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Math Help - polynomial roots

  1. #1
    Senior Member sfspitfire23's Avatar
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    polynomial roots

    Two questions that I know the answers to

    1. How many polynomials of deg 3 are there in \mathbb{Z}_5[X]? The answer is 500.

    2. Find the roots of (x-4)(x-5) in \mathbb{Z}_6. Answer= 4,5,1,2

    I know for question 2 the 1,2 comes from the fact that its in Z5, but I cant seem to apply that fact to get 1,2 myself. For question 1, what is the methodology of finding 500? I know that each coefficient can only be 0-4...
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    For 1), notice that we only have 4 choices for the coefficient of x^3 because it has to be nonzero. We have 5 choices for all the other coefficients, hence we get 4 \times 5^3 = 500.

    For question 2), just try all the elements of \mathbb{Z}_6!
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