Let $p(x) = 1+a_1x+a_2x^2+\cdots+a_nx^n$ be a polynomial where $a_1,\ldots,a_n$ are integers, and $a_1 + \cdots + a_n$ is even. Prove that there is no integer x such that $p(x) =0$.
I have started this by trying to examine this case when it is only $1+a_1x$ but since $a_1$ cannot be $1$ (it has to add evenly), this will not work without the x value being a fraction.
When you try to examine the next value of $1+a_1x+a_2x^2$ you get the same problem.
How would I start this problem?