Find the greatest common divisor of the polynomials $ \ f(x)=x^{3}+2 \ \ and \ \ g(x)=x- 1 \ $ over the field $ \ \ \mathbb{Z}_{3}$

Question

Find the greatest common divisor of the polynomials $ \ f(x)=x^{3}+2 \ \ and \ \ g(x)=x- 1 \ $ over the field $ \ \ \mathbb{Z}_{3}$. $$ $$ We have $ \ f(x)=x^{3}+2 \ \ and \ \ g(x)=x-1=x+2 \ \ \ mod \ (3) \ $ . Now $ \ x^{3}+2=(x+2)(x^{2}-2x+4) \ =(x+2)(x^{2}+x+1)$. (since -2=1 and 4=1 in $ \ \mathbb{Z}_{3} $ ). So according to me the G.C.D =(x+2) . Am I right ? Any help please

Answer 1

It's easier if you write $x^3+2=x^3-1=(x-1)(x^2+x+1)$, so it becomes apparent that $x-1$ is a divisor, hence the greatest common divisor.

Anyway, your computation is correct, using $x^3+2=x^3+8=(x+2)(x^2-2x+4)$.