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 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 a counterexample.

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.)

Impasta Feb 8, 2020