 Hello, welcome to our last video on factoring. So this one leads us back to the AC method, and then we're going to be doing X diagrams. So a typical trinomial, as it says here, is of the form AX squared plus BX plus C. So the A we've done before has always been a 1, where now we're going to be doing things that are other than 1. The first step of the AC method then is to multiply your A and your C together. So in example number one, we have 6X squared plus 17X plus 5. So my A in this case is going to be 6, my B in this case is going to be 5. So when I go to make my X for my diagram, I'm going to multiply the 6 by the 5, so in the top I'm going to put a 30. Then as we've been doing before, the middle number here, my middle coefficient, which is 17, is going to go into the bottom part of it. So I need two numbers that will multiply to 30 and add to 17. So that will be 15 and 2, or 2 and 15, order does not matter. I typically put the smaller one first, but that's just me. So I'm going to rewrite my polynomial now. So this is where here I'm going to use these two side numbers are going to be used to break up the middle term. So that's going to look like 6X squared plus 2X from one of my side numbers plus 15X from my other side number plus 5. Okay, now when I factor by grouping, just like we were doing in a previous video, I now need my GCF. So the GCF of my first two factors is 2X. When I factor that out, that leaves me with 3X plus 1. Now, remember my goal is to have a 3X plus 1 left in my second set of terms. And if I factor a 5 out, which is the GCF, I have a 3X plus 1 left. Okay, looking at this set as one term, this set as another term, these two have a 3X plus 1 in common, so I can factor that out. And when I factor that out, that leaves me with a 2X plus 5. And that is my trinomial factored. These can be some of the most frustrating polynomials to try to factor, because if you ever learn guessing and checking or anything like that, oh, you just can take forever trying to do some of these problems. Okay, my next one. So my A in this case is 3, my B or my C in this case is negative 28. Okay, so when I multiply negative 28 by 3, I get a negative 84. I haven't factored that one in a long time. Then my bottom number again is going to be my middle coefficient. So that is a 5. Okay, so let me bring up the calculator and again show you how we can use this. So in my y equals, I'm going to go ahead and type negative 84, 84, and divide that by X. Okay, when I go to look at my table, I'm probably going to have to scroll around a little bit here. So I need two numbers that will add to 5. Okay, so none of these are going to work. I'm probably going to want to go down, because I'm going to want the bigger number to be positive. I probably could have gone the other way too. And I think we're getting close. So if I add 6 and negative 14, that gives me negative 8. If I add 7 and negative 12, that gives me a negative 5. So I think if I flip those signs, I should be safe. So that means I'm going to do a negative 7 and a positive 12. And a word of advice here. If you have one that's negative one that's positive, stick the negative one on the left. Okay, that's just word of wisdom. So I'm going to use these two numbers then to break up my middle term. So that's going to look like 3X squared minus 7X plus 12X plus 5. All right. Oops, plus 5. I don't know where that plus 5 came from. Was it painted into my original polynomial? Sorry, that was a minus 28. Okay. Now I can factor by grouping. So group the first two together. Group the last two together. And what can I factor out of the first two? Well, that's just an X. And that leaves me with 3X minus 7. Plus looking at the second two, I can factor a 4 out. And that leaves me with a 3X minus 7. Looking again at this is one term. This is another term. They both have a 3X minus 7 in common. Factor that out. That leaves me with an X plus 4. Okay, fantastic. Last one of this way. And then we'll do one that's a little bit more visual again with the box method that we did a long time ago. So my first term and my last term, so that'll be a 4 times a 10. So that'll give me a 40 at the top and a negative 13 at the bottom. So because this number's positive and this number's negative, that means they're both going to need to be negative. So I think a negative 5 and a negative 8 is going to work there. So my whole trick about putting the negative 1 on the left isn't really going to fly, but that's okay. We'll see how to deal with that too. So breaking up my middle term, that gives me 4X squared minus 5X minus 8X and then minus a 10. Got my constant term right that time. Then we can group the first two together and the last two together. And again, watch that negative sign that's floating in the middle. Notice that 4X squared minus 5X only have an X in common. Factor that out. That leaves me with a 4X minus 5. Now I want a 4X minus 5 left, so that means I need to factor a negative 2 out of my second set. So I have a 4X minus 5 left there. And I see a 4X minus 5 in both of them. When I factor that out, that leaves me with an X minus 2. Okay, fantastic. So these trinomials, you know, even though they're ugly, you can factor them and it is possible to factor them most of the time anyway. There are some that are not factorable, of course. Okay, so again, we're going to do the same way, but then we're going to put it together a little bit differently in the end. Let's set up our X like we've been doing. At the top, again, goes our A and our C. So multiply 2 by negative 15. That gives me a negative 30. And my middle term is technically a negative 1 for that coefficient. So I need two numbers that multiply to negative 30 and add to negative 1. So that would be a negative 6 and a positive 5. Okay, this is where things get a little bit different. So what I'm going to do then is I'm going to set up my box like we did with multiplying polynomials. And in the upper left corner, I'm going to put my 2X squared, since that's my first term. In my lower right-hand corner here, I'm going to put a minus 15, because that's my constant term. And then in my opposite corners, I'm going to stick a negative 6X and a 5X, just like we were doing up above by breaking up that middle term. Okay, so now I need to look across my first row. So what do a 2X squared and a 5X have in common? Well, they have an X in common. When I factor that out, what am I left with? Well, I'm left with a 2X and a 5. So let's go at the top. Now you can ask yourself the question then, what do I have to multiply the 2X by that's sitting here to get to the negative 6X that I've got down below? Well, that's going to be a negative 3. And then our gut check, that's always a good thing to have, is if I multiply a negative 3 by a positive 5, do I get negative 15? And the answer is yes. So those are my factors. Looking across the top up here, my one factor is going to be 2X plus 5. And looking across the side over here, my other factor is going to be X minus 3. All right, last one. We'll do one more like this. So let's set up my X. I've got a 3 and I've got a negative 4. Multiply those together. That gives me a negative 12. My number at the bottom is negative 11 because that's my middle term. Two numbers that multiply to negative 12 and add to negative 11 are a negative 12 and a positive 1. Okay, that works. So setting up my box. Again, if the box doesn't float your boat then use the method we were using up above. Hopefully one of the two makes sense to you. All right, so 3X squared was my first term so that needs to go in this box. Negative 4 was my last term so that needs to go in that box. And then in my opposite two, I'm going to put my two numbers from my middle part here. So that's a negative 12X and 11X. Again, the order of these two does not matter whatsoever. So I need to look at my first row. So what do a 3X squared and a 1X have in common? Well, they have an X. If I factor that out, that leaves me with a 3X and a 1. And then what do I multiply the 3X by to get the negative 12X that's sitting there? That must be a negative 4. And then my gut check is a negative 4 times positive 1 equal to negative 4. That works. So that means my factors are 3X plus 1 and X minus 4. And that would be my final answer. Okay, try these out. Keep practicing them and good luck.