In this post on Art Of Problem Solving, it is claimed that the following group isomorphism holds:
$$(\mathbb{Z}[x], +) \times \mathbb{Z} \times \mathbb{Z} \cong (\mathbb{Z}[x], +) \times \mathbb{Z}$$
where $(\mathbb{Z}[x], +)$ is the additive subgroup of $\mathbb{Z}[x]$. Any ideas on how one could show this?
I tried for a while but I can't for the life of me find an injective homomorphism between the two. My best idea would be to go through a multiplicative group first (because constructing an isomorphism from $(\mathbb{Z}[x], +)$ to $(\mathbb{Q}^+, \cdot)$ is much easier, the latter being a multiplicative group) but I haven't gotten far with that.
Any hints?