Problem

Source: Chinese TST

Tags: algebra, polynomial, modular arithmetic, absolute value, algebra proposed



Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) = x^n + a_{1}x^{n - 1} + \cdots + a_{n}$ such that (1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$; (2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees; (3) for any integers $ x, |f(x)|$ isn't prime numbers.