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 3 of 4: Force the factor degree

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In plain words

The coefficient 5 is the obstruction: modulo 3 it cannot be supplied by the earlier divisible terms.

b_0c_{n-1}+\cdots+b_{n-2}c_1+b_{n-1}c_0=5

Detailed analysis

Reducing the coefficient equation modulo 3 leaves $b_{n-1}c_0$ congruent to 2 modulo 3, so $b_{n-1}$ is nonzero. Thus $g$ has degree at least $n-1$ ; because $h$ is nonconstant, its degree is exactly 1 and $g$ has degree $n-1$ .