Question 1173277: 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.)
Answer by ikleyn(52800)   (Show Source):
You can put this solution on YOUR website! .
If you consider the ring of polynomials with real coefficients, it is NOT necessary
i.e., the first fact not necessary implies the second fact.
If you consider the ring of polynomial with complex number coefficients,
and if the mentioned roots of the polynomial f(x) all have multiplicities lesser than
the corresponding roots of g(x), then it IS necessary: the first fact implies the second.
But if the condition about multiplicities is not held, then the first fact does not necessary implies the second fact.
Example : g(x) = (x-(-5)) * (x-(-3)) * (x-2) * (x-4) * (x-8)
f(x) = (x-(-3))^2 * (x-2)^3 * (x-8)^9
Over the complex domain, the necessary additional information is about MULTIPLICITIES of the roots.
If multiplicities of all corresponding/relevant roots of g(x) are greater than or equal to those of the polynomial f(x),
then the divisibility g(x) = 0 ( mod f(x) ) takes place;
if not - the divisibility does not take place.
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