How many times the parabola $y=x^2$ intersects the origin? The answer of this question is 1, right? but I'm about to study algebraic curves by this book
and I was surprised by this theorem:

If I'm right the immediate  corollary of that is $y=x^2$ intersects the origin two times.
Then he continues given an example:


I don't understand this, anyone can help me?
Thanks a lot

You have to be a bit careful there. If we want to find the intersection of $y=x^2$ and the line $y=0$ within this framework, we would (or, at least, could) say
$$
y = p(x) = x^2\\
g(x,y) = y-x^2
$$
The theorem then tells us that as long as $\mathbf{y-p(x)}$ is not a factor of $\mathbf{g(x,y)}$, there are two intersections of the origin. Since $g(x,y)=y-p(x)$ (or, if you put things together a bit differently, $-(y-p(x))$) the theorem doesn't apply.
