I am a graduate teaching assistant at a larger state university teaching a college algebra class. Today we begin our decent into polynomials and one of the facts we will soon get to is the connection between roots of a polynomial and linear factors. We do not teach polynomial long division (explicitly avoid it in fact), which is normally how I could convince students that $x - a$ is a factor of a polynomial $p(x)$ if and only if $p(a) = 0$.
My question is how can I convince my students of this fact without appealing to division? Keep in mind that these are largely college freshman who do not necessarily care for explanations. While they would take my word on something and memorize it if I told them it was necessary, I still like to convince them this is indeed mathematical fact and not something that has come out of thin air.