Ambiguous Factoring

Fun with factoring, part I–Difference of Cubes

I wish I could remember the teacher’s blog that alerted me to this. I can see the page in my memory, but I can’t find it. I will happily include a link if anyone can tell me the blog. It was written in February, I think, a year ago.

The blog entry involved an interesting order of operations ambiguity when factoring a difference of two sixths as a difference of two cubes. I note that Wikipedia gives patterns for factoring differences of 4, 5, 6, and 7th powers, but in precalc and earlier we limit ourselves to squares and cubes.

Here’s an example of the same polynomial factored as a difference of two cubes and a difference of two squares:

So factoring it as a difference of squares allows for a second factoring round as a sum and difference of cubes, which fully factors the polynomial. Factoring it as a difference of cubes first allows for one factor to be further broken down, but not the other.

The additional factors on the squares side are indeed factors of the unfactorable term on the cubes side.

Most instructions for sum and difference of cubes explain that the second term is “unfactorable”> Clearly, this isn’t the case. We may not be able to factor it with our usual methods, but the term is factorable. We can only find the factors if we take the right road? Is that how it works?

I thought it was interesting, anyway, and gave the exercise to my precalc students, telling them I had no idea why the difference, but I’d find out if I could. Meanwhile, wasn’t it kind of cool? And the neat thing about precalc was that most of the kids were able to tangle with the math, realize the ambiguity and likewise be interested.

The blog entry I read and can’t find didn’t have an answer, either. In googling, I could only find one site that specifically mentioned order of operations when the cubes were also squares. Most didn’t mention it at all.

Interlude: Khan Academy Rant

While wandering round the web looking for sites that might address this ambiguity, I stumbled onto Khan Academy’s explanation for factoring the sum of cubes. While I think the fuss about Khan is little more than a reminder that great riches come to those who never need it and really don’t deserve it, I’ve never looked at his pedestrian videos closely enough to be flatly offended until now. Please understand that I am not the sort who goes through a blow by blow of the faults with the video, so it’s best to go see it yourself.

Khan starts with a sum of two cubes. Then he says, in order to do this, “this is really just something you need to know…I’d argue whether you REALLY need to know this, but to actually do this problem, it’s something you just need to know…”

Now, I want to be clear that math teachers will occasionally say “here’s something you need to know”–give a formula or fact that’s needed to explain or prove some other concept, when the students aren’t quite ready to grasp that fact. If you’re the type of teacher who just says “Sum of cubes, here’s the pattern”, that’s fine, too, if the students don’t have the math ready to understand it. However, Khan is doing the “here’s something you need to know” routine about the subject at hand. That’s not only weird, it’s bad. If you’re doing direct instruction without the why, then you say “Here’s what a sum of cubes looks like, here’s the pattern you use” and work some examples, Khan’s saying “well, in order to do this problem, here’s this little thing you need to know.” It’s unclear, and of course, Khan then undercuts it, saying that as far as he’s concerned, it’s debatable whether the student will need it at all. Simply dreadful lecture technique, and he doesn’t make it at all clear what he’s doing.

Worse, instead of just giving the pattern, Khan decides to “explain it”. You know, he’s not one of those schlubs that tells math without showing it.

It gets worse. Instead of just giving the pattern—because Khan, he don’t want to be one of those teachers, the ones that just show how but not why—the guy sets out to explain the pattern. Not prove it, god forbid, just explain where it comes from.


He just says “…it’s something you need to know. Let’s take the term” and he comes up with a2 – ab + b2 OUT OF THE BLUE. The student has absolutely no idea where this term comes from. Then he comes up with (a+b), again, not giving any reason. Then he multiples them and, in effect, says “Wow when we multiply these two terms that I seemed to just have pulled right out of my posterior, the product looks very similar to the problem we started with, the one I’m supposed to be showing you how to factor.”

Notice that, in the sequence of lessons, Khan has already covered division. But instead of doing a simple algebraic proof, he seemingly at random comes up with two factors. There’s no logical progression, no use of concepts already covered. Perhaps Khan does a better job when he thinks the students should actually know how to do the math.

As if that’s not bad enough, Khan finishes the multiplication of (a+b)(a2 – ab + b2) to get a3+b3 and then, I kid you not, says “Another way to look at this is to say that (a+b)(a2 – ab + b2) = a3+b3

Um, that’s not another way. That’s the same way.

Anyway. Speaking of ways. Rant over. But it’s truly dreadful lecture, and yet more proof that Khan is a fad, not a teacher.

I am hoping to have my students prove the pattern as part of our polynomial division section. This is the logic I’m planning on having them discover—let me know if I’ve missed something.

Fun with Factoring, Part II: Quadratic with Irrational Coefficients

On Friday, I set my precalc students an interesting problem, as part of a lesson on solving difficult equations:

I used the area model approach:


But two different groups of students came up with a solution that looked different on the face of it, yet had the same solution values:


So presumably one of them isn’t factored completely? But which one?

Here’s the solution using the quadratic formula, which is the method the math book used.


I was really quite surprised when the students came up with the second method, which they did with guess and check. Everyone agrees that the area model solution is “cleaner”, but I readily admitted that I didn’t know why we could get two solutions.

I don’t have any ego about appearing uncertain in front of my students. I know more than enough math, particularly for someone who isn’t a mathematician, and I”m a good problem solver. But I’d love to find out more about this. How were we able to get the same solutions but different factors?


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