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 2 of 4: Propagate divisibility by 3

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In plain words

The nondivisible constant coefficient prevents cancellation unless the next coefficient of g is also divisible by 3.

3=b_0c_0

Detailed analysis

Since the constant term is 3, one of $b_0,c_0$ is divisible by 3; interchange the factors so $3|b_0$ and $3\nmid c_0$ . The zero $x$ coefficient then gives $3|b_1$ , and repeating the same comparison gives $3|b_{n-2}$ .