I'm reading a book on plane algebraic curves. It's a brazilian portuguese book, so I guess the title would be uninteresting to others, but if you want to know the title, just tell me $\tiny \text{and we can trade for some nudes.}$
At analytic geometry classes, I learned that conics can be represented in an equation such as:
$$ax^2+2hxy+by^2+2gx+2fy+c=0\tag{1}$$
And then we use the method of change of coordinates to simplify it and check if they are of one of the following forms:
$$\frac{x^2}{a^2} \pm \frac{y^2}{b^2}=1 \quad \quad \quad \quad y^2=4ax\tag{2}$$
At the book I mentioned earlier, it is said that each of these polynomial equations is in irreducible form. From the faint knowledge I have from algebra, an irreducible polynomial is a non-constant polynomial that can't be factored in a product of two non-constant polynomials. Is there an analogy between these two notions? That is: Is change of coordinates to obtain $(2)$, in some way, analog to factoring $(1)$?
I am aware that the curve in $(1)$ is not really the same as the curves given in $(2)$, they are the same by means of isometry. But I'm curious if change of coordinates is in any meaningful way similar to the factorization of polynomials.