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# Part 1: Let and be polynomials. Suppose for exactly three values of : namely, and . Suppose

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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  is not divisible by .

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.)

Thank you guys so much! Feb 5, 2020
edited by DragonLord  Feb 5, 2020

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Try watching and learning from this clip:

Feb 5, 2020
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Thank you, Guest!

Feb 5, 2020
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I think I get how to do this problem now

Feb 5, 2020