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.
1Let Ak*X^k = general term of P.
2So A0 = odd to satisfy P(0) = 0.
3The sum of the remaining Ak must be even to satisfy P(1) = odd, which requires that the number of oddvalued Ak is even.
4If x is even all terms except A0 are even and their sum is odd, ie, not zero.
5If 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 20110413 18:14:00 