Intermediate Value Theorem for Polynomials

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Discussion Overview

The discussion centers around the Intermediate Value Theorem (IVT) specifically in the context of polynomial functions. Participants are exploring the proof of the IVT, its implications, and the continuity of polynomials.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Homework-related

Main Points Raised

  • One participant asserts that if P is a polynomial function and P(a) and P(b) have opposite signs, then there exists at least one value c between a and b for which P(c) = 0, referencing the IVT.
  • Another participant questions the need for further discussion, suggesting that the IVT itself should suffice to address the problem.
  • A different participant clarifies that they are seeking a proof of the IVT, indicating that their professor mentioned the existence of such a proof, which they find challenging.
  • One participant suggests that searching online for proofs of the IVT will yield many results and emphasizes that proving the continuity of polynomials is straightforward, while also noting that the topic is more aligned with analysis/calculus rather than number theory.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the necessity of proving the IVT, with some believing it is already established and others seeking a deeper understanding of its proof.

Contextual Notes

There is an assumption that participants are familiar with the concepts of continuity and the IVT, but the discussion does not delve into specific mathematical steps or definitions that may be relevant to the proof.

Rhythmer
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Prove: if P is a polynomial function and P(a) and P(b) have opposite signs, then there exists at least one value c between a and b for which P(c) = 0
 
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You have the IVT, so what's the problem?
 
matt grime said:
You have the IVT, so what's the problem?
I need to prove the IVT. Our professor told us there is such proof but it's not an easy one.
 
If you insert some words like;

proof of the intermediate value theorem

into google you get lots of proofs. All you need to do is justify that polynomials are continuous, and that is easy. Also this has nothing to do with number theory. It is analysis/calculus.
 

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