In the textbook that I'm using the division algorithm over $F[x]$ is stated as the following:
Let $F$ be a field and let $f(x)$, $g(x) \in F[x]$ with $g(x) \neq 0$. Then there exist unique polynomials $q(x)$ and $r(x)$ in $F[x]$ such that $f(x) = g(x)q(x) + r(x)$ and either $r(x) = 0$ or $deg(r(x)) < deg(g(x))$.
If we consider $\mathbb{Q}[x]$, and $f(x) = 7$, $g(x) = 2$ then $q(x) = 3$ and $r(x) = 1$ but in this case $r(x) \neq 0$ and $deg(r(x))\nless deg(g(x))$ since both of them are $0$.
Edit: I had $\mathbb{Z}[x]$ before but I was reminded $\mathbb{Z}$ isn't a field.