I'm curious if there's a technical term for the following concept. Let's say we do a curve fit to a set of points and it conforms to the equation $y = 1.44x^2 + 82x + 5$
Obviously as a human, I'm well aware that this is just a basic quadratic function and which coefficient is A, B, and C. But in more explicit terms, we're matching up the determined equation with the general form, like so. $$y=Ax^2+Bx+C\\y = 1.44x^2 + 82x + 5$$ Where we know by inspection that only $x^2$ terms are going to be involved with A, x terms with B, and so on.$$A = 1.44\\B=82\\C=5$$ of course, one can formally take the first and second equations, say that they're equal, and group the terms appropriately, and call it a day.
Naturally, this is one of those things human beings are pretty good at. I understand the what and how of what's going on here, and usually do this stuff by inspection rather than show all of the steps. I'm not asking how this process actually works mathematically, I know that.
What I'm asking is whether there is there a name for this particular process of grouping the terms and "applying" a coefficient variable to them? Is there a better way to explain it to someone less experienced than "So all of the $x^2$ terms can be added, and no other terms go toward those, so the coefficient on one side must be equal to the corresponding terms on the other side." and so on.
If I understand you correctly, you’re wondering why $$ Ax^2 + Bx + C = Lx^2 + Mx+ N \quad \text{for all } x $$ allows us to conclude that $A=L$, $B=M$, $C=N$.
This has nothing to do with the pattern recognition capabilities of human beings. It’s the linear independence of the polynomials $\{1,x,x^2\}$ that makes this process of “equating coefficients” legitimate.
You can perform the same sort of reasoning with any set of linearly independent polynomials. For example, if $$ A(1-x)^2 + Bx(1-x) + Cx^2 = L(1-x)^2 + Mx(1-x) + Nx^2 $$ you can again equate coefficients and conclude that $A=L$, $B=M$, $C=N$. And, again, this works because the three polynomials $\{(1-x)^2, x(1-x), x^2\}$ are linearly independent.
So, I think the terms you’re looking for are “equating coefficients” and maybe “linear independence”.