How can we show that as groups $\mathbb Z[x] \times \mathbb Z \cong \mathbb Z[x] \times \mathbb Z \times \mathbb Z$?

Question

How can we show that as groups $\mathbb Z[x] \times \mathbb Z \cong \mathbb Z[x] \times \mathbb Z \times \mathbb Z$?

I tried coming up with an isomorphism but I am stuck. One map I tried was $(p(x), n) \mapsto (p(x), n, \deg(p(x)))$, but this is not a homomorphism.

Answer 1

Hint: Use the fact that$$\begin{array}{ccc}\mathbb{Z}[x]\times\mathbb Z&\longrightarrow&\mathbb{Z}[x]\\\bigl(p(x),k\bigr)&\mapsto&k+xp(x)\end{array}$$is a group isomorphism.

Answer 2

Hint Show that $$a_nx^n+...+a_1x+a_0 \to (a_nx^{n-1}+a_{n-1}x^{n-2}+...+a_1 ,a_0)$$ is an isomorphism between $ \mathbb Z[x]$ and $\mathbb Z[x] \times \mathbb Z$.