By “complex zeros”, do you mean both complex and real zeros, or just complex non-real zeros?

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- Feb 24th 2008, 12:56 PM #1

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## Problem 46

1)Let be a monic polynomial* with integer coefficients with . Prove that if the sum of all coefficients and the product of all the complex zeros (counting multiplicity) are both odd then the polynomial has not integer zeros.

*)Leading term is 1.

- Feb 28th 2008, 04:52 AM #2

- Feb 28th 2008, 06:21 AM #3

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- Feb 28th 2008, 02:57 PM #4

- Feb 28th 2008, 07:28 PM #5

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- Oct 20th 2008, 02:45 PM #6

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- Jan 22nd 2009, 08:46 PM #7

- Jan 22nd 2009, 11:01 PM #8

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- Feb 9th 2009, 07:02 PM #9
Proof by contradiction: Assume is a polynomial with integer roots.

First note that is the product of all the roots and is the sum of coefficients.

Let , where are all the roots to .

So and since is odd, this implies each is odd too.

But and since is odd, this implies that each is odd which implies each is even.

Hence a contradiction and therefore there are no integer solutions.