help with remainder theorem problem,pls?

Question

pls-help with the remainder theorem im finding difficult to solve-

Given that (2-3x) is a factor of a0 + a1x +a2x² +a3x³, show that a0 is a multiple of 2 while a3 is a multiple of 3.

IMP side note-the numbers behind the a is always subscripted.

Answer 1

Since (2-3x)) is a factor of a0 + a1x +a2x² +a3x³, it follows r = 2/3 is a root of the given polynomial. Then r is a rational root to it's lowest terms 2/3. Assuming the coefficientes are integers, it follows from the rational roots theorem that 2 divdes a0 and 3 divides a3, which is exactly the desired conclusion.