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# Started working, do not know where to continue.

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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!

May 15, 2020