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 4 of 4: Rule out the integer root
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In plain words

An integer root would make the value zero, while parity says every integer value is odd.

xn+5xn−1≡xn−1(x+1)≡0(mod2)x^n+5x^{n-1}\equiv x^{n-1}(x+1)\equiv0\pmod2
Detailed analysis

Since the polynomial is monic and h is linear with integer coefficients, h would give an integer root of f. But for every integer x the displayed congruence makes f(x) odd, because the remaining constant term is 3. Therefore f has no integer root, a contradiction.