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 1 of 4: Assume a factorization
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In plain words

Compare coefficients from the constant term upward.

f(x)=g(x)h(x)f(x)=g(x)h(x)
Detailed analysis

Assume a factorization into nonconstant integer polynomials. Write the coefficients of g and h as b_i and c_i, with missing coefficients treated as zero.