Math Is Fun Forum

Oran2009

Member

Registered: 2016-06-13

Posts: 21

Polynomial Proof

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 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.)

How to do this problem?

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Polynomial Proof

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}