Problem 1

Let

f(x)=x^n+5x^{n-1}+3

, where

n>1

is an integer. Prove that

f(x)

cannot be expressed as the product of two non-constant polynomials with integer coefficients.

Step 1 of 3: Put the polynomial in monic form

In plain words

A large next-to-leading coefficient prevents a nontrivial factorization.

f(x)=x^n+5x^{n-1}+0x^{n-2}+\cdots+0x+3

Detailed analysis

The polynomial is monic, and the coefficient immediately below the leading term is 5. All other nonleading coefficients have total absolute value 3.