Fundamental Theorem of Algebra Application Let $f(x)=a_{k}x^k+a_{k-1}x^{k-1}+a_{0}$ be a polynomial with integer coefficients and $a_{k}$ nonzero.
Prove that if $a_{0}\geq2$, $f(n)$ is not prime for some integer $n$.
Any ideas?
 A: Proof by contradiction. Consider $f(na_0)$ for integers $n$. Observe that $a_0 \mid f(ka_0)$. Hence if all of these numbers are prime, then $a_0$ must be prime and these values are either equal to $a_0$ or $-a_0$. 
Thus infinitely many of them are equal, which contradicts FTA which says that there are at most $k$ roots to the equation $f(x)= \pm a_0$
A: Hint: for what $n$ is $f(n)$ divisible by $a_0$?
Doesn't have much to do with the FT of A though.
A: Fundamental theorem of algebra does not easily work here, because certain factorizations like
$$
x^2 + 1 = (x + i)(x - i)
$$
or
$$
x^2 - 2 = (x + \sqrt{2}) (x - \sqrt{2})
$$
don't give much insight into whether this complex number is prime or not.

Instead, try showing that if $f(ka_0)$ for any integer $k$ is prime, then $f(ka_0)$ is constant, and hence $f$ is constant.
A: We can in fact generalize and consider any polynomial $f(x) \in \Bbb Z[x]$,
$f(x) = \sum_0^k a_j x^j \tag{1}$
with $a_k \ne 0$ and $a_0 \ge 2$.  We observe that for any integer $m$ we may write
$f(ma_0) = \sum_0^k a_j m^ja_0^j = a_0 (\sum_1^k a_j m^ja_0^{j - 1})+ a_0 = a_0((\sum_1^k a_j m^j a_0^{j - 1}) + 1), \tag{2}$
and note that, for $m$ sufficiently large we have
$\vert (\sum_1^k a_j m^j a_0^{j - 1}) + 1 \vert > 1 \tag{3}$
since $(\sum_1^k a_j m^j a_0^{j - 1}) + 1$, being a polynomial of degree $k$ in $m$, becomes arbitrarily large as $m \to \pm \infty$; thus for $m$ large enough the expression $(\sum_1^k a_j m^j a_0^{j - 1}) + 1$ cannot be a unit in $\Bbb Z$; hence $f(ma_0)$, being reducible in $\Bbb Z$, is not prime.  QED.
Note: this demonstration isn't really a "fundamental theory (sic) of algebra application", but it nevertheless establishes the requisite result.  Incidentally, I think the OP must have meant "Fundamental Theorem of Algebra Applicaton" in the title. End of Note.
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!
