Checking irreducibility of $3x+6$ in $\mathbb Q[x]$ and $\mathbb Z[x]$

Question

Any hint How should I check whether $3x+6$ is irreducible in :
1.) $\mathbb Q[x]$
2.) $\mathbb Z[x]$

Answer 1

Hint: If $$ 3x+6=p(x)q(x) $$

is a factoring then clearly $$ \deg(p)+\deg(q)=\deg(3x+6)=1 $$

since $\mathbb{Z},\mathbb{Q}$ does not have zero divisors.

This imply that one of $p$ and $q$ have degree $0$ while the other have degree $1$.

Without loss of generality $\deg(p)=0$ and $\deg(q)=1$.

Now - can you see if such factoring exists over $\mathbb{Z}$ ? over $\mathbb{Q}$ ?

Answer 2

Can we write $3x + 6 = 3(x + 2)$ in either ring? Is $3$ or $x+2$ a unit in either ring?

Answer 3

Just check the definitions.

1) You can write $P(x) = 3 \times (x + 2)$, and $3$ is not a unit in $\mathbb{Z}$ (neither is $x+2)$).

2) Polynomials of degree 1 are irreducible in $\mathbb{k}[X]$, where $\mathbb{k}$ is any field (I suggest you to check why).