<s>[INST] Find the greatest common divisor (gcd) of $f(x) = x^2 + 1$ and $g(x) = x^6 + x^3 + x + 1$ Find the greatest common divisor (gcd) of $f(x) = x^2 + 1$ and $g(x) = x^6 + x^3 + x + 1$.
Since $x^6 + x^3 + x + 1 = (x^2 + 1)(x^4 - x^2 + x + 1)$, $\mathrm{gcd}[f(x),g(x)] = x^2 + 1$.
My question is how could I JUSTIFY that the answer is ACTUAL gcd of $f(x)$ and $g(x)$. thanks
 [/INST] To justify, you could say that it divides g(x) and f(x) to the rational polynomials such that they have no other common divisor, or common factor. So the answer is proved.
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