Basic Query about Polynomials over a field - Do their roots have to be in that field?

Basic Query about Polynomials over a field - Do their roots have to be in
that field?

This one is a rather basic query . I have only recently started studying
Fields and in particular finite fields. My question is - do the roots of a
polynomial over a given field have to be inside that particular field ?
For ex : - consider polynomial : -
$f(x) = x^2 + x + 1$ over $GF(2)$ then does something like $f(7)$ make
sense here ? Or are $f(1)$ and $f(0)$ the only ones which make sense ?
When defining a polynomial over a field it is mentioned that the
coefficients have to be in the field over which the polynomial is
defined.But I am confused if it also applies to the values which $x$ can
take in $f(x)$.I am studying these concepts in the study of rings so
usually we deal with operations between different polynomial.
According to me the answer is no because if that were the case $x - 7$
would be a factor of $f(x)$ and $x - 7$ is itself a polynomial which
doesn't quite make sense since the field is $GF(2)$.
One of my sources of confusion is the concept of extension field. Suppose
we consider the extension field $GF(2^4)$ now can we speak of roots other
than $0$ or $1$ for the same polynomial. (I.e can $x$ take up $16$ values
instead of $2$ in $f(x)$ )
Thanks.