 Okay, before we continue with our polynomial long division, because we're going to do one really large one right now, just to finish off a long division of polynomials, just a quick little update. At the beginning of this series, I mentioned that I was going to grow the beard again, and as of October 1st, I decided to kick dad into gear. So I'm going to be bearded chicho again, and instead of a clink cut chicho. So you're slowly going to see beard growing. And for anyone that can grow a beard, or who has thought about growing a beard, I highly recommend living with a full beard for at least one year in North America anyway, at least in North America. I'm not sure how it works out in the rest of the world, because your perspective changes your interaction with the world around you changes, because other people's perspective of you changes. It's just a side note, because when you grow a beard, other people see you as bearded, but when you look out out of your eyes, you don't see a beard. So you're still the same person as you are. When other people look at you, they see a beard instead of a clean cut. So their perspective of you changes, and that affects you in a certain way, and your perspective of your surroundings changes. So again, just a side note, if you've ever thought about, or if you can grow a beard, and if you've ever thought about growing a beard, I highly recommend living with a beard for at least one year in Canada and the US. I'm not sure what the situation is like in the rest of the world. Couple things before we start off on the long polynomial, the long division, right? I already mentioned that you have to put your terms in descending order, right? So if you have X to the power of four, that has to go first and then X to the power of three, two, one, zero, or your constant, right? So all the X's, all your terms have to be in descending order, both for your product, your divisor, and your dividend, and your divisor, okay? So all your terms have to be in descending order. Another thing that you have to keep in mind, which goes back to series two, really, where we talked about, you know, adding like terms, right? If you have X to the power of two, you can't add that to X to the power of three. This comes, this is really important when you're doing long division of polynomials, because sometimes you're missing terms. For example, you could get a polynomial where you have X to the power of four and then X to the power of two, so you're missing your X to the power of three. What happens is you might not have your terms lining up with the proper order. One thing you can do is just keep in mind that, you know, you can't add like terms, so the highest order goes first when you're adding it. The other thing you can do, which is what we're going to do, is put a place marker for the terms that are missing. And the place marker would just be zero X to the power of whatever that is missing. The reason we're going to do this is because it comes in really handy with synthetic division. Synthetic division, you can do it without place markers, and we will talk about that. But it comes in super handy for synthetic division. And we're going to do it for polynomial long division as well, okay? So we're going to put place markers for any terms that we're missing when we're putting them in descending order, okay? So for example, if we have the following, now this isn't the one that we're going to complete, okay? This is just to show you how you place the place marker or put in the place marker, okay? So if we had the following long division that we had to do. 12 X to the power of four minus 5X squared plus five divided by 3X squared plus 2X minus one, okay? So they ask you to do that long division, right? So you're going to set it up in your long division format. And again, you're going to look for, what do you multiply 3X squared by to give you 12X to the power of four? Well, that's just going to be 4X squared, right? So you're going to put that thing up there. So we multiply 3X squared by 4X squared, and we're going to get 12X to the power of four. Now four are dividend here. We didn't have the X cubed term, and we didn't have the X term, right? We're missing the X cubed term and we're missing the X term. So what we did when we laid it out here, we put in 0X cubed and we put in 0X here, and those are our place markers. That way we don't make a mistake of adding 8X cubed with my negative 5X squared, right? Because if we didn't put those place markers there, this guy would be here, and the five would be here, and we might make a mistake of doing that, okay? So what we do, we put place markers there. That way everything lines up with descending orders. Now, if your first term in your numerator, you're in the top here, it was a really high order, let's say it was X to the power of 27, the next one at X to the power of 12, you're not going to sit there and put zeros, place markers for all the descending orders going down, right? All you're going to do is make a mental note that from series two, you already know that you can only add like terms, right? So you have to keep in mind that you cannot add terms that don't have the same power, you can only add like terms, and you would forget about putting the place markers in. And just keep that in mind, and when you're doing your subtraction or addition, what you would do is just make sure you're adding like terms, right? Right now, and the odds are you're not going to get too many questions or you're not going to deal with too many polynomials, where you have X to the power of 27 and X to the power of 11 or something like this, right? You're not going to be missing too many terms between your terms, right? So what we do initially right now, just to get the process going. So we learn how to do it and then later on for more complex polynomials, we can just keep a mental note that you can only add like terms. So for these types of polynomials, all we do, questions that are low power, all we do we put place markers here, okay? So what we can do right now is we put the place markers and multiply the whole thing out. And if you had anything missing in your divisor as well, right? In your denominator, you would also put place markers in the denominator as well, right? So what we do is multiply the four X squared by, this guy, this guy, this guy. We get this term here, change all the signs and add them. So when you change all the signs and add your terms together, your first term just kill each other, this one add it together. Zero X cubed is just zero, right? So all you're doing is just taking negative X cubed and just bringing it down, right? The other term, you add negative five X squared plus four X squared. It's gonna be negative X squared. And what I do, as soon as I finish that, I bring all the terms down, okay? Over on this side, I bring everything down. That way, you don't have to continuously look up, right? So again, what you're gonna, your question you're gonna be asking yourself is what are you gonna multiply three X squared by to give you negative X cubed, okay? Now, these three and eight don't go together very evenly, right? And some people have a problem with this. So one thing you gotta keep in mind is you can always get fractions up here. And we're basically gonna stop doing this right now. But what I'm gonna show you is how to figure out what you multiply three X squared by to give you negative X cubed, okay? And this is the way you should be really thinking about it, right? So what you're asking yourself is what, the box is what, you know? What are you gonna multiply three X squared by to give you negative eight X cubed? So all you do to figure out what goes inside the box is divide both sides of the equation by three X squared. And your result is gonna go there, right? Negative eight X cubed divided by three X squared. The X squared reduces the X cubed down to X. So it just kills down, kills the power in the three to one. And these guys, you can't reduce anymore. So it just stays negative eight on top and three in the bottom. So what goes up top is gonna be negative eight X divided by three. So what you can do is figure out what you multiply three X squared by to give you negative eight X cubed. And you just put that thing there just by doing, you know, you wouldn't do this exactly at the bottom of, you know, your long division format. You would take this and do it on the side, right? That way you leave enough, you know, you're leaving your space, your workspace empty so you can continue your work. So that's how you figure out if you have fractions, if you have things that don't multiply evenly together, that's how you figure out, you know, what you're gonna multiply and buy to give you your, you know, terms that you need to be able to eliminate your first term in, you know, whatever the process is when you're doing the long division, okay? So keep this in mind. It's just basically, you know, putting in, you know, putting in place markers, putting in place markers where you're missing, you're missing the X, you know, the X terms, whatever they might be. It might be X to the power of four, it might be X to the power of three that you're missing X, that you're missing or whatever it is that you're missing. We're gonna put zero X to the power of whatever it is that we're missing to put place markers. That way our terms line up so we can add or add and subtract them, okay? Now, let's, you know, go find a bigger space and do, you know, a large polynomial long division and we're gonna write out the division statement and we're also gonna look at it graphically to show what it looks like graphically. What it means graphically when you're writing out the division statement, which is just basically taking one function, multiplying by another function and then adding another function to give you your, you know, numerator, your original function or whatever you're starting with in here, okay?