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Lesson 6.3 of Exploding Dots

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OK, I have a serious confession to make. I've been pulling the wool over your eyes all this time. This long division in base x actually has a serious problem. I've been choosing very nice examples that just happen, by coincidence, to work.

Actually, I've been kind of lying all this time. It's not as magical as it looks. For example, let me show you this problem. I'll show you what I've been avoiding all this time.

Let's try something like x³ - 3x + 2 divided by x + 2. Can you see what I've been sneakily avoiding all this time? Negatives. All right, so I can draw it. I can at least draw the top line. That will be an x³. And it will be another x². But it will be three anti x's, the three antidots, and 2. Grand. I'm in an 1←x machine.

All right, um, let's see. And I'm dividing by x + 2. What does x + 2 look like? One dot and two dots. And the question is, can I see any one dot next to two dots anywhere in this picture? And I look at it, and I weep. Oh, dear. I can't. I can't. So like I said, I've actually been fooling you all this time. This beautiful work of advanced high school algebra polynomial division actually doesn't work, in general.

Unless—unless. Unless you have a flash of insight of some kind. Let's see. What can we do? I don't see any one dot next to two dots. I mean, I see one dot over here and two dots way over yonder. But that's no good. I need them to be right next to each other.

So the question could be, any smart ideas? And yes, actually, I bet you're thinking right now— maybe or maybe not, I don't know— um, but I see that one dot there. Wouldn't it be nice to just unexplode it and make some more dots there? That would be grand. And that's a very good, brilliant idea. And I genuinely mean that as good and brilliant.

Just turns out not to be helpful because it's an 1←x machine. X—you don't know what x is in my head. If I told you it was a 1←10 machine, I would know to unexplode and draw 10 dots there. Or if I told you it was a 1←2 machine, you'd know to unexplode to draw two dots there. The thing is you don't know how many dots to draw there.

So it was a good idea. I genuinely mean that. Just turns out not to be helpful right now. Truly stuck. Truly stuck. Is there any way to move forward with this one? Or am I lying about previously lying?

Oh, oh, messing with our minds now. Can we make this work? Or can we not? That's an interesting challenge.

Video Details

Duration: 2 minutes and 37 seconds
Language: English
Genre: None
Views: 2
Posted by: jamestanton on Jun 9, 2018

This is the 14th lesson of the Exploding Dots story for Global Math Week 2017 (

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