 And welcome back. Today we're going to talk about polynomial long division. Done a couple of videos in the past hour about adding, subtracting, multiplying, and dividing. And so now what we're doing is we're dividing. I'm going to do a couple of different videos on this. I'm going to show you polynomial long division. And then I'm going to also show you synthetic division. And then I'm also going to do synthetic substitution and all sorts of different stuff. But I'll do separate videos for those ones. So okay, let's get into polynomial long division. Anyway, what we're doing is we're taking a polynomial divided by a polynomial, just like you're adding, subtracting, and multiplying. Now we're doing division, just doing all the basic operations. So we're taking this polynomial divided by this one. This is a cubic. Make sure you notice the higher power here. Cubic trinomial divided by a binomial, a linear binomial. So now from this, what we want to do is we want to rewrite it. Rewrite it just like you would for regular long division. So notice that what we have here is going to be what we're dividing by. This is called the dividend. Sorry, I had to remember the word. This is the dividend. This is the divisor. This is the number that's going to go inside the house. That's what's commonly used. This is going to go inside the house. Now notice that I actually rewrote this. This was not in standard form. You always want your higher powers first. So notice that the 2y to the third is going to come first. Negative y squared comes second. And then you're 25 last. And then what you're dividing by always goes on the outside of the house. So the setup for long division is actually the very, very same as what you learned in elementary school. Now what we're going to do is we're going to start dividing. Now again, you ask yourself the same questions. Now this might look really difficult at first, but if you know how to do long division, you know how to do this. You know all of the steps. The steps are the exact same as regular long division. Of course it's going to be more complicated because we're multiplying and adding polynomials and variables and numbers and all sorts of different stuff. But anyway, stick with me. So here we go. What we've got to do is figure out what do I need to multiply to get really, really close to 2y to the third. So y times what gives me 2y to the third? Well, 1 times 2 is going to give me 2. Let's get rid of that too. To get to 2 and then y times what to give me y to the third? Well I would need y squared. So 2y squared times y is going to give me 2y to the third. And then also, when you do your long division, this number times what you're dividing by always gives you your number under here. But since we're dealing with polynomials here, what we've got to also consider is this negative 3 here. So this 2y squared doesn't only just multiply times y. We also have to multiply times this negative 3 here. So we also got to take 2y squared times negative 3 for a negative 6y squared. Now you can already start to see how complicated this process can be. You have to really keep track of how to multiply, how to divide, how to add everything together, things like that. So to recap on this, what we're doing, y times what is going to give us really close to 2y to the third? And we figured out to be 2y squared is going to be that number. And now, once we get this number down here with long division, what do we do? We simply just subtract these numbers. So here we go. Now I'm going to put parentheses around this and subtract. This is something you don't normally do with long division, but I do this for this because in the second step, I'll show you here in a moment. 2y to the third minus 2y to the third is going to be 0. Not a big deal. Everything goes away. That's fine. The second step here, this is what always tricks everybody up. Negative y squared minus a negative 6y squared. That's why you got to put parentheses around it. That's why you got to put that subtraction sign out there because if you notice minus a negative 6y squared, we are actually adding 6y squared. So this is negative y squared plus 6y squared to get a positive 5y squared. That right there, that tricks everybody up because of that. So what I always suggest, I always tell students, make sure put parentheses around what you're subtracting. Put this subtraction sign out front. Make sure and do that each every time. 100%, 90% of the mistakes that I see that are made is because this is improperly used. These parentheses and this subtraction, they're not put up there. Students try to skip that step to try to save time or something like that, and they end up doing the process completely incorrect just because of this one simple little step here of changing that sign. So make sure you put the parentheses and that subtraction outside. That's a big, big part of this. Now what we do, just like long division, after you subtract these, you just bring down the next number. So in this case, I'm going to bring down the positive 25 just like you would for regular long division. And now we start the whole process over again. Now we go back here. Y times what is going to give me 5y squared? Y times what gives me 5y squared. So that's going to be a positive 5y times y is going to give me y squared. So 5 times 5y is going to be 5y squared. And then also I'll remember to multiply by this negative 3 here. So 5y times negative 3 is a negative 15y. But parentheses, please, please, please put parentheses around what you're subtracting. All right, now this problem, I chose this for a couple of different reasons because there's actually something weird that happens here. Okay, this 5y squared minus 5y squared, that's zero. Okay, nothing fancy about that. But here's where it gets a little bit tricky. 25 minus negative 15y. Now hold on a minute. 25 and this is going to be a positive 15y. That makes no sense. Those are not going to combine. Okay, you should realize that right away. Those are not going to combine. Well, what the heck, what are we going to do with this? Actually, it's relatively simple. If they do not combine, just rewrite it. Just rewrite it as you would see it. Okay, so this is a positive 15, positive 15y. 25 is going to just come right out here. If they do not combine, then don't force them to combine. Okay, don't force that upon them. Okay, this is going to be a positive 15y and then 25 is just going to come down there with it. And basically what I'm going to do, you can actually think of it this way. The 25 got brought down too early. Okay, notice that there's a gap right here. We have to the third power to the second power, but there's no why between them. So you could also say that the 25 was brought down too early. And so therefore, we cannot combine these two numbers in this second step here, okay? So just make sure that might happen every once in a while, especially if you have a gap with your variables, like I said, to the third, to the second. There should be a why to the first power here somewhere, but of course it's not there. So sometimes you might bring numbers down too early, which is okay. There's nothing wrong with that. If they don't combine, just rewrite them. Not a big deal. All right, then we start the process all over again. So I need to figure out why times what is going to give me 15y. Well, I already have the why. I just need the 15. So plus 15. So 15 times y is 15y. 15 times negative 3 is a negative 45. Okay, notice that these numbers here will combine. Those are both constants. So those will combine. Get that negative out front, okay? 15y squared, off y squared. 15y, excuse me, minus 15y is zero. That's what we want. 25 plus 45 is going to be a nice even 70. Okay. And then from there, I have nothing else to bring down, so I'm done. This is your remainder. This is your remainder. Okay, remember back to your middle school days that if you had any number down here, that was your remainder. So usually you would put the r70 or something like that. Well, we're going to do something a little bit different. We're going to make this a fractional remainder. Okay? So this is going to be plus 70 over what we're dividing by of y minus 3. Okay, and then this here, this top part is our solution. Okay? So when we take this polynomial, this trinomial, divided by this binomial, this is what we get. Okay? Doesn't divide evenly, but hey, that's the process. Alrighty. Now, this is a little bit of a lengthy process. You might need to watch the video again to go over all the finer things. Just to recap a little bit, a couple of things. If you know how to do long division, you know how to do this process. Okay? So going into this, I know it looks confusing, but just remember, you already know how to do this. So you're unlike some other mathematics that you might be learning, you actually know this process. Okay? So take that into account while you're doing this. Second thing, make sure, please, please, please, use parentheses here when you're subtracting, and remember that you're subtracting the second term. This is not a negative seven. This is going to be a positive five. Okay? That right there is the number one mistake that I see students make. The third thing, if you get a number for a remainder, that's going to be a fractional remainder, in this case, 70 over y minus three. The y minus three is what we're dividing by. Okay? Anyway, that is polynomial long division. Thank you for watching, and we'll see you next time.