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.)

 May 31, 2020

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.


For the original poster: We realize that homework can be challenging. If you wish to receive help from the staff or other students, we encourage you to use the resources that the online classes provide. Please don't ask or search online for homework help. We understand that it's common in today's information age to look for resources online, and in some contexts, that's a great thing! However, it's against our Honor Code. You can ask for help on the message boards, and you can learn from the official solution after you submit your answer.

 Jul 10, 2020

15 Online Users