Somebody can help me with this polynomial problem? I tried something but i am not really sure if i can finish with that. Thank you!
Let $P$ a polynomial with integer coefficients for which exists $2$ integer numbers, one odd, one even such that values of the polynomial in these values are odd. Show that the polynomial cannot have integer zeros.
I tried to use the contradiction and purpose we have zeros integer numbers. But i don't know how to elaborate that.
I used $P=a_nX^n+\cdots+a_0$ and $a,b\in Z,a=2k,b=2k+1,k\in Z$. Then we have $P(a)=2k+1$ and $P(b)=2k+1$. If we say $P$ has integer zeros let $a,b$ to be zeros. But actually $P(a)$ and $P(b)$ are even, contradiction?