Quoting Henry Baker <hbaker1@pipeline.com>:
The usual trig solution for a monic real cubic eqn with 3 real roots has the high school teacher pulling an obscure trig identity out of a hat in deus ex machina fashion, leaving the high school student feeling inadequate, helpless, irritable, and (s)he probably then tunes out of the rest of the presentation & starts texting on his/her cellphone.
Those cell phones are really a plague on contemporary civilization comparable to the biblical locusts. I have always admired the trig solution, and it is not so obscure if you think of multiple angle formulas (not to mention Tchebycheff polynomials. Did Tchebycheff know that was what he was doing? One hopes so.) The cubic version is but a modest extension of the double angle formula, but how many trig or geometry courses get that far? I once got into very hot water (fired, in fact) for showing a similar derivation to a *college* trig class. Now that cell phones have graphics capabilities (and certainly more familiar computers), one can graph the coefficients of the polynomial as functions of the roots. For a (monic) quadratic polynomial, the remaining coefficients are the sum and the product of the roots, so you get a family of lines r1 + r2 = (coef of x), and a family of hyperbolas r1 x r2 = const. The pairwise crossings show the roots in a very symmetric way: straddling the mean, real when the curves actually intersect, horrendous numerical instability at double roota because of the tangency, and so on. I once heard while an undergraduate that there were tables of roots of cubic equations, although I am not sure whether Jahnke and Emde had them. Or even graphs. Well, following the quadratic analogy, monicize the equation and then make the sum of the roots 1, not 0, by the usual shift, and then graph r1 x r2 + r2 x r3 + r3 x r1 = (coefficient of x), and r1 x r2 x r3 = (const) (another hyperbola) using plane trilinear coordinate graph paper. Again it is possible to read the roots off the intersections. This is much too complicated for a course, but I have used it to amuse my numerical analysis students. I'd have to go look up the reference, but the british Mathematical Gazette (I think) had a nice article matching the triangle you mention to the critical points of the cubic. Centroid = point of inflection = center of 180 degree rotational symmetry, for example. But one really wonders what students can be expected to endure. -hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos