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Math Help - Polynomials

  1. #1
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    Polynomials

    Let a(x) and b(x) be in F[x]. I need to prove that if a(x) and b(x) determine the same function, and if the number of elements in F exceeds the degree of a(x) as well as the degree of b(x), then a(x) = b(x).

    I know what if a(x) and b(x) determine the same function means. It means that a(x) - b(x) gives you the zero function. I dont know what if the number of elements in F exceeds the degree of a(x) as well as the degree of b(x) means. I also do not see how either of the two can conclude a(x) = b(x). Can anyone help me?
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  2. #2
    Junior Member Nehushtan's Avatar
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    Re: Polynomials

    The terminology is certainly confusing. I think what you mean by “zero function” is the constant function on F whose image is 0, not the zero polynomial in F[x]. In other words, a(x) and b(x) determine the same function on F iff a=b considered as functions a,b:F\to F.

    An example to show what I mean. Suppose F=\mathbb F_2=\{0,1\}, a(x)=x, b(x)=x^2. Then, considered as functions F\to F, a and b are equal (to the identity function on F), but a(x)\ne b(x) as polynomials in F[x].

    In the example, the number of elements of F is not greater than the degree of a(x). What you are asked to show is that if |F|>\deg a,\deg b, then a(x)=b(x) as polynomials in F[x] if a=b as functions F\to F.
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