Calculating a value in $\mathbb{Z^3}[x]$

Question

Let $\mathbb{Z^3}= \mathbb{Z} \times \mathbb{Z} \ \times \mathbb{Z}$.
Let $f(X) = X^2 + X +1 \in\mathbb{Z^3}[x]$; $1=(1,1,1)\in \mathbb Z^3$.
Let $x = (1, 0, 2) \in \mathbb{Z^3}$.
I am to find the value of $f(x)$. My answer is $(3,1,7)$. Is it correct?

Answer 1

Algebraic operations in a product of rings like $\mathbb Z \times \mathbb Z \times \mathbb Z$ are defined componentwise. So, if $x = (1,0,2)$, then, indeed,

$$x^2+x+1 = (1^2,0^2,2^2) + (1,0,2) + (1,1,1) = (3,1,7)$$

That is, your answer is correct.