gcd and lcm - finding the greatest common divisor of two polynomial, I'm stuck

I'm trying to find the greates common divisor of two polynomials. The polynomials are:
$$\begin{align*} p_1 &= x^3+3x+1\\ p_2 &= x^4+1 \end{align*}$$
Matlab is telling me that the GCD is 1, and that's also what I was expected.

However when I try to do It by hand it dosn't equal to 1, so what am I doing wrong?(I'm using Long division and writing it as Euclidian algorithm).

$$\begin{align*} x^4+1 &= (x^3+3x+1) \cdot x -(3x^2-x+1)\\ x^3+3x+1 &= (-3x^2-x+1) \cdot (\frac{-x}{3}+\frac{1}{9}) + (\frac{31x}{9}+\frac{8}{9})\\ -3x^2-x+1 &= (\frac{31x}{9}+\frac{8}{9})\cdot (\frac{-27x}{31} - \frac{63}{961})+ \frac{1017}{961}\\ \frac{31x}{9}+\frac{8}{9}&=\frac{1017}{961}\cdot (\frac{29791x}{9153}+\frac{7688}{9153})+\frac{8}{9}\\ \frac{1017}{961} &= \frac{8}{9} \cdot \frac{9153}{7688} + 0 \end{align*}$$

so the GCD should be $= \frac{8}{9}$??