To Show that, $f$ cannot be expressed as the product of two polynomials with integer coefficients and degree $>1.$

Question

$\textbf{Problem}$- Let, $f(x)=x^n+ax+p$, a $\in \mathbb{Z}.$ $p$ is a prime. Also, $p>|a|+1$. Show that, $f$ cannot be expressed as the product of two polynomials with integer coefficients and degree $>1.$

For this problem, which is a classical polynomial problem, I have been trying to apply the Vieta s theorem and irreducible criterion,but I can't structure my proof. A little bit rigorous proof is appreciable. Thank you.

This problem has been lying unanswered... Kindly help ...I don't know why this problem is lying unanswered... Is there is any typo problem in the problem?

Answer 1

If you could factor the polynomial it must be in the form. :

$$ (x^{bb}+... + b_1x + b_0) (x^{cc} + ... + c_1x+c_0)$$

$$bb+cc = n$$ $$ b_0 c_0 = p$$ $$ b_1c_0 + b_0 c_1 = a$$

$$ p> |b_1c_0 + b_0 c_1|+1$$

Since p is prime you must iterate through the 4 possibilities for $b_0$ and $c_0$ : {1,p}, {-1,-p}, {p,1}, {-p,-1}.

Plug these 4 possibilities into the inequality above and you'll see that the inequality cannot be satisfied with integer values for the $b_1$ and $c_1$. Picking $b_1$ and $c_1$ from {1,-1} gets you $|b_1c_0 + b_0c_1|$ only as small as $p-1$.