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

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- February 24th 2008, 01:56 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.

- February 28th 2008, 05:52 AM #2

- February 28th 2008, 07:21 AM #3

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- February 28th 2008, 03:57 PM #4

- February 28th 2008, 08:28 PM #5

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- October 20th 2008, 03:45 PM #6

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- January 22nd 2009, 09:46 PM #7

- January 23rd 2009, 12:01 AM #8

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- February 9th 2009, 08:02 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.