Is the given polynomial irreducible:
(x^2)+x-2 in Z3, Z7?
which shows that the polynomial IS reducible in Z3[x].
(Okay, so the red coloring didn't work in the quote! You still get the idea.)
PS: That x = 1 produces implies that x - 1 is a factor of the polynomial in Z3[x]. By doing the long division you can show that . Where is the x + 2 factor in the above list? Well, this would imply a root of x = -2 = 1 (mod 3), so really x = 1 again. This says that: