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02-24-2008, 02:56 PM
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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.
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Last edited by ThePerfectHacker; 02-28-2008 at 09:29 PM.
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02-28-2008, 06:52 AM
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By “complex zeros”, do you mean both complex and real zeros, or just complex non-real zeros? | | The following users thank JaneBennet for this useful post: | | |
02-28-2008, 08:21 AM
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Quote:
Originally Posted by JaneBennet  By “complex zeros”, do you mean both complex and real zeros, or just complex non-real zeros? | I mean all zeros. For example, take  then the zeros are  . Does saying "zeros in  " make it better?
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02-28-2008, 04:57 PM
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I must not understand the question. My first thought is:  .
The sum of the zeros is 9, the product of the zeros is 15, and all three zeros are integers. | | The following users thank Henderson for this useful post: | | |
02-28-2008, 09:28 PM
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Thank you. I was inventing this problem, I should have been more careful. I fixed it above. Tell me if it is okay now.
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10-20-2008, 04:45 PM
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By considering the equation  and the fact that f(1) and f(0) are both odd implies the truth of your statement | | Thread Tools | | | | Display Modes |  Linear Mode | 
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