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

[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.

 

[mfb]

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

 

[auru]

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

 

[haruspex]

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.

 

[auru]

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.