An odd polynomial

Let p(x) be a polynomial with integer coefficients.  Show that, if the constant term is odd, and the sum of all the coefficients is odd, then p has no integer roots.  (That is, if p(x) = a₀ + a₁x + ... + a_nxⁿ, a₀ is odd, and a₀ + a₁ + ... + a_n is odd, then there is no integer k such that p(k) = 0.)