 Welcome to a three-part series on factoring. So today we are going to start with the greatest common factor, which is also known as the GCF, which means the largest factor that both terms have in common. So we're going to have to factor some of these numbers. I start with numbers and then I move into some expressions here. So the first one we want to find the GCF of is 16 and 24. Now some of you may be able to look at 16 and 24 and pick out the greatest common factor. That's fantastic, but I'm going to also walk you through then how you can do this by hand. So let's look at 16. So 16 can be factored as 1 and 16. That's always the easiest one to forget. It can also be factored as 2 and 8. It can be factored as 4 and 4. So those are my three different ways I can factor 16. Do the same thing for 24. I'm getting more room on that one. 24 can be factored as 1 and 24. It can be factored as 2 and 12. It can be factored as 3 and 8. And it can be factored as 4 and 6. So comparing these two, so let's go ahead and list them out. So 16, our factors are 1, 2, 4, 8, and 16. 24, our factors are 1, 2, 3, 4, 6, 8, 12, and 24. So comparing these two, it looks to me like the biggest number that is on both of these lists is 8. So the GCF of 8 and 16 is 24. Okay, now if it's difficult for you to come up with these factors on your own, we can use the calculator. So let me bring up the calculator here. What you want to do then is what I described down here below. If you want to factor a number, you want to go to your Y equals menu. So click on Y equals, type in your number. So let's say we want to factor 16, so 16, and then divide that by X. Okay, now what this will do is it will factor this number. So if we go to second table, you will see all these different factors. So zero error because nothing times zero will give you 16, so ignore that one. 1, 16, 2, 8, 3, oh isn't this one lovely, 5.33333. Okay, ignore anything that has a decimal. We're only looking for whole numbers here. Then we have 4 and 4, 5 and 3.2, 6 and 2.66667. So those we're going to ignore, and then you can keep tagging down to get more values. But obviously, so here's 8 and 2, so things are going to start to repeat. So just toggle up and down your table in order to see what those factors are, and that will help you then with your calculator. Okay, right, fantastic. Okay, now let's go to some expressions. So these have some X's in them. So we're going to do the same thing. I'm going to write these out. So let's write out 2X squared. So factors of 2 are 1 and 2, obviously. Okay, so 1 times 2 times X squared is better known as X times X. And you don't have to put this 1 every time, by the way. I just put it in there. I guess to be consistent with what I did up above. Then 8X cubed, let's go ahead and write out what that looks like. So 8 can be factored as 2 times 2 times 2. Just to break that one down into prime factors. Then X cubed is X times X times X. So there's 3X's there. Alright, and then again let's circle what they have in common. Let me do that with a different color this time. So I see a 2 in common with both of them. I see 1X and I see another X. Okay, so my GCF of this one then is 2X squared. So you'll notice we're doing something different with our coefficients than we are with our variables. So with our coefficients you want the biggest number that goes into both of them. With our variables you want the same thing, but you notice it's actually the smaller of the two exponents. Okay, so something to keep in mind for the future. Okay, building up a little bit more, we've got 3X times X minus 5 and 7 times X minus 5. Your first instinct might be to multiply this out. Don't do that. It's already factored. So to list out my factors, I really don't have to do anything. So this would look like 3X. Yeah, let me not even put an equal sign there. That doesn't make any sense. All right, so this would just be 3 times X times X minus 5. I'll put that in parentheses. And this one's just 7 times X minus 5. What do both of them have in common? I see an X minus 5 in both of them. So let's go ahead and do that. If you were to multiply it out, you'd have to refactor it and then you get right back to where you started. That doesn't make any sense. So for this one, the GCF is X minus 5, which is technically a factor of both of them. Okay, all right, moving along. So now let's put this idea of the GCF into some actual, they look more like expressions, so they get a little bit harder here. Okay, so 36X plus 16. So I'm looking at 36X as one term and 16 as another term. So I want to think about what is the biggest number that goes in a bull 36 and 16? So we'll deal with the coefficients first and then we'll look at the variables next. So the biggest number that goes in a bull 36 and 16, leave is 8. So, oh no, 8 doesn't go into 36. It would be 4. So we can factor a 4 out. And you notice this one has an X. This one does not. So these don't have any variables in common. So our GCF is just going to be a number. Okay, so I put my parentheses here. Let me make that 4 a little bit better. Okay, so I got a 4 here. So 4 times what will now give me this 36X? Well, 4 times 9 is 36. And then I need an X there to make my X. And I got my plus sign. Okay, 4 times what will give me 16? Well, that's a 4. And then this thing is now factored. Now, how do we know we're done? How do I know that 8 that I said originally wasn't correct or was not my greatest common factor? Well, that's where you've got to look at what's left inside of these parentheses. So 9X plus 4. Do 9 and 4 have anything in common? No. So that's how I know I'm done. That's how I know I did pick the greatest common factor to start with. So what I'm saying is if you saw a 2 in common at first, you could take the 2 out. But then you'd have to notice that whatever you have left has another 2 in common. Okay, so that's where you've got to always go back and double check what you did. Okay, and another way also to check this is you can multiply this back out like we did with the last set of videos. So you could multiply this thing back through to check our answer. So this is our check. So when you multiply 4 by 9X, you get 36X. You multiply 4 by a positive 4. You get a positive 16. So it checks. Okay, good. Next one. This one's got some Y's in it. So again, we want to look at 9Y squared is one term and 3Y is another term. So let's look at the coefficients first. So 9 and 3, 3 goes into both of them. And let's look at the variables next. I see a Y squared and I see a Y. So they both have a Y in common. It's a smaller of the two exponents. Okay, ask ourselves the same questions. So 3Y times what will give me the 9Y squared? Well, 3 times 3 will give me the 9. And then Y times Y will give me a Y squared. I need a plus sign here because you don't want to lose your terms. You notice here I had back in this number 4, I had two terms and I end up with two terms of my parentheses. This one I have two terms. I better have two terms of my parentheses. So that way when I multiply them back out, they will check. So 3Y times what will give us 3Y? Well, 1. Again, multiply it back out and check it and make sure that it works. Okay, next one. So this one looks like 2X times X plus 5 minus 3 times X plus 5. Again, do not multiply this back out. You're going to be doing too much work. So look at this as the first term and this as the second term. So what do those two terms have in common? What's their GCF? Well, like we did up above, I see an X plus 5 in both of them. So let's factor an X plus 5 out. Now you'll notice I put all that in parentheses and that's because there's two terms with that one instead of just the one that we had up above. Okay, now what do we have left? So if I take an X plus 5 out of my first term, let's cross that off, I have left a 2X. If I take an X plus 5 out of my second term, what do I have left? A minus 3. So that is my factored form. Okay, again you can multiply this back out and make sure you get back to where you started, but you cannot be careful when you multiply this one back out. You may have to multiply both of them out to make sure they are actually equivalent. Fabulous. Alright, last concept here is what's called a difference of squares. So I put over here some perfect squares from the book and how they got them. Okay, so these are kind of good to talk in the back of your mind. You don't have to memorize them but just know. And then they say, so what you have to do is you have to identify the values of A and B. So basically what, looking at number 7 here, what when you square it will give you X squared and what when you square it will give you 100. So that will give you your A, that will give you your B. Okay, so this is technically A squared and B squared. And then 2, so it is a difference of squares. You better make sure you have a minus sign in here and then kind of follow in this pattern. So what when you square it will give you X squared. Well that's just an X. Set up my next set of parentheses here, an X. What when you square it will give you 100? That's a 10. So we know we're going to have a 10 on each one. And then we just need a minus and a plus or a plus and a minus. It doesn't matter what order you go in. So minus and plus. Okay, then when you multiply this back out, you'll notice that you get X squared plus 10 X minus 10 X. So those middle two terms cancel, which is why this works for a difference of squares. Okay. All right, next example. So this one gets a little bit trickier. So we got to think to ourselves what when we square it is going to give us 4 Y squared. So again, let's take the number and the variable separate for the coefficient. So what when I square it will give me 4? Well, that would be a 2 because 2 squared is 4. Looking at what they have up here on our chart. What squared will give me Y squared? Well, that's just going to be a Y. So that means my A in this case is 2 Y. So that's going to go with the front of both of these sets of parentheses. Okay, do the same thing for the next one. This one's just a number, so that's a little bit easier. So what squared will give me 25? And that would be a 5. 5 times 5 is 25. Give 1 a minus, give 1 a plus. Doesn't matter what order, you're factored. Again, how do I know I'm done? Well, a 2 and a 5 don't have anything in common. So I can't factor either of these two terms anymore. Okay, then the last one. So I see 16 m squared and 49 n squared. So what when I square it will give me 16 m squared? Well, what squared will give me 16? That would be a 4. What squared will give me an m squared? That would be an m. And then doing the same thing over here. So what squared will give me a 49? That would be a 7. What squared will give me an n squared? That will give me an n. Okay, so going to throw these into the equation Oh, did anybody notice what happened with this one? So you notice number 7 had a minus. Number 8 had a minus. What does number 9 have? It has a plus. So this is a sum of cubes or a sum of squares. Not a difference of squares. So this one does not factor. Man, so we just did all that work for nothing. I just wanted you guys to see how this one would maybe, you know, it looks perfectly innocent. I see squares everywhere, but it's got a plus in there instead of a minus. So this one does not factor. If we had tried to factor it as 4 m minus 7 n and 4 m plus 7 n, if you multiply that back out, you're not going to get back to where you started. So, you know that one's wrong. Okay, fantastic. So we'll do more factoring then very soon. Thank you for listening.