 So let's take a look at another topic, which is known as factoring. So this is something you've probably already seen in other contexts, but here's a reminder of how this works. First of all, this is based on the distributive property that a times quantity b plus or minus c is a times b plus or minus a times c. Now the important thing to observe here is on the left-hand side, again, the type of expression that we have is determined by the last thing that we do. So over here on the left-hand side, parentheses say do this stuff first, then I have a multiplication. So over on the left-hand side, I have a product. Over on the right-hand side, again, the type of expression is determined by the last thing you do. So here I have a multiply, I have a multiply, order of operations say do that first, and then I have an addition or subtraction. So on the left-hand side, I have a product. On the right-hand side, I have a sum or difference. And here's the key thing to remember. When we use the distributive property to go from a product to a sum or difference, we say that we're expanding the expression, and if we use the distributive property to go from a sum or difference to a product, we are factoring. In general, when we say to factor, what we want to do is we actually want to factor into expressions where the coefficients are rational numbers. Their coefficients are rational coefficients. So let's take a look at this. And here's probably the one and most important type of factoring, which is our removing of a common factor. So a little bit of analysis goes a long way, and the thing to notice here is the type of expression we have, again, let's check it out, order of operations, exponents first, exponents first, multiply next, multiply next. Finally, last thing we do is we subtract. So this is a difference, and so we want to use the distributive property to transform a difference into a product. So for convenience, I might write out my distributive property this way, A times B minus C is the same as AB minus AC, and I'll go ahead and compare my two expressions. So paper is cheap, understanding is priceless. So let's write out each of my terms. So 6x cubed, that's really 6x cubed is x times x times x, and x squared is 12x times x. So these are my terms written out, paper is cheap. And let's take a look at this. So I'm going to compare the distributive property to what I have, and the thing I might see is that I need something that's going to appear in both of the terms of my difference, and I have this xx in both terms. So A is this xx, and then B is whatever is going to be left over. So if I ignore the xx, I have a 6x left over here, and in the other expression, I have a 12 left over, and so I can remove a common factor of xx. And so I'll take that common factor out, and so there's my distributive property, AAB times AC. I'll remove that common factor, and what I have left over, 6x minus 12, and it's helpful, but not absolutely necessary to remove any common constant factors as well. So here, both 6 and 12 are multiples of 6, 12 is 6 times 2. So again, I have that additional common factor to both. Both of these have a 6 in them, so I can remove that 6 as well. And finally, I'll restore my exponential notation, xx is x squared, and there's my factorization. Well, let's take a look at another example. Here I have a couple of terms there, and again, paper is cheap, understanding is priceless, and again, I have a sum of things. The last thing I do here is I add things. I have a sum, and I can factor by turning this sum into a product. So let's go ahead and take a look at that, and I have this x cubed y, xxxy, xy squared, xyy, xy cubed, xxyy, and I'll write things out without exponents, and I'll look for things that I have in common to all of them. And here we have to be a little bit more careful. Let's see, everything has at least one y, and everything has at least one x in them. So it appears that the common factor is going to be xy. So xy appears in this, xy leaves xx, xy leaves y, xy from here leaves xyy, and this xy appears in all three terms. So I can factor that out from all three terms, leaving this and restoring my exponents looks something like that. And again, quit check, this is an important idea. I have transformed this sum into a product. So again, last thing I do on the right-hand side, parentheses, do this stuff first times xy. Well, let's take a look at something else. So again, it is helpful to look at the expression and see what we have. Order of operation says do stuff inside parentheses first. Order of operation says do the implied multiplications next. Order of operation says the last thing I do is I'm going to add these things together. So I'm adding, and I can factor. So I want to transform this sum into a product. And if only I found something that was a common factor in both terms that I'm adding. Oh wait, x minus 4 appears in both. So I can remove x minus 4 from both. And so that common factor x minus 4 comes out. What's left over x plus 8 plus x squared plus 4x minus 5. So that's x minus 4 times the rest of this. So I can do a little bit of algebraic cleanup. This x squared plus 4x minus 5 plus x plus 8. So that's going to be x squared x plus 4x that's 5x. 8 minus 5 is 3. And so there's my factorization of this expression equal to this. Now, here's an important thing to check. I should at the end of factoring have a product. And so on the right hand side, again order of operation says do this first, do this first, multiply, and that's the last thing we have to do. This is actually a product, so we have done a factorization. And I'll emphasize that by giving a bad example at the very last. So again, I want to transform a sum or difference into a product. And in this case, I might say, ah, well, those first two terms have this common factor of x. So I'll remove it. And there's my factorization. Well, no, not really. It is true that x squared minus 5x minus 8 can be written x times x minus 5 minus 8. The problem is it's not a factorization. Again, let's take a look at the type of expression we have. Parenthesis say do this first, then order of operation says do the multiplication, and then do the subtraction. The type of expression is determined by the last thing that we do. And so the last thing that we did is subtract. This is a difference. This is a subtraction. It is not a product, and because it's not a product, it's not a factorization. So we have not factored. We have turned this difference into a difference.