Question

Is$\mathbf{2}\mathit{x}\mathbf{+}\mathbf{2}$ irreducible in$\mathrm{\mathbb{Z}}\left[x\right]$ ? Why not?

Short answer

No, $2x+2$is not irreducible in$\mathrm{\mathbb{Z}}\left[x\right]$.

Step-by-step solution

Prove that 2x+2 is reducible in ℤ[x]

Clearly, $2x+2$can be written as $2\left(x+1\right)$.

That is $2x+2=2\left(x+1\right)$.

Here, neither 2 nor $x+1$is a unit.

Hence,$2x+2$ is reducible in$\mathrm{\mathbb{Z}}\left[x\right]$ .

Conclusion

Thus, it can be concluded that $2x+2$is reducible in $\mathrm{\mathbb{Z}}\left[x\right]$.

Hence the statement is false.