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Monic Greatest Common Divisor

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Homework Statement



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

Homework Equations



The Euclidean Algorithm.

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.
 
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Answers and Replies

  • #2
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(6x2 + 6) = 6(x2 + 1)?
Your factorization of b has an error.
 
  • #3
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I have fixed it. I'm still unsure how to find the monic greatest common divisor.
 
  • #4
haruspex
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6x2 + 6 which is not monic. How would I go about finding the monic greatest common divisor.
To minimise your embarrassment, I feel it is best to let you think a bit more about that.
 
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To minimise your embarrassment, I feel it is best to let you think a bit more about that.
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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