help interpreting an abstract algebra test question This is a take-home test problem, and I don't want help solving it, just understanding what it's asking.  I've asked my prof a couple times, but she's either unwilling or unable to give me a straight answer, and I don't understand why.  The problem:


  Suppose that $f(x)=x^n+a_{n-1} x^{n-1}+\cdots+a_0\in \mathbb{Z}[x]$. If $r$ is rational and $x-r$ divides $f(x)$, prove that $r$ is an integer. 


My question:  It's my understanding that when you say "$t$ divides $s$" you need to specify what domain you're in.  So they say that "$x-r$ divides $f(x)$", I'm confused about the domain, since $f(x)$ being an element of $\mathbb{Z}[x]$ is implying polynomials with integer coefficients, while $r$ being rational doesn't seem to fit with that.  Are they saying that $f(x) = (x-r)$ (another element of $\mathbb{Z}[x]$), or is it $f(x) = (x-r)$ (an element of $\mathbb{Q}[x]$)?
You are told that $f(x)$ is in $\mathbb{Z}[x]$, and that $f(x)$ is $x-r$ times an element of $\mathbb{Q}[x]$.  You are to deduce that $r$ must be an integer. The fact that $f(x)$ is monic (the lead coefficient is $1$) is crucial.
In a general Euclidean domain, given x,y in $R$, there are $q, r$ in $R$ with: $$a=bq+r $$ .

IF $r=0$ , then we say $b$ divides $a$. From Wiki's Euclidean Domain. $\mathbb Z[x]$ is a Euclidean domain. Notice that $\mathbb Z$ is embedded in $\mathbb Z[x]$ as the constant polynomials, so that integers are actually part of $\mathbb Z[x]$. IF $r$ is Rational; $r=p/q$ , you can always multiply through by $q$ to get an integer.