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

• auru
In summary, the conversation is about finding the monic greatest common divisor of two polynomials using the Euclidean Algorithm. The attempt at a solution involves factorizing the polynomials and identifying a common divisor, but there is confusion about how to find the monic greatest common divisor. It is later clarified that the common divisor can be divided by a constant to attain the monic form.
auru

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.

Last edited:
(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.

mfb
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.

What is a Monic Greatest Common Divisor?

A Monic Greatest Common Divisor (GCD) is the highest common factor of two or more polynomials with the leading coefficient of 1. It is the largest polynomial that can divide evenly into the given polynomials.

How do you find the Monic Greatest Common Divisor?

The Monic GCD can be found by first factoring each polynomial into its irreducible factors. Then, the GCD is the product of the common irreducible factors with the smallest degree. If there are no common factors, then the GCD is 1.

What is the difference between a Monic GCD and a regular GCD?

The main difference is that a Monic GCD has a leading coefficient of 1, while a regular GCD can have any leading coefficient. This means that a Monic GCD is unique and can be used to simplify polynomial expressions without changing their overall value.

Why is the Monic GCD important in polynomial computations?

The Monic GCD is important because it allows us to reduce a polynomial expression into its simplest form. This can help in solving equations, finding common denominators, and simplifying complex expressions.

Can the Monic GCD be used for more than two polynomials?

Yes, the Monic GCD can be used for any number of polynomials. The same process of factoring and finding the common factors with the smallest degree can be applied to find the Monic GCD for multiple polynomials.

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