MathLabs

Problem 1

Let f(x)=xn+5xn−1+3f(x)=x^n+5x^{n-1}+3, where n>1n>1 is an integer. Prove that f(x)f(x) cannot be expressed as the product of two non-constant polynomials with integer coefficients.
Step 2 of 3: Check Perron’s inequality
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In plain words

The margin is one, so the criterion is strict rather than borderline.

5>1+35>1+3
Detailed analysis

The strict inequality required by Perron’s criterion holds: the next-to-leading coefficient exceeds one plus the sum of absolute values of every lower coefficient.