Which of the following polynomials in Z_3 [x] is irreducible? (a) p (x) = x ^ 3 + x + 1 (b) p (x) = x ^ 4 + 1 (c) Factorize the polynomials that are not irreducible.

Step 1
Useful result:
If f(x) in ${\displaystyle F\left[x\right]}$ be a polynomial over a field F of degree two or three.
Then f(x) is irreducible if and only if it has no zeros.
(a) Given ${\displaystyle p\left(x\right)=x^3+x+1}$
${\displaystyle Z_3=\{0,1,2\}}$
${\displaystyle p\left(0\right)=1,p\left(1\right)=3\text{mod}3=0\text{mod}3=0,p\left(2\right)=11\text{mod}3=2\text{mod}3=2}$
Here p(x) has zero over ${\displaystyle Z_3\left[x\right]}$.
Therefore, given p(x) is reducible over ${\displaystyle Z_3\left[x\right]}$.
Step 2
(b) Given ${\displaystyle p\left(x\right)=x^4+1}$
This is a reducible polynomial over ${\displaystyle Z_3\left[x\right]}$.
(c) Factorization of given polynomials is as follows:
For (a), ${\displaystyle p\left(x\right)=x^3+x+1=(x+2)(x^2+x+2)\text{mod}3}$
For (b), ${\displaystyle p\left(x\right)=x^4+1=(x^2+x+2)(x^2+2x+2)\text{mod}3}$