I am writing this post to clear up some confusion I have with complex roots of polynomials. I am learning about extensions of fields (e.g. from $\mathbb R$ to $\mathbb C$) and have come to realize that I clearly do not understand the concept of roots.
Apparently naively, I thought roots were any values of $x$ in a polynomial $a(x)$ of the form $a(x) = a_0+a_1 x+a_2 x^2 + ... + a_n x^n$ that results in $a(r)=0$. Using graphs as a basic visual understanding of this, I interpreted this as saying "any value of $x$ that sends $y$ to $0$, where $y$ is $a(x)$.
Employing https://www.geogebra.org/m/z374cvsr to generate some graphs, I quickly realized that something is wrong with this interpretation.
In line with the idea of field extensions from $\mathbb R$ to $\mathbb C$, I generated a random quadratic polynomial that has no roots in $\mathbb R$:
Now, in line with the idea of extension, I took this $2D$ graph to a $3D$ representation where the imaginary axis is now in play. However, I also added a plane that intersects through all ordered triplets that have $y=0$ and $x$ and $z$ (the imaginary number value) can be whatever...i.e. $(x,0,z)$.
Firstly, I notice from this picture that the visual interpretation of the polynomial has changed; it is no longer just a line but a surface. In fact, for convenience, at first glance it seems like this polynomial is acting like a function that takes in two values (namely an $x$ value and a $z$ value) and spits out a $y$ value.
In this photo, you will see that I have circled, in yellow, the complex roots of this polynomial. However, I have also traced in a segmented orange line all points of this surface that intersect the plane defined by $(x,y=0,z)$.
Now, using my prior understanding of root, this would seem to imply that this arbitrary quadratic polynomial (which has degree $2$) has an infinite number of roots...because it exhibits an infinite number of points $(x,z)$ that result in a $y$ value of $0$. Obviously, I know that this cannot be true because of a prior theorem my book has outlined...specifically:
If $a(x)$ has degree $n$, it has at most $n$ roots.
(I am aware of the fundamental theorem of algebra, but my book has largely glossed over this). So my question is the following:
Why are the two yellow-circled points the actual roots of this polynomial but the other points on the orange, segmented lines are not?