Just for Fun … Creative Factoring

I’m teaching a class focused on individualized learning and flexible pacing.  One student in that class took a test on factoring in our intermediate algebra course.  In the process, I experienced something very enjoyable — a creative way to factor a polynomial.

Here is the situation:

Problem:   Factor r^4 – 16

Student:   (r – 2)(r³ + 2r² + 4r + 8)

Initially, I found this a bit confusing; I was not expecting to see a proposed factor with 4 terms.  In the materials, we focus on patterns to factor binomials involving the difference of squares.  So, I asked the student why he did this; his answer was “it checks”.  [This is exactly what I tell students when they ask WHY we factor a polynomial in a specific manner.]

After a quick transition from confusion to mathematical thinking, I looked more closely at the cubic factor.  Sure enough, it factors to produce:

Correct answer:   (r – 2)(r +2)(r² + 4)

This particular student (planning to be an engineer of some sort) had a creativity I would like to see more of.  The only negative feedback I had to deliver was “Finish the factoring”.

I found this to be just a lot of fun (though I doubt this student enjoyed it as much as I did, though he did enjoy it).  Mathematical fun is meant to be shared.  In 40 years, I’ve not seen a student do this; it’s too good of a thing to keep to myself.

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