Factoring a Quadratic Polynomial

This is the Classroom Tips & Techniques article for the May, 2011 Maplesoft Reporter, which, after publication, finds its way into the Maple Application Center. The article takes the liberty to rail against the stress placed on a particular manipulative skill in the precalculus curriculum, and likewise, I take the liberty to post it as a blog. The windmill at which I tilt is the "skill" of factoring a quadratic polynomial by inspection, a technique in which I find little intrinsic value.

My guess is that for historic reasons, factoring a quadratic was the way to obtain its zeros. The essence of the concept one would want a student to absorb is the factor-remainder theorem, so finding zeros becomes important. But demanding that students learn about the factor-remainder theorem via the travail of factoring a quadratic by inspection seems to me rather senseless, given that shortly, the student learns to complete the square and thereby obtain the roots of a quadratic equation. In fact, the quadratic formula is derived by completing the square, and not by "factoring."

I remember in my high school math curriculum (mid 1950s) that I learned to multiply and divide large numbers via the addition and subtraction of their logarithms. This material disappeared from the curriculum as soon as calculators became a commodity in the 1970s. If the curriculum can change in one way because of technology, why, using the same technology, can't it change in another? The higher cognitive merit is in understanding the relationship between zeros and factors. It isn't really necessary to torment students with factoring-by-inspection as a way of finding zeros. There are other ways - either use an electronic technology or use the quadratic formula.

Indeed, consider how a cubic is factored in the precalculus curriculum. First, zeros are found; then from the zeros, the factors are written. Just the opposite of the process imposed in the quadratic case!

Finally, I would conjecture that in some appropriate space, the set of quadratics that can be factored by "inspection" has measure zero. Any quadratic worth factoring probably doesn't yield to "inspection" anyway.

OK, now that I've ranted, I'd like to finish with a tool I just wrote, a tool for helping a student with the task of factoring a quadratic polynomial. I never thought I'd find myself creating such an applet, but I get asked about this as part of many of the webinars I present for Maplesoft. Just recently, after again getting that question, I found I just couldn't let it go. I kept thinking about what it takes for a student to master the appropriate skill. So, the tool I built reflects the way I think about the task.

Factoring a Quadratic Polynomial by Inspection

To factor a polynomial in the form , the "factor-pairs" of both a and c must be determined. It's not enough just to get the divisors of these coefficients; one must determine the pairs of divisors whose product equals that coefficient. Then, these factor-pairs must be arranged in linear factors in such a way as to reproduce the given quadratic. What I believe students struggle with is holding the various combinations in their heads while they test to see if the product of their linear factors equals the given quadratic.

Indeed, if the linear factors are of the form , then the mental arithmetic that the student must master is deciding if a sum or difference of the products  and  can equal the coefficient b. (What this has to do with understanding the factor-remainder theorem is beyond me, but this is the stumbling block for many a weary student. Oops, there I am back on my high-horse again.)

The tool I constructed (shown below, and available in the Maple document) is not modeled on a "drill-and-practice" philosophy. (At one stage of my teaching career, I was a strong advocate of such a heavy-handed approach to learning, but as I "got older" I mellowed, realizing that bludgeoning a student into mastery of a skill was counter-productive.)

Instead, the tool found below simply presents the factor-pairs, the linear factors so determined, and the expansion of the products of these factors. It's my belief that a student struggling with the task of factoring a quadratic will more likely see the "pattern" in these calculations than in a tool that asks for the right answer and simply tells the student that the answer is right or wrong.

The Tool