By “complex zeros”, do you mean both complex and real zeros, or just complex non-real zeros?
1)Let be a monic polynomial* with integer coefficients with . Prove that if the sum of all coefficients and the product of all the complex zeros (counting multiplicity) are both odd then the polynomial has not integer zeros.
*)Leading term is 1.
Proof by contradiction: Assume is a polynomial with integer roots.
First note that is the product of all the roots and is the sum of coefficients.
Let , where are all the roots to .
So and since is odd, this implies each is odd too.
But and since is odd, this implies that each is odd which implies each is even.
Hence a contradiction and therefore there are no integer solutions.