1. ## Greatest common divisor

$\displaystyle f(X)=X^4+X^3+2X+2$ and $\displaystyle g(X)=X^3+X^2+X+1$.
$\displaystyle f(X)$ and $\displaystyle g(X)$ are both polynomials over $\displaystyle \mathbb{Z}_5$. Compute their greatest common divisor.

My notes aren't helping me here; never done an example like this before, so a detailed explanation of the method would be greatly appreciated

2. Originally Posted by chella182
$\displaystyle f(X)=X^4+X^3+2X+2$ and $\displaystyle g(X)=X^3+X^2+X+1$.
$\displaystyle f(X)$ and $\displaystyle g(X)$ are both polynomials over $\displaystyle \mathbb{Z}_5$. Compute their greatest common divisor.

My notes aren't helping me here; never done an example like this before, so a detailed explanation of the method would be greatly appreciated
I'd start by factorising the two functions. To do this bracket the first 2 terms and the last two terms, and proceed to factorise.

3. Originally Posted by Debsta
I'd start by factorising the two functions. To do this bracket the first 2 terms and the last two terms, and proceed to factorise.
I've never heard of that method of factorising... care to explain it?

4. Originally Posted by chella182
I've never heard of that method of factorising... care to explain it?
Doesn't always work but is a method to try when you have 4 terms (it does help in both your cases)
eg
$\displaystyle x^4+x^3+2x+2 = (x^4+x^3) + (2x+2) = x^3(x+1) + 2(x+1) = (x+1) (x^3+2)$

5. Originally Posted by Debsta
Doesn't always work but is a method to try when you have 4 terms (it does help in both your cases)
eg
$\displaystyle x^4+x^3+2x+2 = (x^4+x^3) + (2x+2) = x^3(x+1) + 2(x+1) = (x+1) (x^3+2)$
Wow, thanks! Not sure if my lecturer will appreciate the method if it doesn't always work (he's like that), but it's worth a shot anyway!