Math Is Fun Forum

Thank you! I did it the same way!

So are you saying that g(x) doesnt have to be divisible by f(x)?  Should we take multiplicity of roots into account?  Also, for part 2, I'm thinking that the degree of f(x) has to be 3 and the degree of g(x) has to be 5, using the fundamental theorem of algebra, so there won't be multiplicity of roots.

I'm not sure how to do part 2 though. Do we need the fundamental theorem of algebra?

The first problem or the second one?

Yes, I got 6x^5-15x^4+10x^3
Thanks!

Thank you!  I also have two other problems I need help on:
1) Find a polynomial f(x) of degree 5 such that both of these properties hold: f(x)-1 is divisible by (x-1)^3 and f(x) is divisible by x^3
2)
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)?

Thank you in advance!!!

yup, I understand that!

I proved that a>0, but I'm having trouble proving that a<-4

Thank you! Its right now.

I think the only roots are

I still don't understand the 3rd question with Vieta's formula either