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 3 of 3: Apply the criterion
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In plain words

Irreducibility is exactly the requested conclusion.

f(x) is irreducible over Zf(x)\text{ is irreducible over }\mathbb Z
Detailed analysis

Perron’s criterion says a monic integer polynomial satisfying the checked inequality is irreducible over the integers. Hence f cannot be a product of two nonconstant integer polynomials.