In The Biggest Box we created a cubic polynomial to model the volume of a box. We found that the graph of this function had three x-intercepts. We also found that these x-intercepts could be predicted based on the factored form of the equation. So, the factored form is just as useful for solving and graphing cubic polynomials as it was for quadratics (MP 7). I will begin today's class with a brief discussion to draw out these points.
I plan to tell students that we'll begin by practicing some expansion of cubic polynomials before we move on to factoring. Then I write the six equations from section 1 of Factoring Cubics on the board and ask them to quickly use the distributive law to re-write them in expanded form. This should take about 10 minutes, and I encourage them to check their answers with one another along the way.
Everyone should have noticed that in a number of cases the expanded form is very brief. In fact, it simplifies to a mere sum or difference of two cubes! (I intentionally include some other cases so they don't get the impression that this always happens.) As students finish, I'll ask one of them to put his or her solution on the board next to the factored form. By the time our 10 minutes is up, we should have solutions to almost all of them written on the board and ready for discussion.
We'll focus on the examples of sums/differences of cubes and try to determine the pattern inherent in the factors. First, we have a linear and a quadratic factor. Second, the signs of the various terms follow a definite pattern. Third, the coefficients on each term follow a definite pattern.
I will formalize these patterns on the whiteboard under the heading: "How to Factor a Sum or Difference of Two Cubes". I'll also remind the students to be sure to copy this down into their notes!
How can we use this pattern? For a class example, I'll use the equation x^3 - 64. Supposing our rule for factoring a difference of two cubes is valid, what would the factors be. Once the class has given me the factors, we'll multiply them carefully to check.
The remainder of the lesson will be spent with the students working individually or in small groups to determine the factored form of the equations in section 3 of the resource.
I will spend my time helping students to apply the patterns we have just discussed. This takes special care whenever the coefficient on x^3 is something other than 1. Very few students will recognize on their own that 8x^3 is equivalent to (2x)^3. (Unless, of course, this is familiar to them from previous experience factoring the difference of two squares. I'd be sure to bring up this connection explicitly at some point.)
I will also point out to students that once they've reduced the equation to one linear & one quadratic factor, they should attempt to factor the quadratic, too.
Finally, I intentionally included some equations that do not fit this simple pattern. In this case, I've chosen functions with three integer roots to keep things simpler. These ones may pose a healthy challenge to your stronger students, and you can feel free to give lots of hints to those who are struggling. (MP 1)