<s>[INST] Finding $\gcd(x^3, x^3+x+1)$ Suppose $f(x) = x^3$ and $g(x) = x^3+x+1$. I am trying to show that the $\gcd(f,g)= 1$ but am running into some trouble...
My attempt: (Division algorithm)
$x^3+x+1 = (1)(x^3)+(x+1)$
$x^3 = x^2(x+1)-x^2$
But I cannot go further than this, what is going wrong here?
 [/INST] Just keep going.  You found that
$$
x^3+x+1 = (1)(x^3) + (x+1)
$$
Now observe that
$$
x^3 = (x^2-x+1)(x+1) - (1)
$$
What can you conclude?
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