# Monic Greatest Common Divisor

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

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

haruspex
Homework Helper
Gold Member
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.

• mfb
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.