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Math Help - quick question

  1. #1
    Junior Member
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    quick question

    show that if f1(x) and g1(x) are congruent as polynomials modulo n and f2(x) and g2(x) are congruent as polynomials modulo n, then

    a) (f1+f2)(x) and (g1+g2)(x) are congruent as polynomials modulo n.
    b) (f1f2)(x) and (g1g2)(x) are congruent as polynomials modulo n.

    thanks in advance.
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  2. #2
    MHF Contributor

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    We are to assume here that f(x), for integer x is integer valued? That is the same as saying the coefficients are integers.

    "Congruent as polynomials modulo n" means that for every integer x, f1(x)= g1(x)+ kn for some integer k. Similarly, f2(x)= g2(x)+ hn.

    Then f1(x)+ f2(x)= g1(x)+ kn+ g2(x)+ hn= (g1(x)+ g2(x))+ (k+ h)n.

    f1(x)f2(x)= (g1(x)+ kn)(g2(x)+ hn)= g1(x)g2(x)+ hng1(x)+ kng2(x)+ hkn^2= g1(x)g2(x)+ (hg1(x)+ kg2(x)+ khn)n.
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