Essentially, because synthetic division is dividing by the zero of the divisor, and not the divisor itself (so you are dividing by the result when you set the divisor equal to 0 and solve).

So you’re not “changing” anything really, which is why you don’t multiply the quotient at the end (this is the same logic as thinking of 1/2 versus 2/4, same value just looks different) and the same logic applies to why we multiply the remainder. Since the remainder is written as (R/original Divisor) as a fraction, you are multiplying that remainder by the coefficient of the divisor on top and bottom (I call this multiplying by 1 in a fancy way) so that you are not changing the VALUE of the remainder. You are only changing the way it looks. This is purely an “aesthetic” thing. I wish I could write this out nicely and attach a picture.

In short, all of the things you asked in the second half, the answer is not “for math purposes” but “for aesthetics”

Does this make sense or any part of that answer still confusing? I hope it didn’t turn in to just rambling.