 Okay, today we are going to work on what are called x diagrams and this will help us with factoring. So we're going to be needing numbers that are going to add to a certain number as well as multiply to a certain number. So I want you to look at this x diagram, paying attention to the 6 and the negative 1 versus the negative 3 and the positive 2. And think about what happens with these numbers. Okay, so after giving you a couple seconds there to think anyway, hopefully you notice that if you multiply negative 3 and positive 2, you get the negative 6 and if you add negative 3 plus 2, you get the negative 1. So the number at the top is always going to be our two side numbers multiplied. The number on the bottom is going to be our side two numbers added. So let's check this one. So 1 times 7 gives us 7 and 1 plus 7 gives us 8. Okay, so that works. All right, now comes the fun part. So I give you the numbers at the top and the bottom and you have to come up with the numbers on the side. So I need two numbers that multiply to 20 and add to 9. Okay, if you're good with factoring, these might come to you pretty easily but let's go ahead and bring up the calculator. And in my y equals, I'm going to put 20 divided by x. And again, we're going to take a look at our table. So second table and when that comes up then I'm going to need to scroll down because I need these numbers to add to positive 9. So I think a 4 and a 5 will work. So over here I'm going to put a 4 and a 5. Okay, again, you know, you can do 5 or 4, 4 and 5 doesn't really matter. Okay, multiply to 12 and to negative 7. So two numbers that multiply to 12 and add to negative 7 are negative 3 and negative 4. Again, if that didn't come to you fairly easily, go look on your calculator and look down your table and see what happens there. Multiply to negative 15 and add to negative 8. Okay, so looking at this, let's see, so we're going to need two numbers that will multiply to negative 15 and add to negative 8. So let's think about how negative 15 works. So that will be, let's just look at the calculator. Y equals, I want to insert a negative sign. So this will be negative 15, let's see if I want to insert any more. Insert 15 and I can delete that too. Don't mind me while I've calculated our issues here. Okay, negative 15 over X and we can go to our table. And if you scroll around your table, you're not going to find a combination here that's going to work. So unfortunately, 15 only factors is 1 and 15 and 3 and 5, but none of those are going to work when we have negatives involved. So this one here is not factorable. And that's going to happen from time to time, so just be prepared for that. Okay, so now if I want to make an X diagram to factor an actual trinomial. So this is a polynomial, it's got three terms, so just why I called it a trinomial. And the leading coefficient here is 1. Okay, so as we saw with the factoring patterns then, you need to put the number that's multiplying, so that's your last number here at the top. So that is 18. And the adding number needs to go on the bottom, just like we've been doing here. So our adding number is negative 9. So negative 9 needs to go at the bottom. So I need two numbers that multiply to 18 and add to negative 9. So that's going to end up being a negative 6 and a negative 3. And again, if those don't come easy to you, just go back to your calculator, practice that kind of thing. Alright, now the best part here of setting up our X is we have our factors. Because this is an X squared here at the front, that means our first term of each of our factors is going to be an X. And then our X diagram here sets us right up. So X minus 6, X minus 3, and we are factored. Okay, next one. So again, let's go ahead and set up our X. And at the top, I'm going to put my constant term, so that's negative 42. At the bottom, I'm going to put a negative 1, because there's technically a 1 sitting there. So I need two numbers that multiply to negative 42 and add to negative 1. So that's going to be a negative 7 and a positive 6. Okay, so that means my factors are going to be X minus 7 and X plus 6. Okay, so like I said, once you have your diagram set up, it's actually pretty easy to go ahead and factor. Okay, so going back to our idea of the greatest common factor and applying that to factoring by grouping, we want to group the first two terms together. I'll do those here. And then we want to group the last two terms together. Do that in a different color. So we want to factor the GCF out of my first two terms. So I have a 3X cubed and a negative 7X. So my GCF there is going to be an X. And that's going to leave me with 3X squared minus 7. I've got my plus sign in between there. What's the GCF of 12X squared minus 28? My second set of parentheses. Well, let's see, that'll be a 4. Goes into 12 and 28. And that leaves me with 3X squared minus 7. So it's the whole goal of these problems. You notice if you treat this as one term and this is another term, look at what they have in common. They both have a 3X squared minus 7. So I can factor that 3X squared minus 7 out. Start my new set of parentheses. So if I take that out of my first set, that leaves me with just an X. If I take that out of my second set, that leaves me with a plus 4. So that's then how this is going to factor. 3X squared minus 7 and X plus 4. I know I'm done because the 3 and the 7 don't have any factors in common. They're prime and they're also not perfect squares. So I know I can't factor that any further. Okay, my next one, I'm going to go ahead and factor group the first two together. Then I'm going to group the second two together. And you notice I'm watching this negative sign right here by this 6. You definitely got to be careful with some of those things there. Okay, whoops, lost my parentheses. Okay, so taking out the GCF of the first two, that's an X. So that leaves me with X minus 3. I had this minus sign in here, don't forget, and I kind of wrote over it here. But remember I want an X minus 3 left. So that means I better take a negative 6 out of those second two terms. If I took a positive 6 out, my signs would be off. So I need to take a negative 6 out. So that way I have my X minus 3 left again. So looking at this is one term, this is another term. They both have an X minus 3 in common. And that leaves me with an X minus 6. And those would be my factors. Okay, keep practicing this, and good luck.