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It is it's "root", it's "zero", and to answer your question, yes, it is not the same
They are not. Factoring the expressions allows the extraction of potential similar components in the numerator and denominator, meaning that they can be simplified. I understand what I am talking about because this is the last thing I learned in my algebra 2 class before break.
What does QED mean?
well im taking pre-calculus, so there is something i have learned that you may not have: even if the numerator and the denominator share a factor, they sort of cancel out, but not really, as seen above, with (x-1)²/(x-1) being different
quod erat demonstrandum, meaning "that which was to be demonstrated", i.e. “im done proving the thing i was trying to”
all good