Question: I need to factor the polynomial $x^5+3x^4+x^3+x^2+3$ into the product of the irreducible ones in the ring $\mathbb{Z}\_5[x]$.  

The problem is I don't see any whole roots (I tried every possible divisor of 3) in the given polynomial. Does it mean that it is already irreducible?

Answer: 
Actually, $2$, $3$, and $4$ are roots of that polynomial in $\mathbb Z\_5[x]$. So, your polynomial can be written as$$(x^2+ax+b)(x-2)(x-3)(x-4).$$Since $b\times(-4)=3$, $b=3$. It is not hard now to see that $a=2$.
Hint: If it factors nontrivially, one factor will have degree 1 (then it has a root in ${\Bbb Z}\_5$) or one factor will have degree 2, by the degree formula. The irreducible polynomials of degree 2 in ${\Bbb Z}\_5[x]$ can be enumerated.