 Let's find the quotient and the remainder when three x cubed plus four x squared plus x plus seven is divided by x squared plus one. So to do long division, you're probably gonna wanna get some space on your paper and list your quotient, excuse me, list your numerator, the dividend and descending order. So start with three x cubed the biggest power, then the next biggest power, then the next power, and then the last one. And then we're also gonna divide this by x squared plus one. Now, if you're wondering, why is there such a huge gap right here? It's that when terms are missing, you might wanna keep track of that here. And so, for example, this doesn't have an x term. So if you want to, you could actually insert a zero x just to be very clear that there's a term that's missing here. I'm just gonna leave a blank, I kinda like a hockey player missing a couple teeth there. There's just gonna be a gap between the x squared and the plus one right there. So now what we wanna do is we wanna consider how many times does the leading term of our divisor divide into the leading term of the dividend? So we think of three x cubed divided by x squared and that's gonna give us a three x. We're then gonna record that three x on the top of our division bar right here. And since it's an x, I'm gonna write it above the term that it corresponds to. So the three x will go above the x. Then we're gonna take our partial quotient three x and times it by x squared plus one. This is going to give us three x cubed plus three x like so, and we're gonna record this down below. So we got a three x cubed and we get a three x. Notice I left this gap right here the same way I had this gap right here. I wanna line up like terms in the same column. So notice what's gonna happen here. You're gonna have a three x cubed, they will cancel out. You should always have the leading terms cancel out if you chose the right quotient up here. Then you're gonna get a four x squared that cancels out with nothing, right? Then you just have a zero x squared right there. So those just are just gonna cancel, I guess there's no cancellation there. You're gonna get a four x squared. And then the next thing here is you're gonna get an x minus a three x which is a negative two x. And then drop down to seven, you're gonna get a plus seven. One of the most dangerous things you should watch out for when you do polynomial division is you have to remember that this negative sign distributes on to all the pieces. Now, the way that human eyes work is we really have this good focus, the central vision, right? We can focus on one thing and then our peripheral vision can notice something like the negative sign. We remember to subtract it. But the problem is when this thing gets long and our eyes start focusing on the three x right there, our peripheral vision's not as good as our central vision. So the negative sign in our peripheral vision might be kind of blurry. We might forget we're supposed to subtract. So one thing you can do is you can just distribute the negative sign if you so choose. This is sort of a stylistic thing. We could have written this instead as, without the parentheses, we might have just put a negative right here and a negative right here to remember to subtract. That's perfectly acceptable. That's not the style that I form. But as you're learning, that might be necessary to do that. So you don't forget to subtract the three x right there. Well, polynomial division is kind of like shampoo in your hair. You have to rinse and repeat. Once you have the first step done, we do the next step. We have to then consider four x squared, the new leading term by the x squared, the leading term of the divisor. This will give us a four. We're then gonna record this on the top right here. We get a four. Next, we're then gonna take four times x squared plus one. This will give us four x squared plus four. And then record that down here, four x squared plus four. And then we subtract that from above. I'm just kind of erasing my scratch work there. The four x squareds will cancel out. Then you're gonna get a negative two x. And then you're gonna get seven minus four, which is actually going to be a plus three. And so this right here turns out to be our remainder because notice a linear polynomial is too small to be divided by a quadratic polynomial. So our remainder is here, a linear polynomial negative two x plus three. And so we might record our answer in the following way. So we have our quotient which is gonna be three x plus four. We have our remainder which is negative two x plus three. But I like to record the answer in the following way. So we started off with, of course, three x cubed plus four x squared plus x plus seven over x squared plus one. And then this improper fraction reduces to be three x plus four, our quotient. And then our remainder negative two x plus three over our divisor x squared plus one. So this would then give us the quotient in this situation. Let's take a look at another example. Let's take the quotient. Let's find the quotient of x to the fourth minus three x cubed plus two x minus five divided by x squared minus x plus one. You'll notice with your dividend there, x to the fourth minus three x cubed. There's no quadratic term. I'm gonna leave a space open for that two x minus five. If you prefer, you can actually write in a zero placeholder if that helps you. But again, I just like to leave it a gap right there. So I remember there's a quadratic term and then you're gonna have x squared minus x plus one. And so let's do the division here. We're gonna take x to the fourth divided by x squared. That's gonna be an x squared. And then I'm gonna record that in my x squared column. So it's good thing I left a gap there. I needed it already. Then we're gonna take, we're then gonna take x squared and we're gonna times it by x squared minus x plus one. And doing that is gonna give me x to the fourth minus x cubed plus x squared. And then we subtract that from above. The x to the fourth will cancel out. We're gonna get a negative three x cubed plus x cubed. That gives us a negative two x cubed. And then we're gonna get nothing minus x squared, which is gonna be a negative x squared, which you can see right there. Then we're gonna bring down the next term, which is gonna be a two x. We don't need to bring down all of the terms because as this is a trinomial, we only have to deal with three terms at a time. We then rinse and repeat this process. We ask ourselves, how many times does x squared divide into negative two x cubed? We're just looking at the monomial there. Subtracting the powers of x, you're gonna get a negative two x, which we'll record over the x column. We're then going to take negative two x and times it by our divisor, which doing that term by term, we're gonna get a negative two x cubed. We're gonna get a positive two x squared and then we're gonna get a negative two x. Subtract this from above. The first terms will cancel, so you get a zero x cubed. You're gonna get negative x squared minus two x squared. That's a negative three x squared. And then you're gonna get two x plus two x, which is a four x, and then bring down the negative five. We rinse and repeat. So this next stage, we wanna consider what is negative three x squared divided by x squared. That's, of course, gonna be a negative three. We now have our quotient. Our quotient will be x squared minus two x minus three. Then we're gonna take our divisor and times it by negative three. That gives us negative three x squared. That gives us positive three x, and that gives us negative three. Subtract this from above right here. We're gonna see that the x squareds cancel out. We're gonna get four x minus three x, which is an x. We're gonna get negative five plus three, which is a negative two. And therefore, x minus two is going to be the remainder here. So summarizing what we found out in this division right here, we found out that x to the fourth minus three x cubed plus two x minus five, all over, let me draw that bar again, all over x squared minus x plus one. This was equal to our quotient, which was x squared minus two x minus three plus the remainder of x minus two over our divisor x squared minus x plus one. I'm gonna zoom out so you can see all of our scratch work on one slide right here. Pause the video right now if you need to look at it over for a few seconds because otherwise we're moving on to the next example. So in this example, we're gonna take the quartic polynomial eight x to the fourth plus six x squared minus three x plus one. We're gonna divide it by two x squared minus x plus two. So let's first write down our dividend, the numerator, eight x to the fourth. There's no x cubed term, so I'm gonna leave a space. We get six x squared minus three x plus one. And we're gonna divide this by two x squared minus x plus two. And so let's go through this process. Eight x to the fourth divided by two x squared, that's going to give us a four x squared, which I record in the x squared column. Times everything in the divisor by four x squared. We're gonna get eight x to the fourth minus four x cubed plus eight x squared. And then subtract that from above that x to the fourth will cancel. We're gonna take no x cubes plus four x cubes that's a four x cube right there. We're gonna get a six x squared minus eight x square that's a minus two x squared. And then we're gonna bring down the negative three x so we can do the next step right here. Next, we ask ourselves how many times is two x squared divided into four x cubed? That is going to happen two x times four divided by two is two x cubed divided by x squared is an x. We're then gonna take our divisor and times it by two x. This is gonna give us four x cubed. The leading term should be identical. Then we're gonna get negative two x squared and then we're gonna get a four x right here. Subtract this from above. The four x cubes cancel, but then the negative two x squared also cancel out. And then we get a three x minus a four x which is a negative seven x bring down the one. So you'll see here that there was cancellation of the x cubes and with the x squares that sometimes happens and we just move on. That actually means we get a skip a step which is great. But then we look at this term right here negative seven x plus one, that's too small. We have a linear polynomial divided by a quadratic polynomial. That means we're done. So our quotient was four x squared plus two x. Our remainder is gonna be negative seven x plus one. Let's write this as a mixed polynomial. So what we've then see here is the following eight x to the fourth plus six x squared minus three x plus one all over two x squared minus x plus two. This thing can be written as improper fraction can be written as a mixed polynomial four x squared plus two x, that's our quotient. And then we add to that our remainder term negative seven x plus one over the divisor two x squared minus x plus two. And like on the last one, I'm gonna zoom out so that you can take a look at all of this at once. And this gives us some examples of polynomial division, so-called long division of polynomials. It's very similar to integer polynomials. And we just do it step by step by step. It's a recursive algorithm that every time you find a part of the quotient, then you subtract that multiple from the dividend and tell you eventually shrink down to get something too small, which is gonna be the remainder when you do this long division.