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Homework Help: Monic Greatest Common Divisor

  1. Jan 8, 2015 #1
    1. The problem statement, all variables and given/known data

    Find the monic greatest common divisor of two polynomials a = 6x6 + 12x5 - 6x4 -12x +12 and b = 3x4 - 3.

    2. Relevant equations

    The Euclidean Algorithm.

    3. The attempt at a solution

    Applying the Euclidean Algorithm, I have

    a = 6x6 + 12x5 - 6x4 -12x +12 = (3x4 - 3)(2x2 + 4x -2) + (6x2 + 6)

    b = 3x4 - 3 = (6x2 + 6)(##\frac {1}{2}##x2 - ##\frac {1}{2}##)

    Now a monic polynomial has a leading coefficient of degree 1. Here, we have a common divisor of 6x2 + 6 which is not monic. How would I go about finding the monic greatest common divisor.
     
    Last edited: Jan 8, 2015
  2. jcsd
  3. Jan 8, 2015 #2

    mfb

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    2017 Award

    Staff: Mentor

    (6x2 + 6) = 6(x2 + 1)?
    Your factorization of b has an error.
     
  4. Jan 8, 2015 #3
    I have fixed it. I'm still unsure how to find the monic greatest common divisor.
     
  5. Jan 8, 2015 #4

    haruspex

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    Science Advisor
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    Gold Member

    To minimise your embarrassment, I feel it is best to let you think a bit more about that.
     
  6. Jan 8, 2015 #5
    Why would I be embarrassed when I don't initially understand something? It may not be initially obvious to me, hence why I have asked for help.

    As it turns out, in the general case I am able to divide the common divisor by a constant to attain the monic.
     
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