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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.)

 Aug 3, 2019
 #1
avatar+105689 
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Stop being so lazy.

Edit it and get rid of the meaninglless dollar signs.

 Aug 3, 2019
 #2
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The dollar signs are there because it is how people write in Latex code.

 Aug 6, 2019
edited by Guest  Aug 6, 2019
edited by Guest  Aug 6, 2019
edited by Guest  Aug 8, 2019
edited by Guest  Aug 8, 2019
edited by Guest  Aug 15, 2019
edited by Guest  Aug 15, 2019
edited by Guest  Aug 15, 2019
 #3
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Wouldn't the final quadratic be \(\frac{a}{b}(x^2+3x-10)\) where the entire quadratic is multiplied by a/b, instead of just x^2?

Guest Aug 8, 2019

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