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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 bt 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 arbritary 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.)

 Jun 3, 2020
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For the community: Please don't submit solutions to this problem. This is a homework problem for an online course that does not allow students to search for answers to specific problems outside of the school.

 

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 Jul 10, 2020

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