 Alright, welcome back. Today we're going to talk about factoring by grouping. Now what we're going to be doing today is you can factor, you can take these polynomials that we have here and we can factor these polynomials by grouping them into smaller groups so that they are easier to factor. Now this factoring process is going to be useful for solving, it's going to be useful for graphing, it's going to be useful for all sorts of different stuff that we do with polynomials. But we've got to learn the basics before we get to the more advanced stuff. So here we go. So what I'm going to do is I'm going to take this polynomial and I'm going to split it into two groups. Again that's what we call factoring by grouping. I'm going to split it into two groups. 2x to the third plus x squared. Now depending on the difficulty level of the problem, for the most part when you split this up it's these two left terms are going to be your one group and then these two right terms are going to be your second group over here. For the most part that's kind of how you split them up. So that's kind of what this little bar here is. I'm splitting the left two terms and the right two terms into two separate groups. I'm going to factor them individually and that's why we call it factoring by grouping. But anyway, here we go. When you factor, you want to look for something in common. So in this case right here I see an x squared that is in common. Now this x squared I'm going to take that out of both of these terms. I see two of the x's here, two of the x's here. So I'm going to take that out of both terms. Now when we factor something out we are dividing this out. That's technically what we're doing. So we're taking 2x to the third and we're dividing out x squared to get 2x. We're taking x squared dividing out the x squared to get 1. x squared divided by x squared is 1. A common mistake that happens is that usually when I say take out or factor out a lot of students just assume, oh you just take it out and you wouldn't have anything right here. We're technically dividing so when you take x squared divided by x squared you just have a 1 left over. Anyway, now onto the next one. Notice over here that I don't have any variables in common but I do have numbers that are in common. The 8 and the 4 are both of those are divisible by a positive 4. So the positive 4 is what I'm going to factor out. What I have left is 2x plus 1. Now that's one step for factoring and I'm going to do a couple more steps. Now notice here that this 2x plus 1 right here, 2x plus 1 and I have another parenthesis over here that's 2x plus 1. Now that's on purpose. That's actually designed that way. Now what we're going to do here is we're going to factor one more time. It's the exact same factoring that we did up here twice but now it's with kind of a larger group. It's with the parenthesis and it says just single numbers. So notice that this term over here and this term over here both of them have a 2x plus 1. So that is what I'm going to factor out. I'm going to take 2x plus 1 and I'm going to factor that out just like up here. Just like up here I had x's in common, x squared in common so I factor that out. I brought it out front. Just like over here I had a 4 in common so I brought that out front. It's the exact same step. We're taking this 2x plus 1 since it's in both terms. I am factoring it out front and then what do I have left over? I have an x squared left over and I have a plus 4 left over. Now that right there is the completely factored form of this polynomial. Now one thing that you usually have to check is that right here you want to check to see if this is factorable. This actually, no, is not factorous. Sometimes with these x squareds here you'll be able to factor this but in this case we won't be able to factor this so that's right there as good as we get. Right here is as good as we get with this one. That's as far as we can factor that. Now with the second example that I have over here, go a little bit faster through this. Just kind of another example for you guys to see. Alright so I'm going to look at these two here. Actually I'm going to rewrite this. Let me rewrite this. Split that up into negative 25x plus 25. The reason I chose this example is for this second part over here. I'll show you in a minute. Over here I see variables in common. I see an x squared in common so that means I have x minus 1 is what is left over after I factor that out. Over here I see 25 in common. Now not only do I see a 25, I see a negative 25. Now when we factor, we always want our beginning number here to be positive. Now if I factor out just a 25, let me show you. If I factor out a 25, this is what it would look like. If I just factor out a 25, a positive 25 because that's what I see in common there. Now okay that's all fine and dandy I can factor that out but then look. I have an x minus 1 and a negative x plus 1. Those are not the same. Like over here, I want them to be the exact same. If I look at them here, they're not the same so I can't factor them out. Actually what I did is I did an incorrect factoring there. Now I kind of did that on purpose to show you why we need to do it this way. Instead of factoring out a positive 25, I want to factor out a negative 25. Now what happens is you take this, you take negative 25x divided by negative 25. The result is x. Positive 25 divided by negative 25 is a negative 1. When I factor out a negative 25, it changes with the signs that are here going to be a little bit. It changes these signs a little bit but notice that when I change those signs, this group and this group right there are now actually the same. That's kind of the point. That's what I wanted. Now I take that group since that's what I have in common. I'm going to factor that out one more time. And left over, I have an x squared minus 25. Now I said with the last example, whenever you see these x squared, you've got to be kind of curious because can I factor this? In this example over here, no, I could not factor that. But this example over here, I can actually factor that. This is what we call a difference of a square number and a square number, difference of two squares. I can actually factor that. What that's going to factor to is x minus 5, x plus 5. Think of it this way. x times x gives me back to x squared. Negative 5 times 5 gives me negative 25. That works. Now those are not the only two multiplications that you do. You also do, what I'm doing here is I'm doing my foil. I'm doing my outers and minors. I'm checking to see if I did this correctly. First and last was the x times x to get x squared. Last is negative 5 times 5 to get negative 25. Now I'm going to try the outers and the inners. The outers and the inners here. So the outers, x times 5 is 5x. Inners are negative 5 times x, which is a negative 5x. Positive 5x, negative 5x makes a zero. I don't have an x term right here in the middle. So that actually makes sense to factor that way. It makes sense that I have those type of factors. So I was just checking to make sure I did that correctly. So again, that's what we call a difference of two squares type of factoring. And again, whenever you have those x squares, you always have to check to see if you can factor that. This one over here, we could factor. This one over here, we could not factor. If it was a negative 4, then yes, we would be able to factor. But in this case, we can't. That's factoring by grouping. Just make sure you have to do multiple steps of graphing, or graphing, multiple steps of factoring. When you split them up, it's usually the left two and the right two. Usually that's what it's going to be, depending on the difficulty level of the problem. Alrighty. Thank you for watching the video, and we'll see you next time.