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Math Help - Polynomials

  1. #1
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    Polynomials

    Please help me with this proof! Thank you!

    Let F be a field. Let f(x), g(x) exist in F[x] be monic polynomials such that f(x)|g(x)
    and deg g(x) is less than or equal to deg f(x). Prove that f(x) = g(x).
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  2. #2
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    Quote Originally Posted by mathgirl1188 View Post
    Please help me with this proof! Thank you!

    Let F be a field. Let f(x), g(x) exist in F[x] be monic polynomials such that f(x)|g(x)
    and deg g(x) is less than or equal to deg f(x). Prove that f(x) = g(x).

    f(x)\mid g(x)\Longrightarrow g(x)=f(x)h(x)\Longrightarrow deg(g)=deg(f)+deg(h) (why?) , but we're given that

    deg(x)\leq deg(f)\Longrightarrow deg(f)+deg(h)\leq deg(f)\Longrightarrow deg(h)=0 (why?), and comparing

    superior coefficients in g(x)=f(x)h(x) we solve the problem.

    Tonio
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