Cubic equations

Sometimes going beyond 2 is extremely difficult.

Our tutoring consultant routinely takes his students through the Quadratic Equation.  This will give you the solutions to any polynomial equation of degree two, that is, whose highest power is the square.  It is guaranteed to work (provided, of course, one makes no mistakes in the algebra or arithmetic).

But little is said about the corresponding equation for cubics, where the highest power is of degree three.  There is an equation, sometimes known as Cardano’s formula, which will give a solution of a cubic.  However, it requires some preparation to set up and a fair bit of complex analysis to use.  It is not mentioned in High School mathematics and, indeed, none of us saw it as science undergraduates.  A couple of weeks ago our tutor decided to attack it and work out its proper employment.  After some effort he found himself looking for the trigonometric formulas for the functions of one-third of a given angle.  He thought he was making progress.  Then, looking up a reference, he found that such a formula existed, kind of.  But it required the solution of a cubic equation, which made using it to solve a cubic equation kind of pointless.  The examples he’d seen of the success of Cardano’s formula relied on the problems being special ones.

He then determined to work things from the other end.  He would choose three solutions, all irrational and thus not amenable to the special methods taught in school, and work out their equation.  After proving that a couple of promising types of solution sets would not work, he found a reference showing that what he sought was impossible.  It was equivalent to trisecting the angle, an ancient problem in geometry.  It took until the nineteenth century for mathematicians to show it could not be done (under the standard restrictions).

For someone who has apparently wasted a good deal of time, our tutor is remarkably content.  After all, he has gotten some useful of exercise in algebra at a level well above what’s called for in tutoring sessions, the mental equivalent of a good run by the river.  And he has gotten the right answer, twice, in areas of mathematics where other workers have failed.

This got us wondering what other results have been produced by mathematicians trying to do something later found to be impossible.  There is a general formula for the solutions of a cubic, as we mentioned; we understand that one exists for quartics, but none are  possible for fifth-degree equations or higher.  No doubt they have been sought for, but something called Galois Theory showed them not to exist.

Our tutor does not intend to attack aspects of higher-order polynomials with his pen and paper now, however.  He would rather learn Galois.  There’s no point in recreating all of mathematics.