This is part of the “Memorize or Melt” series, for more information go here if you haven’t read the introductory post on this topic.

At some point in a student’s math course work she is taught about how to factor quadratics. Typically a factoring problem will look like

Factor \(x^{2} + 6x + 5\).

The student is expected to churn out $$(x+5)(x+1)$$ by recognizing that \(5+1 = 6\) while \(5\times 1 = 5\). Thus, the emphasis is on being clever in recognizing two integers (not necessarily distinct) \(a,b\) such that \(a + b = p\) and \(a\cdot b = q\) where \(p,q\) are constants in \(x^{2} + px + q\). There is similar logic for when the \(p,q\) are not both positive.

Being able to factor quadratics is a good technical skill for any math student to develop. Unfortunately, the way factoring of quadratics is taught, if it is taught via the examples above, is not very useful. The types of examples where the student can cleverly recognize some combination of integers that multiply to \(p\) and add to \(q\) are entirely too specific. That technique only works in a special case. If one were to ask a student to factor \(x^{2} + 5.3x + 2.3\) he would not be able to use this technique effectively, if at all.

So, why are these types of exercises in so many Algebra textbooks? What is the benefit in having a student go through these motions?

If one were to venture a guess, the exercises are probably in Algebra textbooks because they are in other Algebra textbooks. This type of feedback loop is a common occurrence in textbook authoring — and not that it’s entirely a bad thing either. If an author can pull together lots of good ideas from multiple sources, then she presumably has a textbook of good lessons, ideas, and exposition.

But the benefit of these types of factoring exercises? It is not entirely clear to this author. There is one small positive side effect — it helps develop some sense of number. However, there are plenty of more effective ways to develop a sense of number; and by the time that a student has reached this level of mathematics study this shouldn’t be an issue, especially on the magnitude offered by the aforementioned factoring problems. Beyond that miniscule benefit, there is no real point in having students do these exercises. What happens if the student is asked to factor \(x^{2} + 50x + 576\)? Is she expected to sit and tinker until she “discovers” that \(32\times 18 = 576\) and \(32 + 18 = 50\)? And what else did she gain out of this?

The relative truth, also sometimes called an opinion, is that this type of factoring exercise teaches little and wastes a lot of precious education time. There is no need to teach this just as there is no need to teach the quadratic formula. These are all redundant. Factoring of quadratics should be taught simultaneously with root finding. In other words, the exercise

Factor \(x^{2} – 4x – 5\).

should be taught as an equivalent problem to

Solve \(x^{2} – 4x – 5 = 0\).

A cursory overview on how to teach solving quadratics can be found here: solving quadratics.

Once the student has been able to find the zeros of \(x^{2} – 4x – 5\) (namely \(\{-1,5\}\)) then it is just a matter for the instructor to teach that \(x^{2} – 4x – 5 = (x+1)(x-5)\). And that should be all there is to it! There are several immediate benefits of correctly providing one proper mechanical lesson in finding zeros of a quadratic:

• the instructor gains valuable time to focus on concepts and other topics;
• the instructor can spend more time explaining, conceptually what a factor is rather than spending time on a drill which, in effect, requires the student to have vapidly memorized another procedure;
• the student can legitimately factor \(x^{2} + 5.3x + 2.3\) without relying on having memorized the quadratic formula (or the use of a calculator for that matter);
• the instructor can easily introduce complex solutions of quadratics and show that factoring \(x^{2} + 1\) is procedurally exactly the same as factoring \(x^{2}-1\);
• the instructor can reinforce the notion of a “root of multiplicity”;
• students are not overwhelmed with an arsenal of nuanced techniques; instead they have one technique that they can leverage and do a lot with.

I strongly encourage instructors to steer away from factoring tricks and problems which have limited use. Instead I encourage instructors to teach students what a factor is and to show them that procedurally finding factors for a quadratic is equivalent to finding zeros of a quadratic.