 This video is called Factoring When A Does Not Equal One. So what's different about this problem to factor than what you have been used to is this number right here in front of the n squared isn't one. It's something bigger. Now this does make the problem a little more complicated. It's going to be a little bit longer. You're probably not going to like it as much. I hate to say that because you're going to have to be willing to do what we call guess and check, which means you'll have to write something down. You're going to have to be willing to erase it and try again. So I highly recommend you use pencil when you solve problems like this and dig deep into the depths of your heart and get a lot of patience. Okay? So in a problem like this, first thing, ask yourself if you can take out a GCF. I can't think of a number that can divide out of 6, 23, and 7 easily. I don't have an n in all three terms, so I don't have a GCF to take out. So now I'm going to ask myself what multiplies to give me 7, adds to give me 23, and multiplies to give me 6. Well I know the 7 is 7 times 1 and the 6 could either be 3 times 2 or 6 times 1. So what I have to do now or what takes a little time is figure out what combination of 3 and 2, 6 and 1 and 7 and 1 works for me. This is where the guess and check comes in. Well fill out as much as you can. So start with your two parentheses and I'm going to go ahead and put the 7 and the 1 in here because the reason I did that is because to multiply to give me 7, I really only have one choice, the 7 and the 1, but the multiply to give me 6, I have more than one choice so that's where I'm going to play around a little bit to see what works. I'm also going to go ahead and the n squared becomes n and n, so that will help me a little bit. Now all I have left is to figure out what numbers go in front of the n. Will it be the 3 and the 2, the 2 and the 3, the 6 and the 1, or the 1 and the 6 and then I have to figure out what signs go in front of the 7 and the 1. We'll take it one step at a time. I'm going to save the signs for the end. Alright? So here's all you can do is take a guess. Okay? I'm going to start at the beginning with 3 and 2. Now what we have to do is foil and when we foil, we want to decide if this will get us back to where we started. So we're looking to get this positive 23. Let's see what happens. 3n times 2n is 6n squared. That's good so far because it gets me back to the beginning. 3n times 1 is 3n. 7 times 2n is 14n and 7 times 1 is 7. All I did there was foil. And now tell me, is there any way that the 3 and the 14 can add up to be a positive 23? There just isn't. So I know that using 3 and 2 here just didn't work out. That happens. Let's try the next one. Let's just keep going in order and try the 2 and the 3. So now let's foil and see what happens. When we multiply the first 2n times 3n we get 6n squared. So that's a good start. Then when we multiply the outsides 2n times 1 we get 2n. 7 times 3 we get 21n. That's the insides. And then 7 times 1 the last we get 7. So now is there any way that a 2 and a 21 can add up to a positive 23? There sure is. They both have to be positive. So I can go ahead and put my 2 in here, my 3 in here and for 2n to be positive that came from 2n times 3 and gave you a positive 6n squared. 2n times a positive 1 gives us the 2n and a positive 7 times a positive 3 gives us that positive 21. So this would be our final answer. We were lucky. It only took us 2 chances before we stumbled on the right one. It could have taken up to 4 because I would have had 3 and 2 or 2 and 3, 6 and 1 or 1 and 6. Let's try a second example. Same thing as before. The 7, we're looking to see what multiplies to give me 7 adds to give me negative 26 to give me negative 28. Well the 7 can be 7 and 1 and then the 8 will either be 4 and 2 or 8 and 1. So like the last problem, we can have up to 4 tries before we get it right. Sorry about that, the phone rang. We can have up to 4 tries to get it right. You just have to start somewhere. So let's split up the 7x and the 1x. I'm going to do that because the 7 really only has one choice. This is where all the choices come from. So you just have to be brave and try something. Let's go ahead and try the 4 and the 2. So when you foil and you do the first 7x times 1x, you get 7x squared. Then when you do the outside 7x times 2 you get 14x. The inside 4 times 1x is 4x and the last 4 times 2 is 8. There is no way that 14 and 4 can add up to a negative 26. So that tells me my guess of 4 and 2 is wrong. Let's try, excuse me, sorry about that. I hate it when that happens. Let's try 2 and 4. Let's switch it around. So when we foil, 7x times 1x is 7x squared. Outside 7x times 4 is 28x. The inside 2 times 1x is 2x and the last 2 times 4 gives me an 8. Is there any way a 28x and a 2x can add up to a negative 26? There is. I have a positive 2 and a negative 28 will add to negative 26. So I feel really really good about putting a 28, I'm sorry about putting a 2 here and a 4 here. I just now have to really be careful to get my signs right. I needed the 2x to be positive. Well, how did I get 2x? Not there. I got up by doing the 2 times 1x and for it to be positive I have to put a plus. Then I needed the 28x to be negative. So when I do 7x times 4x I need that 4 to be negative to get my negative 28x. So the quantity of 7x plus 2 times the quantity of 1x minus 4 is my answer. If we're through let's try video number 3. Before I factor this I can definitely take out a GCF. I can divide out a 2 from each of these terms. So I take out my 2 and what am I left with? 5d squared minus 14d minus 3. I just made this problem a lot simpler on myself because the numbers that I have to factor with are smaller and they have less factors. So ask yourself what multiplies to give you 5 adds to give you negative 14 and multiplies to give you negative 3. Well 5 can only be 5 and 1. 3 can only be 3 and 1. So let's go ahead and do our 2. Sorry that's annoying. Let's go ahead and do our 2. Make our parentheses. Let's go ahead and have the 5d and the 1d be put in place. Now all I have to do is figure out will I have 3 and 1 or 1 and 3. So you just have to kind of go with your gut and try something. It really doesn't matter. I'll pick 3 and 1. Let's go ahead and foil these 2 binomials and see if we get back to the 5d squared minus 14d minus 3. So when we do the first 5d times 1d is 5d squared. The outsides 5d times 1 is 5d. The insides 3 times 1 is 3d. The last 3 times 1 is 3. I don't think there's any way that a 5d and a 3d can add to a negative 14. So I think I must have chosen wrong. So that's okay. There's only one other choice. So I think I'll get it right this time. Instead of 3 and 1 let's try the placement of 1 and 3. So when I multiply my firsts 5d times 1d will be 5d squared. My outside 5d times 3 is 15d. My inside 1 times 1 is 1d. And my last 1 times 3 is 3. Is there any way that the 15d and the 1d can add up to a negative 14? There sure is. When the 1d is positive and the 15d is negative. So I am ready to put the 1 and the 3 in my pots. And now all I have to do is put the signs right. Be very careful. I need the 15d to be negative. Well how did I multiply to get 15? It was when I did 5d times 3. So I needed a negative there. So the 5d times negative 3 becomes my negative 15. And it looks like I needed the 1d to be positive. Well what did I multiply together to get the 1d? It was this one and this 1d and since I needed to be positive I'll put a plus. So my answer don't forget about that 2 that you factored out at the very beginning. It's 2 times the quantity of 5d plus 1 times the quantity of d minus 3. That's the problem for the video. We're going to factor 6y squared plus 15y plus 6. Again I think I can factor out a GCF. I can take out a 3 from each of these terms. If I can do that it's really important because not only will it be completely factored but it makes this part to factor this out quite a bit easier. So now we're looking to see what multiplies to give me 2, adds to give me 5 and multiplies to give me 2. Well for this 2 my only choices are 2 and 1 and for the last term 2 my only choices are 2 and 1. So let's go ahead and bring down the 3. Make my 2 parentheses. Let's go ahead and place my first terms and now I have 2 choices. Will it be 2 and 1 or will it be 1 and 2? I am going to play a hunch simply because I've done this for a long time. You guys the more you do the more you'll kind of be able to guess better at what order these will go in. I'm going to go ahead and put in, I'm going to guess a 1 and a 2. So let's foil and see what happens. 2y times 1y is 2y squared. The outsides 2y times 2 is 4y. The insides 1 times 1y is 1y and 1 times 2 when you multiply the last is 2. Is there any way a 4y and a 1y can multiply together to give you a positive 5? There sure is if they're both positive. So I'm going to put my 1 here and 2 here and now I'm simply going to be very careful picking my signs. This one will actually be fairly easy because both the 4y and the 1y need to be positive so I know I have pluses for both of my signs. So my answer, don't forget about the 3 you took out at the beginning is 3 times the quantity of 2y plus 1 times the quantity of y plus 2.