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.
1-Let A
k*X^k = general term of P.
2-So A
0 = 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 A
k is even.
4-If x is even all terms except A
0 are even and their sum is odd, ie, not zero.
5-If x is odd, then A
k*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 |
even odds (spoiler)
