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

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.
