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.

P(0) is equal to the constant of the polynomial. As it is given that P(0) is odd, the constant will be odd.

P(1) is equal to the sum of the integer coefficients plus the constant. Following the laws of addition and subtraction, since the constant is given as odd and P(1) is given as odd, the sum of the integer coefficients must be even.

As the sum of the integer coefficients is even, all the terms with integer coefficients must be even or there is an even number of terms that are odd.

As zero is an even number and the constant is odd, following the laws of addition and subtraction, the sum of the terms must be odd. In order for the sum of the terms to be odd, following the laws of addition and subtraction, an odd number of terms must be odd.

Ergo, since the number of terms that are odd can not be both odd and even, P can not have any integer zeroes.

Posted by Dej Mar on 2011-04-13 23:24:00