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).

What I did so far:

I realized the factors are f(x) = a(x+3)(x-4)(x-8), and for g(x) = b(x+3)(x-4)(x-8)(x+5)(x-2),

Dividing, I got \(\frac{a(x^{2}+3x-10)}{b}\)

I don't know where to continue from here, to prove or disprove that \(\frac{a(x^{2}+3x-10)}{b}\) has no remainder.

Thanks!

shad0w May 15, 2020