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Odd and Integer Zeros (Posted on 2011-04-13) Difficulty: 1 of 5
P(x) is a polynomial with integer coefficients such that each of P(0) and P(1) is odd.

Prove that P has no integer zeros.

No Solution Yet Submitted by K Sengupta    
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even odds (spoiler) | Comment 3 of 4 |

1-Let Ak*X^k = general term of P.

2-So A0 = odd to satisfy P(0) = 0.

3-The sum of the remaining Ak must be even to satisfy P(1) = odd, which requires that the number of odd-valued Ak is even.

4-If x is even all terms except A0 are even and their sum is odd, ie, not zero.

5-If x is odd, then Ak*X^k is odd only when Ak is odd. But there is an even number of occasions when Ak is odd, by 3. So that sum is even, and the sum where Ak is even is even, and A0 is odd, so the total sum is odd and therefore not zero.

  Posted by xdog on 2011-04-13 18:14:00
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