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# Polynomial Question

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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.)

-------Thanks!

Mar 12, 2020

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Part 1:  Yes.

f(x)  =  a(x +3)(x - 4)(x - 8)             g(x)  =  b(x + 5)(x + 3)(x - 2)(x - 4)(x - 8)

g(x) / (x)  =  [ b(x + 5)(x + 3)(x - 2)(x - 4)(x - 8) ] / [ a(x + 3)(x - 4)(x - 8) ]  =  (a/b)(x + 5)(x - 2)

Part 2: Each of the x-term factors of f(x) must be x-term factors of g(x).

Mar 12, 2020