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Thread: Polynomial questions

  1. #1 Polynomial questions 
    Forum Junior epidecus's Avatar
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    Hello. Have a few questions about polynomials that were never really cleared up in my algebra courses.

    1. Are a polynomial's derivatives and antiderivatives (of all orders) also polynomials?
    2. If a function is equal to a polynomial, but isn't in the ordinary form of one (as in, it looks nothing like the usual sum of powers with coefficients), is it still considered a polynomial function?
    3. The proof of the rational root theorem assumes that and are coprime. Does this need to be taken into account when evaluating the roots this way? It never seems I've had to worry about it when simply using it, so why is it necessary in the elementary proof?


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  3. #2  
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    1. yes (trivially) integrals and derivatives of powers of x are also powers of x.

    2. how do make a function which is equal to a polynomial for all values of the argument and avoid looking like a polynomial?


    for 3. Rational root theorem - Wikipedia, the free encyclopedia


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  4. #3  
    Forum Junior epidecus's Avatar
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    Quote Originally Posted by mathman View Post
    2. How do make a function which is equal to a polynomial for all values of the argument and avoid looking like a polynomial?
    For example, which is true for all reals. I was able to find this info on Wiki so this question's resolved.

    Thanks, but I don't see anything that answers my question.
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  5. #4  
    Forum Professor river_rat's Avatar
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    Those polynomials are called chebyshev polynomials epidecus - are are quite important in applied mathematics
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  6. #5  
    Forum Junior epidecus's Avatar
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    Thanks for the insight, river rat

    Luckily, I found the answer to #3. When I was browsing through the proof, I never put much thought to the fact that the assumption of the coprime operands was fundamental to the proof's method.
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  7. #6  
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    Quote Originally Posted by epidecus View Post
    For example, which is true for all reals.

    (Italics mine.)

    The equality would seem to hold only for magnitudes of x less than or equal to 1.
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  8. #7  
    Forum Junior epidecus's Avatar
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    Quote Originally Posted by tk421 View Post
    The equality would seem to hold only for magnitudes of x less than or equal to 1.
    It holds for all reals. is complex valued for magnitudes of greater than 1. And the cosine function (apparently) seems to have real outputs for complex values.
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  9. #8  
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    Quote Originally Posted by epidecus View Post
    Quote Originally Posted by tk421 View Post
    The equality would seem to hold only for magnitudes of x less than or equal to 1.
    It holds for all reals. is complex valued for magnitudes of greater than 1. And the cosine function (apparently) seems to have real outputs for complex values.
    Ah yes, of course. Thanks for reminding me of the relationships between the hyperbolic trig functions with real arguments and ordinary trig functions with imaginary arguments, and vice-versa.
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