The factor theorem for polynomials is a topic that some students seem to stubbornly refuse to adopt into their mathematical toolkit at A Level. My best guess as to the reason for this, is that it is a mathematical idea where it can be quite hard to 'see what is going on' and so students stick to algebraic division, although this is typically a more time consuming technique with more scope for error (and indeed, it can be equally unclear as to 'what is going on'.)
Since the factor theorem is a necessary, 'must-be' maths fact, when students in my class were due to encounter it, I wanted them to see why it must be so. With this objective, I carefully constructed the following series of questions (perhaps you may want to work through them yourself before continuing.)
Part (a) gives some routine practice of expanding brackets, (b) and (c) apparently some substitution, and part (d)... What I hoped that some students would notice was that I'd given away the answers to (b), (c) and (d) in part (a), since the two cubics in the question are in fact the same! Not many students managed to do this, but the understanding of the idea came for most in the re-analysis of the whole question.
Since (2x - 3)(x + 3)(x - 1) and 2x³ + x² - 12x + 9 are equivalent expressions, ran the reasoning, substituting the same number into either of them will yield the same result. It is fairly easy to see from the factorised form that substituting -3 will give an answer of 0, whereas the expanded form does not reveal this so readily. This leads to the idea that the three roots are related to the three factors and thus, substituting and getting a result of zero indicates you have substituted a root.
I believe the careful sequencing of questions like this can provide this opportunity for students to see must-be maths facts for themselves, which leads to a stronger understanding of the underlying idea.
On reflection of this lesson, I think some students may have benefited from working through two or three questions with this structure before we analysed as a whole class. This would give more time for individuals to notice the connections and to develop their own awareness of the consequential results.