What's important when dividing the following two polynomials
$x^4 + x + 1 \qquad \;\;\,\in \mathbb{Z}/2\mathbb{Z}[x]$
$x^3 - x^2 + 1 \qquad \in \mathbb{Z}/2\mathbb{Z}[x]$
How two calculate the first step
$\quad\,(x^4 + x + 1) : (x^3 - x^2 + 1) = x + 1 \; ...$
$-(x^4 -x^3 + x)$
$\quad \;\;\;x^3 + 1 \qquad \text{is this right?}$
$-(x^3 - x^2 + 1)$
$\quad \;\;\;x^2 \qquad \text{is this right?}$
So $x^4-x^4=0$ and since its residue class $x^4+x^4=0$ as well?