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I Question about the Fundamental Theorem of Algebra

  1. May 20, 2017 #1
    Hi All,

    According to the fundamental theorem of algebra: "every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots".
    My question is: what about polynomials with degree say 2.3 or 3.02, as in the polynomial:
    ## p(x) = x^{2.3} - 5x + 6 ?##
    Do these polynomials take part in the FTA ?

    Best wishes,

    DaTario
     
  2. jcsd
  3. May 20, 2017 #2

    mfb

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    That is not a polynomial. By definition, polynomials only have integer powers of the variable.

    Your function cannot even be defined as continuous function over the whole complex plane.
     
  4. May 20, 2017 #3

    Mark44

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    Further, the function isn't defined on the negative real numbers.
     
  5. May 20, 2017 #4

    PeroK

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    Further, no equation can have 2.3 roots!
     
  6. May 20, 2017 #5

    mfb

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    ##\displaystyle x^{2.3}=e^{2.3 \ln(x)}## can easily be defined for negative real numbers, you just have to choose a branch, and you have to define where to make it discontinuous.
     
  7. May 20, 2017 #6

    Mark44

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    The context of my comment, which I didn't state, was polynomials with real variables. I thought those were what he was asking about in writing p(x) = ... instead of ##p(z) = z^{2.3} - 5z + 6##.

    In any case, this is not a polynomial, as you have already said.
     
  8. May 20, 2017 #7

    mfb

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    The fundamental theorem of algebra works with complex numbers and DaTario mentioned them in the first post as well.
     
  9. May 20, 2017 #8

    Mark44

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    I was so dumbfounded by the 2.3 exponent on what he called a polynomial that I didn't notice that he had mentioned complex coefficients.
     
  10. May 20, 2017 #9
    Thank you all.
    I am satisfied with the comments. I was curious because the graph of the function:
    ## f(x) = x^{2.3} - 5x + 2 ##
    as shown below, suggested to me that we could also have some control over its roots.
    Obs: The command to this plot was:
    Plot[x^2.3 - 5x + 2, {x, -5, 5}].

    power23div10.jpg
     
    Last edited: May 20, 2017
  11. May 21, 2017 #10

    Svein

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  12. May 22, 2017 #11
    I must confess that, when I formulated this question, I was in the spirit of that child that asks if someone can multiply a number by itself 2.3 times.
     
  13. May 22, 2017 #12

    FactChecker

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    And the answer is yes -- with conditions. As long as you can take the 10'th root of x, then (x1/10)23 can be considered and studied. So your question has some interesting aspects.

    No matter how strange you think your question is, there is still a chance that someone has studied it seriously -- and maybe even applied it somewhere. There are also fractional derivatives, which I have a very hard time thinking about.
     
  14. May 22, 2017 #13

    Mark44

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    But this gets us well beyond the concept of multiplication, depending as it does on being able to find the 10th root of a number.
     
  15. May 23, 2017 #14

    Svein

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    It has some practical aspects - in a tempered scale, the ratio between two tones a half tone apart is [itex]\sqrt[12]{2} [/itex].
     
  16. May 24, 2017 #15
    I had contact with fractional derivatives once in my life and it was also "hard time".
     
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