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Factorizing a polynomial over a ring

  1. May 31, 2012 #1
    1. The problem statement, all variables and given/known data
    Factorize [itex]x^2 + x + 8[/itex] in [itex]\mathbb{Z}_{10}[x][/itex] in two different ways


    2. Relevant equations



    3. The attempt at a solution
    I can see that x = 8 = -2 and x = 1 = -9 are roots of the polynomial, so one factorization is (x + 2)(x + 9).

    Is there a systematic way to find all the factorizations?
     
  2. jcsd
  3. May 31, 2012 #2

    HallsofIvy

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    I just looked at the various ways to get a product of 8 mod 10, the look at the sum of those products. I immediately see that 2*9= 18= 8 mod 10 and 2+ 9= 11= 1 mod 10 and that 4*7= 28= 8 mod 10 and that 4+ 7= 11= 1 mod 10.
     
  4. May 31, 2012 #3

    SammyS

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    It looks like the "Master Product rule"form introductory algebra --- only here you use modular arithmetic.

    You are looking for a pair of numbers whose sum is 1 (mod 10) and whose product is 8 (mod 10).
     
  5. May 31, 2012 #4
    So, it's essentially just trial and error?
     
  6. May 31, 2012 #5

    SammyS

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    Yes, but it's somewhat systematic trial & error.
     
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