CAVEAT: There isn’t any *deep* math in this post. There aren’t any lessons or lesson ideas. I was just playing with quadratics today and below includes some of my play.

I’ve been struggling with coming up with a precalculus unit on polynomials that makes some sort of coherent sense. You see, what’s fascinating about precalculus polynomials is that to get at the *fundamental theorem of blahblahblah* (every *n*th degree polynomial has *n* roots, as long as you count *nonreal* roots as well as double/triple/etc. roots), one needs to start allowing inputs to be non-real numbers. To me, this means that we can always break up a polynomial into *n* factors — even if some of those factors are non-real. This took up many hours, and hopefully I’ll post about some of how I’m getting at this idea in an organic way… If I can figure that way out…

However more recently in my play, I had a nice realization.

In precalculus, I want students to realize that *all* quadratics are factorable — as long as you are allowed to factor them over complex numbers instead of integers. (What this means is that is allowed, is allowed, but so are and and . (And for reasons students will discover, things like won’t work — at least not for our definition of polynomials which has real coefficients.)

So here’s the realization… As I started playing with this, I realized that if a student has *any* parabola written in vertex form, they can simply use a sum or difference of squares to put it in factored form in *one step*. I know this isn’t deep. Algebraically it’s trivial. But it’s something I never really recognized until I allowed myself to play.

I mean, it’s possibly (probable, even) that when I taught Algebra II ages ago, I saw this. But I definitely forgot this, because I got such a wonderful *a ha* moment when I saw this!

And seeing this, since students know that all quadratics can be written in vertex form, they can see how they can quickly go from vertex form to factored form.

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Another observation I had… assuming student will have previously figured out why non-real roots to quadratics must come in pairs (if *p+qi* is a root, so is *p-qi*): We can use the box/area method to find the factoring for any not-nice quadratic.

And we can see at the bottom that regardless of which value of *b* you choose, you get the same factoring.

I wasn’t sure if this would also work if the roots of the quadratic were *real*… I suspected it would because I didn’t violate any laws of math when I did the work above. But I had to see it for myself:

As soon as I started doing the math, I saw what beautiful thing was going to happen. Our value for *b* was going to be imaginary! Which made *a+bi* a real value. So lovely. So so so lovely.

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Finally, I wanted to see what the connection between the algebraic work when completing the square and the visual work with the area model. It turns out to be quite nice. The “square” part turns out to be associated with the *real* part of the roots, and the remaining part is the square associated with the *imaginary* part of the roots.

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Will any of this make it’s way into my unit on polynomials? I have no idea. I’m doubtful much of it will. But it still surprises me how I can be amused by something I think I understand well.

Very rarely, I get asked how I come up with ideas for my worksheets. It’s a tough thing to answer — a process I should probably pay attention to. But one thing I know is part of my process for some of them: just playing around. Even with objects that are the most familiar to you. I love asking myself questions. For example, today I wondered *if there was a way to factor any quadratic without using completing the square explicitly or the quadratic formula*. That came in the middle of me trying to figure out how I can get students who have an understanding of quadratics from Algebra II to get a deeper understanding of quadratics in Precalculus. Which meant I was thinking a lot about imaginary numbers.

That’s what got me playing today.

Love it Sam! I want to look at this with my honors pre-calc when we start next week! Thank you for playing with this!

Thanks Sam! I plan to try this with my pre-calc honors when we start next week! Great insight and connections!

I 💜 algebraic numbers! I have a hobby of factoring addresses in my head.

When I went to my niece and nephew’s house, address 437, I wondered out loud if it was prime, then hit my forehead and said “duh, of course not, it’s the difference of squares: 441-4.”

It was the joke of the day.