Writing a polynomial as a product of irreducible elements

Question

Write $2+4x+6x^2+x^3$ as a product of irreducible elements considering an element in each of the following rings. In each of the following rings. $\mathbb{Z}[x], \mathbb{Q}[x], \mathbb{Z}_{3}[x]$

My thinking was first to factorize the polynomial. After factoring I could see how I could transform them into irreducibles which means give that a root that do not exist in this ring. But I am not sure how to start really.. S

Answer 1

This polynomial is irreducible by the Eisenstein's criterion for two first cases.

In the third case it's $x^3-2x-1=(x+1)(...)$

Answer 2

You already have an answer for the first two cases. In $\mathbb{Z}_3[x]$, your polynomial is just $2+x+x^3$, which has a root: $2$. Therefore, it is not irreducible.