What Is the Monic Greatest Common Divisor of Two Given Polynomials?

AI Thread Summary
To find the monic greatest common divisor (GCD) of the polynomials a = 6x^6 + 12x^5 - 6x^4 - 12x + 12 and b = 3x^4 - 3, the Euclidean Algorithm is applied. The common divisor identified is 6x^2 + 6, which is not monic due to its leading coefficient of 6. To convert this to a monic polynomial, it can be divided by its leading coefficient, resulting in x^2 + 1. The discussion emphasizes understanding the process of finding the monic GCD through proper factorization and division by constants. The importance of seeking help when concepts are unclear is also highlighted.
auru
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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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(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.
 
auru said:
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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haruspex said:
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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