+133 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 the answer to Part 1 was "yes", then stating the generalization should be straightforward. If the answer to Part 1 was "no", then try to salvage the idea by imposing extra conditions as needed.)
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