Int Algebra: Factor Sums and Differences of Cubes

A great way to learn!

01/15/2009

~ brittanie

My teacher doesn't dance but I think if she did I would find Algebra a lot more interesting :)

• View all learner reviews »

Exponents and Polynomials
Special Factoring
Factoring Sums and Differences of Cubes Page [1 of 2]
I’m here with my dancing cubes, because in fact cubes are great. I mean, we can not only look at the difference of two perfect cubes, but we can also get the sum of two perfect cubes. And this is great because, if you remember with perfect squares, the sum of two perfect squares is hard to factor, it’s hard to think about how to factor that. But here with the cubes, no problem, we can do them both.
Okay, that’s the good news. We can actually figure out how to factor easily, well, not easily, but we can factor the difference of two perfect cubes and the sum of two perfect cubes. That’s the good news. The bad news is that the factorization is a pain in the thing. Let me show you how this goes. I’m going to write down the factorization and then I’m going to try to convince you that, first of all, the factorization is correct, and then where it all comes from.
Suppose you have the difference of two perfect cubes. So let’s say you have x3 – y3––so that’s the difference of two perfect cubes. How would you factor that? Well, one factor is always going to be just the thing itself without the cube thing there. In fact if you think about it, the same thing holds with the difference of two perfect squares. All right, let’s put in here the x and then –y. Now if you think about it, the difference of two perfect squares works the exact same way, right? I always have a factor that looks like this, but now because it is cubes, I’m going to have some stuff with squares in it. And what I’m going to have actually is x2 + xy + y2 and that’s a whole mouthful of stuff, but that actually is the factorization. And let me try to convince you that in fact this is going to actually give the right answer. Namely, if you were to multiply all this out by distributing––remember everything gets hit with everything else––we should end up just with this. So there should be oodles of cancellation, things should be, it should be the massacre of middle terms, right, everything just dies and there should be no one left to tell future generations what happened. Okay, let’s make sure this really is the case.
Let’s take that x first of all and I’m going to start to distribute it just a little bit. The x times the x2 happily gives me the x3 that I want. And notice that the –y times the +y2 gives me the –y3 that I want. So now I have all these middle terms, in fact let me show you the middle terms. I have to take x times this, that’s the middle term, and then also x times that last term, and then I have to take the –y times this, and the –y times that. Let’s see what happens and see how this conspiracy works. That negative sign is the key to the conspiracy, because if I am going to have cancellation I have to have some negative things and positive things to cancel. Look what happens. When I take that x and multiply it by here I’m going to get a what? I’m going to get a +x2y, but later in life I’m going to multiply these things together and notice what that is. That happily is a –x2y, so this term and this term will cancel each other, these two and these two cancel. Okay, now what about this term here? That’s a +xy2, but then later in life I’m going to have a –xy2, so in fact these will cancel with these and there’s nothing left. So everyone in fact does die off, so this really is the right factorization. And one way of remembering it is to remember that when you have the difference of two perfect cubes you are always going to have the number minus the number. And then what is going to be left is going to be always of the form x2 and then a y2, and then inside there is going to be an xy. And everything has to be positive here; the plus signs have to be here between them, because this minus sign will take care of all the cancellation. So that is the formula for the difference of two cubes.
And actually, the sum of two cubes works the exact same way. First there is going to be a factor, which looks just like this without the cube. So you have an x + y, same theme here; x2, xy, y2, and let’s figure how the signs should go. Just by thinking about it, we can figure out the formula. These last terms have to give me a +y3, so what do you think this should be? Well, obviously, since this is a plus this should be a plus. And now you can just make a guess as to what you think this should be. If I make this a plus here, I will have no cancellations because I will have pluses everywhere. So it must be a minus just to get all that annihilation happening. And in fact that is what it is. So you can see the differences between these two formulas. With the difference of two squares, I have a negative sign here, but then a plus sign here. And with the sum of two squares I’ve got a plus sign here and the negative sign here. So basically whatever you have when it comes to cubes, difference or sum, the same thing will be in this factor and then the opposite thing will be in this factor. Here’s a way of remembering it, but better still will be just to understand this because it’s not that hard.
Let me just do a couple of examples really fast for you so that you can see this in action. So let’s take a look at 8 – t3. And notice that really is the difference of two perfect cubes, because 8 is a perfect cube, it’s 23 minus––and then I’ll write this out to be really pedantic––t3. Now I can use the formula that is right over here, right, there it is, you can see it, the difference of two perfect cubes. Or, we could just think about it together, let us reason together. I know this is
Exponents and Polynomials
Special Factoring
Factoring Sums and Differences of Cubes Page [2 of 2]
going to be a factor that is going to look like 2 – t. Then the other factor is going to look like what? Well, that is going to first start out with the square of this, so that is going to be a 4, 22, and then I’m going to have the products of these two things. Well, that‘s a 2t, and what kind of sign am I going to want? Well, since I’m subtracting here I have the negative sign already there, I better put a positive sign here. And the last term has to be the t2 to top off the extra t to make this a t3. What sign do I want? I already have a negative sign, so I make this a positive. There’s the factorization. And you might want to check this by the way. Certainly, to be honest with you, I always like to check this myself because I’m so bad at arithmetic. This is an 8, this gives me the –t2, everything else should cancel. Here I see a 4t, but then happily here I see a –4t, here I see a +t2, but happily here I see a –t2, everything cancels out, great.
All right, one last one to see if you are really with it. In fact, do you know what I’m going to do, why should I be doing all the work? I think we should share the burden, don’t you think? So here’s 125A3 + 27B3 and you know what? Why don’t you do this one and I’ll just see if I can do it right after you do it. So try it right now and see what you can do. Okay, well, let me see if I can do it as well as you. I notice that this is the difference of two perfect cubes, because in fact this is (5A)3, 53 is actually 125, because it is 52, which is 25 times 5. And this is (3B)3, so this is the sum of two perfect cubes. So how does it go? Well, first of all, I have a factor which just has the sum of those two people uncubed, so 5A + 3B. And then I’m going to have this thing that starts off with the square of this, that is going to be 25A2. Then I’m going to have the product of these things, so it is going to be 15AB, and the last term is going to be this squared, which is going to be a 9B2, you see it? I have the formula there, of course, you see that. But do you see how I am using the formula by taking this thing squared, then these two things by themselves multiplied, and then this thing squared. Notice I am not putting them together with the cubes there, I’m peeling the cubes off because I’m factoring so I’m only looking at the inside now. What about the signs? Again, we can reason through this. The last term has to be positive, this is already positive so this always has to be positive, but I need cancellation to happen all over the place, so this must be the negative. And if you checked your answer you can see that it is negative.
So I hope you got this. And I also hope this is correct. So I hope that we both got it and it’s both right. Anyway, that’s how you factor the sums or differences of two perfect cubes. Not a big problem, see you later.