1. Domains, "real zeros"

I realize this is not exactly Calculus, though it is in a college pre-calc class, so I believe it is probably beyond the High School level.

I'm having trouble understanding why some things need to be simplified to find roots, and some things do not.

For example, when finding the possible roots of a function, ie,
(2x-1)(x-2)
(x+2)(x-2)

My book tells me that to find the roots and asymptotes, I have to solve for what makes the denominator and nominator zero.
The book tells me that in this situation, the domain of x does not equal -2 or 2.

Similarly the roots should be 2 and 1/2, according to the book.

When I asked my teacher, she replied:
The domain restrictions are taken BEFORE we reduce the rational function to the lowest terms.

The vertical asymptote is found AFTER we reduce the rational function to the lowest terms.
The thing is, I can memorize that just fine, but I want to know WHY. Why do some things need to be simplified first, and not others?

Aren't they the same exact thing, whether simplified or not? Why should the answer change depending on that?

2. I am quite sure that I disagree with one of the text or your instructor.
Not sure which, because I find you post a bit hard to follow.

Let me explain. I argue that there is only one root $x = \frac{1}{2}$.
The domain comes first. Therefore $x = 2$ cannot be a root because 2 is not in the domain.