when did polynomial coefficient matching start for solving equations?

Coefficient matching feels rather natural when solving equations and checking dimensions, however in footnote 2 to "Two alternative derivations of Bridgman's theorem" (Berberan-Santos M N, Pogliani L, J. Math Chem 1999, 26, 255-261; p256) it's mentioned that Descartes shifted from line/area/volume ideas to the basic number line view.

Is this where the idea of coefficient matching between the terms when solving polynomials began, or was it already well established for pure number polynomials?

• Could you explain what "coefficient matching" is / was? Aug 7, 2016 at 0:48
• @RoryDaulton It may not be the right term, but I was refering to the case where you get, via two routes, say A=ax^2+bx+c, and A=5*x^2 + 7, so c=7, b=0, a=5 (seems obvious?). The comment in the paper implied that historically you could not add a square (e.g. the area of a room) to the length of a line (length of drive way) to a pure number, and even now, we should not add a probability to an angle, despite them being thought of as pure numbers Aug 7, 2016 at 10:13
• This is so in your example only if the two equalities for A hold for at least 3 values of x. For 2 or fewer, the coefficients may well not match. I believe the original view of polynomials was as expressions taking values, not as mathematical objects of their own as in modern algebra. Aug 7, 2016 at 11:44
• @RoryDaulton, when was that change in view about expressions taking values? In the applied sciences there are still some issue regarding quantity calculus and just when polynomial terms can be matched, and which terms are 'just values'. Aug 8, 2016 at 13:51