Part 1: Let $f(x)$ and $g(x)$ be polynomials. Suppose $f(x)=0$ for exactly three values of $x$: namely, $x=-3,4,$ and $8$. Suppose $g(x)=0$ for exactly five values of $x$: namely, $x=-5,-3,2,4,$ and $8$. Is it necessarily true that $g(x)$ is divisible by $f(x)$? If so, carefully explain why. If not, give an example where $g(x)$ is not divisible by $f(x)$.
Part 2: Generalize: for arbitrary polynomials $f(x)$ and $g(x)$, what do we need to know about the zeroes (including complex zeroes) of $f(x)$ and $g(x)$ to infer that $g(x)$ is divisible by $f(x)$? (If your answer to Part 1 was "yes", then stating the generalization should be straightforward. If your answer to Part 1 was "no", then try to salvage the idea by imposing extra conditions as needed. Either way, prove your generalization.)
Many Thanks!
