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Ignoring Fundamental Theorem of Polynomials (a^n + b^n)

  1. Mar 7, 2008 #1
    1. The problem statement, all variables and given/known data

    Okay, basically why does (a[tex]_{}n[/tex] + b[tex]_{}n[/tex]) ignore the Fundamental Theory of Polynomials?

    2. Relevant equations

    ... I could post them here, but basically when n is odd (a[tex]_{n}[/tex] + b[tex]_{n}[/tex]) = a series that looks like this: (a+b) (a[tex]_{n-1}[/tex] b[tex]_{0}[/tex] - a[tex]_{n-2}[/tex]b +a[tex]_{n-3}[/tex]b[tex]_{2}[/tex] + ... + a[tex]_{2}[/tex]b[tex]_{n-3}[/tex]-a[tex]_{1}[/tex]b[tex]_{n-2}[/tex]+b[tex]_{n-1}[/tex])

    3. The attempt at a solution

    I have done a lot on paper, but basically what I am looking for is WHY it isn't following Pascal's triangle. I have flipped through tons of mathematical journals and haven't slept in the past two days for other various reasons... I am doing this for a research class and has lost its proof after years and years of assuming that it is truth.

    Argh. If anyone can help I would really appreciate it. If anyone has the proof to this that would be great.
     
  2. jcsd
  3. Mar 7, 2008 #2

    Dick

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    I'm not sure what the Fundamental Theorem of Polynomials is, but a^n+b^n factors for odd n but not for even n because a^n=-b^n can be 'solved' for a over the reals for odd n but not for even n. If you work over the complex numbers you can factor the even n case. E.g. a^2+b^2=(a+bi)(a-bi).
     
  4. Mar 7, 2008 #3

    HallsofIvy

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    Pascal's triangle holds for (x+ y)n and this isn't of that kind! If you are doing this for a research class perhaps you have something else in mind. My question would be "Why would you expect it to have anything to do with Pascal's triangle?"
     
  5. Mar 7, 2008 #4

    D H

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    The polynomial [itex](a^n+b^n)[/itex] doesn't "follow Pascal's triangle" because Pascal's triangle doesn't apply to this polynomial. It applies to the polynomial [itex](a+b)^n[/itex], which is obviously different from [itex](a^n+b^n)[/itex] for all n but 1.
     
  6. Mar 7, 2008 #5
    Okay, fair enough.

    I understand that this polynomial does not follow Pascal's triangle and why. That was poorly worded on my part. What I am asking for is the proof for why the factorization of the polynomial exists as it does. I am aware of the fact that it uses imaginary numbers for the even powers. I also know how to derive the (a^n+b^n) formula from the (a^n-b^n) formula. It is complicated and I don't feel like texting the whole thing.

    The fundamental theorem of algebra states that every polynomial has n factors based on its degree, yada yada yada. The way the factorization works (If you make the general form) you notice that it establishes a pattern. However, I don't know the steps to factorize it outside of knowing the formula. I am looking to find where this was derived from.

    God I wish my brain was being coherent enough to explain what I am trying to say or wasn't being totally lazy and could make up the math text. Sorry.
     
  7. Mar 7, 2008 #6
    I am about to go on spring break so will be able to work on this more. I am going to get some sleep and then get back to you all.
     
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