Let $f(x) = a_kx^k + a_{k-1}x^{k-1} + \cdots + a_0$ be a polynomial with integer coefficients where $a_k \neq 0$. Prove that if $a_0 \geq 2$, $f(n)$ is not prime for some integer $z$.
I know I will eventually have to show that a polynomial of degree $k-1$ with real coefficients has at most $k-1$ real roots, but I have not had much luck thus far with induction.
Hint $\ $ If not $\,f(n)\,$ is prime for all $\,n\,$ so $\,a_0 = f(0)= p\,$ is prime. $\,f(pn)\,$ is prime for all $\,n,\,$ and $\,p\mid f(pn),\,$ so $\,f(pn) = p.\,$ So $\,f(px)-p\,$ has infinitely many roots so $\,f(px) = p,\,$ so $\,f(x)=p.$
Hint: Plug in $x = Na_0$ into the polynomial where $N$ is a large integer and see what you get. Note the leading term will dominate.
