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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).

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)?

Apr 24, 2019

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Next time make sure you delete all the meaningless $signs. So that your question presents properly. $$\text{Let F(x) and G(x) be polynomials such that }\\ f(x)=F(x)(x+3)(x-4)(x-8)\\ g(x)=G(x)(x+3)(x-4)(x-8)(x+5)(x-2)\\ \frac{g(x)}{f(x)}=\frac{G(x)(x+3)(x-4)(x-8)(x+5)(x-2)}{F(x)(x+3)(x-4)(x-8)}\\ \frac{g(x)}{f(x)}=\frac{G(x)(x+5)(x-2)}{F(x)}\\$$ g(x) is only divisable by f(x) if F(x) is a factor of G(x)(x+5)(x-2) for instance in the following example g(x) is not divisable by f(x) $$g(x)=7(x+5)(x-2)(x+3)(x-4)(x-8)\\ f(x)=(x^8+x^5-4x)(x+3)(x-4)(x-8)\\$$ . Apr 25, 2019 ### 1+0 Answers #1 +101046 +2 Best Answer Next time make sure you delete all the meaningless$ signs.  So that your question presents properly.

$$\text{Let F(x) and G(x) be polynomials such that }\\ f(x)=F(x)(x+3)(x-4)(x-8)\\ g(x)=G(x)(x+3)(x-4)(x-8)(x+5)(x-2)\\ \frac{g(x)}{f(x)}=\frac{G(x)(x+3)(x-4)(x-8)(x+5)(x-2)}{F(x)(x+3)(x-4)(x-8)}\\ \frac{g(x)}{f(x)}=\frac{G(x)(x+5)(x-2)}{F(x)}\\$$

g(x) is only divisable by f(x) if F(x) is a factor of G(x)(x+5)(x-2)

for instance in the following example g(x) is not divisable by f(x)

$$g(x)=7(x+5)(x-2)(x+3)(x-4)(x-8)\\ f(x)=(x^8+x^5-4x)(x+3)(x-4)(x-8)\\$$

Melody Apr 25, 2019