Monic Greatest Common Divisor

1. Jan 8, 2015

auru

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. Jan 8, 2015

Staff: Mentor

(6x2 + 6) = 6(x2 + 1)?
Your factorization of b has an error.

3. Jan 8, 2015

auru

I have fixed it. I'm still unsure how to find the monic greatest common divisor.

4. Jan 8, 2015

haruspex

To minimise your embarrassment, I feel it is best to let you think a bit more about that.

5. Jan 8, 2015

auru

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.