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.

Results 1 to 9 of 9

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

- Joined
- Nov 2005
- From
- New York City
- Posts
- 10,616
- Thanks
- 10

## 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

- Joined
- Nov 2005
- From
- New York City
- Posts
- 10,616
- Thanks
- 10

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

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

- Joined
- Nov 2005
- From
- New York City
- Posts
- 10,616
- Thanks
- 10

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

- Joined
- Oct 2008
- From
- Guernsey
- Posts
- 69

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

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

- Joined
- Oct 2008
- From
- Guernsey
- Posts
- 69

- 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.