts discussed.

Areas of Agreement / Disagreement

Participants generally agree on the application of the rational roots theorem and the distinction between irrational and complex roots, but there is some confusion expressed by one participant regarding these concepts.

Contextual Notes

The discussion highlights the importance of precise definitions in mathematics, particularly regarding the nature of roots and the implications of the rational roots theorem. Some assumptions about the relationships between irrational and complex roots remain unresolved.

dnt
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this is not a homework question, but rather I feel like there is a contradiction in the theorem and just want clarity. I know the theorem is correct so I am looking for help in where the mistake is in my logic.

take f(x) = x^3 + x^2 - 4x- 7

the rational roots theorem says if there are any rational roots they must be in the set: plus or minus 7 and 1. none of which work using synthetic division. so the logic would dictate there should be no rational roots.

however, when you graph it you can see there is one root. but if this root were irrational, I believe the conjugate is always a root - hence there should be two. so therefore it cannot be an irrational root? therefore it must be rational?

which contradicts my previous statement from the rational roots theorem stating there were no rational zeros in this function.

hence my confusion...can someone clarify where I made my mistake in my logic or which assumption I made that was wrong? thanks!
 
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##\sqrt[3]{3}## is the only real root of ##x^3-3##. ##\sqrt[3]{3}## is not rational.
 
but if this root were irrational, I believe the conjugate is always a root

The conjugate? Are you confusing irrational roots with complex roots?
 
If a polynomial equation, having only real coefficients, has a non-real complex number as root, it must also have the complex conjugate of that number as root.

But "irrational" does NOT mean "complex"!
 
Having checked that f(1),f(-1),f(7),f(-7) are all \neq 0, you've indeed shown (by the rational roots theorem) that f has no rational roots. That is, are no a\in\mathbb Z, b\in\mathbb N such that f\left(\frac{a}{b}\right)=0. As others have pointed out, this result doesn't at all involve talking about complex conjugation (given that \mathbb Q\subseteq\mathbb R).
 
dnt said:
but if this root were irrational, I believe the conjugate is always a root - hence there should be two. so therefore it cannot be an irrational root? therefore it must be rational?

"The conjugate is a root" statement refers to complex conjugation. You are mixing that up with conjugation from rationalizing the denominator.
 
i see - thanks for the help everyone! makes sense. i did get complex confused with irrational.
 

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