Application of Taylor's Theorem in Number Theory I'm working through Alan Baker's book A Concise Introduction to the Theory of Numbers, and there's an assertion in there that confuses me. Here's the quote:

It is easily seen that no polynomial $f(n)$ with integer coefficients can be prime for all $n$ in $\mathbb{N}$, or even for all sufficiently large $n$, unless $f$ is constant. Indeed, by Taylor's theorem, $f(mf(n)+n)$ is divisible by $f(n)$ for all $m$ in $\mathbb{N}$

How is this an application of Taylor's theorem? It's entirely mysterious to me. Thanks in advance for any insight on this.

By Tayor's Theorem
$$f(x) = f(n) + f'(n)(x-n) + \frac{f''(n)}{2!}(x-n)^2 + \cdots + \frac{f^{(k)}(n)}{k!}(x-n)^k + h_k(x)(x-n)^k$$
Now take $k$ big enough so that $h_k(x)=0$ (possible since $f$ is a polynomial).
So 
$$f(x) = f(n) + f'(n)(x-n) + \frac{f''(n)}{2!}(x-n)^2 + \cdots + \frac{f^{(k)}(n)}{k!}(x-n)^k$$
Now plug in $mf(n)+n$ for $x$.  You get an $f(n)$ in every term.
