# Lesson 6.4 of Exploding Dots

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All right, we are stuck on this particular polynomial division.
x³ - 3x + 2—
x³, no x²s, negative 3x's, and 2—
divided by x + 2.
Hmm. Looks like we're stuck.
We thought about unexploding.
Brilliant idea.
Turns out not helpful here,
because I don't know how many dots to draw when I unexplode.
Because I cannot tell you what x is.

What can we do? All right. This is why I love being a mathematics teacher. Because right now, we have a wonderful opportunity to learn a life lesson. In fact, here's the life lesson I want to teach right now. If there's something in life you want, just make it happen and deal with the consequences.

All right, so what do I mean by that? I see one dot, two dots. That's what I'm really wanting here, and I've got that one dot. There's something in life I want right now. What do I want right here? Well, I want two dots to go with it. So my life advice— if there's something in life you want, just make it happen.

There it is, two dots. But deal with the consequences. Now, I can't just change this box. It was meant to be empty. So then technically, I've got to keep this box empty. So how can I keep that box empty and still get what I want?

Well, what if I did this? Whoa. If I put in some antidots with these two dots, that box, with annihilation, is still technically empty. So I haven't changed anything, and I've got something— I've got what I want right there, in fact. All right, so that's at least one group of what I'm looking for.

Bingo. Now, like all brilliant ideas, that was brilliant, that was clever, that was grand. But the real question is, was it helpful? It got us going a little bit, but I'm not quite sure if it really got us all the way there.

Hmm. All right, so what can we do now? Um, [taps] I see those two dots there. Is there something in life I want right now? Is there something in there I want to go with them? You bet there is. What I want in life right now is a dot right there to go with those two dots there, make another one-two pair. Brilliant.

Now, something in life you want? Make it happen, but deal with the consequences. And the only way to deal with that consequence is actually put another antidot to counteract it—brilliant. So now I see two of what I want. One there and one there.

Oh, I'm still not quite sure if it's helpful. Got me a little bit further—feels good. But I'm not quite sure if it's really doing what I want. Because now I've got all these antidots floating around.

Oh, heavens. All right, so it's moments like these, we might just say we really are stuck. Um, or it might be time to go for a little walk and just let your mind rest, and think about this for a while. And then maybe a flash of insight will just come to you. Because actually, [chuckles] I see something.

What if I did this one antidot and those two antidots— one antidot, two antidots. Is that the exact opposite of what I'm looking for? That's an anti-one of what I'm looking for. In fact, there it is again Whoa. One antidot, two antidots— makes another anti-one. And that accounts for everything. Now that feels really good. In fact, I can see the answer just has to be One x², two anti-x's, and 1.

Bingo. I was actually lying about lying earlier on. This method is brilliant. It really does work. So what I'll do now is yet another example to show how this works, just to practice it. It's actually grand. This feels joyful. This is exciting. I just love it. So let me clean the board. We'll do one more example.

All right, board is clean. Let's do another example. Let's do, say, x to the fifth - 1 divided by x - 1. Let's try it. OK. Uh, the numerator. OK, lots of boxes. So there is a one box, an x box, x², x³, x to the fourth, x to the fifth. Uh, one x to the fifth—yep. No x to the fourths, no x³, no x², no x's, and one anti-one.

All right, I'm going to put x - 1. What does that look like? That's an x and an anti-one. So I'm looking for one dot and one antidot next to each other. I see none right now, but I'm not going to panic. Because I have a life lesson under my belt, which is, if there's something in life you want, make it happen.

I would love an antidot to go with this dot, please. Make it happen. Deal with the consequences. Got one of what I want. OK, anything else in life I want right now? Seeing that dot there, I'd love to have an antidot there. Make it happen. Deal with the consequences. Bingo. Dot, antidot—make it happen. Deal with the consequences. Make it happen, deal with the consequences.

Oh, and look, it worked out perfectly. Because there's a final, final copy of what I'm looking for. So this division problem worked out nicely. And I've got this really strange answer. It looks like it's, what, an x to the fourth, and an x to the cubed, and an x squared, and an x + 1. This polynomial division works out magically.

By the way, I need to point out people really do seem to forget that x can be a number in algebra class. For example, I think this problem now— because this works out to be a nice answer— tells me that 17 to the fifth power minus 1, whatever crazy big number that is, is a multiple of 16. A multiple of 16. I claim that's divisible by 16.

Can you see how I could see that from there? Huh. Actually, all this great polynomial work is really great results in number theory. You can tell whether numbers have factors or not by playing with some polynomial work like this. Crazy. All right, grand stuff. Polynomial division, high school algebra here, is just beautiful. Just draw a picture, and it all just falls out magically. Love it.

What can we do? All right. This is why I love being a mathematics teacher. Because right now, we have a wonderful opportunity to learn a life lesson. In fact, here's the life lesson I want to teach right now. If there's something in life you want, just make it happen and deal with the consequences.

All right, so what do I mean by that? I see one dot, two dots. That's what I'm really wanting here, and I've got that one dot. There's something in life I want right now. What do I want right here? Well, I want two dots to go with it. So my life advice— if there's something in life you want, just make it happen.

There it is, two dots. But deal with the consequences. Now, I can't just change this box. It was meant to be empty. So then technically, I've got to keep this box empty. So how can I keep that box empty and still get what I want?

Well, what if I did this? Whoa. If I put in some antidots with these two dots, that box, with annihilation, is still technically empty. So I haven't changed anything, and I've got something— I've got what I want right there, in fact. All right, so that's at least one group of what I'm looking for.

Bingo. Now, like all brilliant ideas, that was brilliant, that was clever, that was grand. But the real question is, was it helpful? It got us going a little bit, but I'm not quite sure if it really got us all the way there.

Hmm. All right, so what can we do now? Um, [taps] I see those two dots there. Is there something in life I want right now? Is there something in there I want to go with them? You bet there is. What I want in life right now is a dot right there to go with those two dots there, make another one-two pair. Brilliant.

Now, something in life you want? Make it happen, but deal with the consequences. And the only way to deal with that consequence is actually put another antidot to counteract it—brilliant. So now I see two of what I want. One there and one there.

Oh, I'm still not quite sure if it's helpful. Got me a little bit further—feels good. But I'm not quite sure if it's really doing what I want. Because now I've got all these antidots floating around.

Oh, heavens. All right, so it's moments like these, we might just say we really are stuck. Um, or it might be time to go for a little walk and just let your mind rest, and think about this for a while. And then maybe a flash of insight will just come to you. Because actually, [chuckles] I see something.

What if I did this one antidot and those two antidots— one antidot, two antidots. Is that the exact opposite of what I'm looking for? That's an anti-one of what I'm looking for. In fact, there it is again Whoa. One antidot, two antidots— makes another anti-one. And that accounts for everything. Now that feels really good. In fact, I can see the answer just has to be One x², two anti-x's, and 1.

Bingo. I was actually lying about lying earlier on. This method is brilliant. It really does work. So what I'll do now is yet another example to show how this works, just to practice it. It's actually grand. This feels joyful. This is exciting. I just love it. So let me clean the board. We'll do one more example.

All right, board is clean. Let's do another example. Let's do, say, x to the fifth - 1 divided by x - 1. Let's try it. OK. Uh, the numerator. OK, lots of boxes. So there is a one box, an x box, x², x³, x to the fourth, x to the fifth. Uh, one x to the fifth—yep. No x to the fourths, no x³, no x², no x's, and one anti-one.

All right, I'm going to put x - 1. What does that look like? That's an x and an anti-one. So I'm looking for one dot and one antidot next to each other. I see none right now, but I'm not going to panic. Because I have a life lesson under my belt, which is, if there's something in life you want, make it happen.

I would love an antidot to go with this dot, please. Make it happen. Deal with the consequences. Got one of what I want. OK, anything else in life I want right now? Seeing that dot there, I'd love to have an antidot there. Make it happen. Deal with the consequences. Bingo. Dot, antidot—make it happen. Deal with the consequences. Make it happen, deal with the consequences.

Oh, and look, it worked out perfectly. Because there's a final, final copy of what I'm looking for. So this division problem worked out nicely. And I've got this really strange answer. It looks like it's, what, an x to the fourth, and an x to the cubed, and an x squared, and an x + 1. This polynomial division works out magically.

By the way, I need to point out people really do seem to forget that x can be a number in algebra class. For example, I think this problem now— because this works out to be a nice answer— tells me that 17 to the fifth power minus 1, whatever crazy big number that is, is a multiple of 16. A multiple of 16. I claim that's divisible by 16.

Can you see how I could see that from there? Huh. Actually, all this great polynomial work is really great results in number theory. You can tell whether numbers have factors or not by playing with some polynomial work like this. Crazy. All right, grand stuff. Polynomial division, high school algebra here, is just beautiful. Just draw a picture, and it all just falls out magically. Love it.