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Math Help - Division in Polynomial Rings

  1. #1
    Member Haven's Avatar
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    Division in Polynomial Rings

    I'm having troubles with this problem

    Let F be a Field, and suppose that a\in F and f(x)\in F[x]. Show  f(a) is the remainder when  f(x) is divided by  x-a

    The case when  f(a)=0 is trivial but I don't know what to do from there.
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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by Haven View Post
    I'm having troubles with this problem

    Let F be a Field, and suppose that a\in F and f(x)\in F[x]. Show  f(a) is the remainder when  f(x) is divided by  x-a

    The case when  f(a)=0 is trivial but I don't know what to do from there.
    If q(x) is the quotient and r is the remainder when f(x) is divided by xa then, by definition, f(x) = (xa)q(x) + r. Now all you have to do is to put x=a.
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  3. #3
    Member Haven's Avatar
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    That is so simple, I can't believe I didn't get that.
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