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Math Help - Division Theorem for Polynomials...

  1. #1
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    Division Theorem for Polynomials...

    Prove the Following Lemma...:

    Suppose p(x) and d(x) are polynomials in F[x] and that d(x) has degree at least 1. There exist unique polynomials q(x), r(x) E F(x) such that r(x) has lower degree than d(x) and
    p(x) = q(x) . d(x) + r(x)



    Please help..I have been struggling to understand this Lemma, but its quite complicated.
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  2. #2
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    Hello,

    Try a proof by contradiction for it...
    This is similar to the Euclidian algorithm.
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  3. #3
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    Quote Originally Posted by vikramtiwari View Post
    Prove the Following Lemma...:

    Suppose p(x) and d(x) are polynomials in F[x] and that d(x) has degree at least 1. There exist unique polynomials q(x), r(x) E F(x) such that r(x) has lower degree than d(x) and
    p(x) = q(x) . d(x) + r(x)
    .
    If d(x) (non-zero) has degree zero then it is a constant polynomial, and there is nothing to show, so it is safe to assume it is a linear polynomial. Consider the set S={ p(x) - l(x).d(x)| l(x) in F[x]}. By well-ordering it means there is a least degree for this set, thus, there exists q(x) so that p(x) - q(x).d(x) = r(x) is smallest degree. Argue that r(x) is either zero polynomial or else deg r(x) < 1.
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