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Math Help - Find zeros

  1. #1
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    Find zeros

    Show that x^2 + 3x + 2 has four zeros in Z_{6}.

    x^2 + 3x + 2 = (x+2)(x+1), by the factor theorem, -1 and -2 are zeros of the poly.

    However, -2 = 4 and -1 = 5 in Z_{6}. But what are the other two?

    Thanks.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by tttcomrader View Post
    Show that x^2 + 3x + 2 has four zeros in Z_{6}.

    x^2 + 3x + 2 = (x+2)(x+1), by the factor theorem, -1 and -2 are zeros of the poly.

    However, -2 = 4 and -1 = 5 in Z_{6}. But what are the other two?

    Thanks.

    since Z_6 is a finite set, if you use tables, you can find it easily..

    ----0---1---2---3---4---5
    f(x) 2---0---0---2---0---0
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  3. #3
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    I just want to mention the interesting fact that even though the polynomial is of degree 2 it has 4 zeros. That might seem impossible but there is a reason for it. That theorem (about polynomials having at most the number of zeros of its degree) is only for fields (or when the division algorithm works). But Z_6 is not a field so it is not supprising why it fails.
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  4. #4
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    I see, so really, I just have to plug the numbers in.

    The zeros are 1 , 2 , 4 , 5.

    Thanks.
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  5. #5
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    Quote Originally Posted by tttcomrader View Post
    I see, so really, I just have to plug the numbers in.

    The zeros are 1 , 2 , 4 , 5.

    Thanks.
    Yes, this is a finite ring so it is possible to check each number. When workin in R, say, there are infinitely many such possibilities so in that case you need to use some analysis on the polynomial without actually guessing for its zeros.
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  6. #6
    MHF Contributor kalagota's Avatar
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    in fact, it is sufficient to know that the set should be an integral domain..
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