>>11876125Sorry if it is already answered, but I can't read an entire thread about high-school maths.
First of all, all questions are troll questions because they try to troll you into answering incorrectly. I assume that you understand that the first three choices are not necessarily true (you can come up with counter examples for each of them). Now, you know that for any polynomials p(x) and q(x) there exist polynomials such that p(x)=s(x)q(x)+r(x). If we take p(x) and (x-3) we get that there exist polynomials such that:
p(x)=(x-3)s(x)+r(x)
Now we evaluate the expression for x=3
p(3)=(3-3)s(3)+r(3), since p(3)=-2 and 3-3=0
-2=r(3)
Now let's talk about the reminder in polynomial division. The reminders rank is always less than the rank of q or else we could also pull another factor out of r which would be a multiple of q and add it to s (reducing the rank of r in the process). q here is (x-3). This means that the rank of q is 1 and according to the previous statement, it is necessary that the rank of r is 0. A polynomial with a rank of 0 is just a constant and its value we have determined already. Therefore r(x)=-2 for all x.
