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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.

March Hare
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  • I'm also aware that it's probably an example of pattern recognition. I didn't know if there were any more specific terms or ways to refer to this particular sort of situation. – March Hare Apr 14 '22 at 04:44

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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”.

bubba
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  • Yes, and treating this process as merely pattern recognition leads to fallacies like this. – ryang Apr 14 '22 at 08:59
  • I think both of those terms are probably pretty reasonable. I don't know why I didn't consider either of them from the get go. I remember when I learned to do this the first time and it kind of blew my mind, and I've been doing it for awhile, so it's kind of automatic for me now, and I don't recall having very much of it explained with many formalisms. – March Hare Apr 14 '22 at 11:58
  • It also doesn't help that I don't usually think of the polynomial terms explicitly as a "basis", even if I know exactly how to use them to set up a matrix to solve for A, B, and C that way, too. – March Hare Apr 15 '22 at 01:45
  • Well, I often use other bases (like Bernstein polynomials or Lagrange polynomials) rather than simple powers. I guess that’s why I’m accustomed to thinking I terms of bases and linear independence. – bubba Apr 15 '22 at 06:14