One of the main sections of the course is devoted to deriving the solution of the general cubic equation. I think of this section as one of the really good sections of the course. We cover the general cubic equation in this class for several reasons. Perhaps most importantly, solving equations is an important historical topic in mathematics. During the Renaissance, Italian mathematics began to blossom as they attacked methods for finding roots of cubic and quartic equations. Moreover, their solutions to these equations show how numbers were thought of quite differently by mathematicians at that time. In particular, Cardano in his Ars Magna solves a quartic equation earlier than solving a linear equation with a negative solution. One reads from this that Cardano considered finding the general (positive real) solution to a quartic equation to be easier than finding a negative solution of a linear equation. This allows us to enter into a discussion of different understandings of numbers, and how numbers which today seem “easy” to deal with, might not have been so in the past. Mathematically, solving the cubic also allows us a natural entry into the complex numbers. In particular, a common fallacy of high school texts is that the complex numbers were developed to solve quadratic equations that didn't have real roots. In reality, however, the complex numbers were first used to find real solutions to cubic equations. Moreover, even to this day, finding these real solutions as sums of radicals appears to require using complex numbers. Thus solving the cubic allows us to enter into a historical discussion of the complex numbers.
In terms of bringing higher mathematics to the high school curriculum, the cubic also allows us to make several profitable investigations. In deriving the solution to the cubic, we are called upon to use several different mathematical moves. Our solution, which mimics one in Cardano's book, begins by trying to associate a cubic of the form x^3 + px = q (p,q positive) to a cube. In starting this process, we can address a similar question involving a square and discover a geometric derivation of the quadratic formula, allowing students to encounter a geometric understanding of the term “completing the square.” When we move to cubics involving a quadratic (x^2) term, we then address another method of solving a quadratic equation, namely by translating the parabola so that its vertex is on the y-axis. This gives the students a third derivation of the quadratic formula, and links together graphs and the algebraic solutions of equations. Having done all of this allows us to reflect upon a great deal of mathematics in a natural way. Moreover, students appear to appreciate these linkages in the class.
The above explains why I teach the cubic and see it as an important part of the class, and here we shall describe, in part, how we teach it. Since part of the purpose of deriving the solution to the general cubic is to reflect upon the derivation of the quadratic formula, we derive the quadratic in three ways. I want prospective teachers to understand that the algebraic derivation of the quadratic formula that they know, may not be the best way to help students understand the quadratic formula. The first several times I taught the cubic solution, I failed pretty miserably at this, and for this reason, the first thing we do is derive the quadratic formula algebraically. I do this by letting the students direct me through the solution while I am at the board. In this case, however, I write down everything they tell me to exactly as they tell me to. This allows me to keep track of the number of times someone has to correct what they have done. Each of the two times I have done this, there were at least 6 errors in terms of dropping a minus sign or making a basic algebraic mistake. Once we have finished, I query them about whether they think their students will be able to do this derivation, given that 10+ senior mathematics education majors managed to make so many mistakes along the way. Now, I talk about geometric methods of deriving the quadratic equation, particularly in special cases. This enables me to show them where the idea of completing the square comes from, and it leads me into the beginning of an attack on solving the general cubic equation, for which we follow Cardano's proof. The first day of the section usually ends with me having completed the geometric derivation of the quadratic equation, and spending the last few minutes linking this idea to manipulatives. This last link is important for the students to see as it links what we are doing back to some of their education classes. I also find it useful as I can talk about using the manipulatives to make algebraic reasoning clear, as opposed to using them for no other purpose than to use them.
The first 20 minutes of the second day on this material is spent in giving a brief history of the solving of the general cubic, together with all of the wonderful duels and releasing of secret methods. This can make the students more interested in what follows, and John suggested this was one of the teaching ideas he got out of the class:
...so the first thing I did was start by talking about Napier, which I never would have done before. I never would have introduced an element of history into a math class.
After this, I link how we will solve the cubic to how we derived the quadratic formula. Moreover, I bring in a cube made up of unifix cubes of different colors, so that I can show how thinking of the volume of a cube of side length u with a corner cube of side length v has two different volume calculations: namely
V = u^3 - v^3 and V = (u-v)^3 + 3uv(u -v)
Setting these equal, we see that if x = (u - v), p = 3uv, and q = u^3 - v^3, then by setting these to volumes equal, we obtain the equation x^3 + px = q. Consequently, we have derived an algebraic equation from a geometric argument, and we use this to solve the cubic. (See the text for the rest of the derivation.) This argument allows me to make mathematical points about the connections between geometry and algebra, and moreover, I can quickly point out how easily forgotten the algebraic expression is without the geometric underpinning.
After we solve the cubic x^3 + px = q with p and q both non-negative (and at this point, we use solve to mean that we find the real solution as the complex can then be found using the quadratic formula), we move to the harder question of what happens when p is negative. In fact, we first look at the cubic that Bombelli examined when he first used complex numbers (x^3 = 15x + 4). Solving this cubic leads us to the discussion of why the complex numbers were first invented. At this point, we spend a day discussing the complex numbers and giving them a geometric meaning (something I am surprised that the students often claim not to have seen before (and probably haven't)).
This is a somewhat cursory discussion of how I cover the cubic equation (or at least the first two-thirds), but I think it gives some insight into my teaching of it.