 So this video we'll talk about division, long division and synthetic division. So we have a polynomial here, P of X over D of X is equal to Q of X plus its remainder, R of X. And if that's true, then we can say that P of X is equal to that D of X that we divided by times the quotient, Q of X, which is what we came up with, plus the remainder. We're going to check a couple of those. So in this case, D of X, again, is a divisor, Q of X is the quotient and R of X is the remainder. So how does that look in polynomial form? Well, I've already set it up out here for you. But remember, the bottom goes on the outside, the top goes on the inside. Remember, this is D of X and this is P of X. We're going to come over here and say, okay, I've got X cubed that needs to be divided by X. Take your first term and divide it by your first term on the outside. And that gives me X squared. I'm going to put my X squared on top and then I'm going to multiply, okay, multiply right here. So X times X squared is going to give me X cubed and then I'm going to multiply here as well. X squared times seven is going to be plus seven X squared. Okay, now remember, we're going to subtract the whole thing. So we've got to distribute that negative. So we can say that this is a negative X cubed and this is a negative seven X squared and then we can add them. So the X cubeds are opposites so they cancel each other out and five X squared minus seven X squared gives me a negative two X squared and just like in long division with numbers you bring your next term down, so minus 17 X and we do it again. We say negative two X squared divided by my X. Now what do I need to multiply by? I have my negative two, the X on the bottom cancels one on the top so I need minus two X and when I multiply again, negative two times X is going to be negative two X squared. Remember, the black is multiplying. Negative two times seven is going to give me minus 14 X and when I come in here, subtract, but that means change the signs and then I can add, then we have this canceling out and negative 17 X plus 14 X is going to be negative three X and bring down the last term. Minus 26, one more time, negative three X divided by X, I can still divide and that gives me negative three to minus three here and again, we're going to multiply in the black, negative three times X is negative three X, negative three times seven is going to be minus 21 and if I change the signs then from my subtraction I can go to an addition and we have this cancels out, negative 26 plus 21 would be minus five and then to check it down here, we're going to take and see if we can get back to this top polynomial, so D of X was X plus seven and Q of X ended up being X squared minus two X minus three because remember this is our Q of X and this is our R of X, D times Q plus R which is a negative five and let's multiply and see what we get, pause here to work it out while I work it out and we get X cubed plus five X squared minus 17 X minus 26 and sure enough that's what my P of X was. So let's try another one, this one you'll notice has some missing terms and I've placed zeros in for those so that things line up nicely. So we didn't have an X term in our divisor and we didn't have an X squared term in our, it's really called a dividend, the polynomial on top, X to the fourth on top divided by the X squared that were on the outside and that gives us X squared and when we multiply, we're going to have X to the fourth and zero X cubed plus five X squared. We're going to subtracts but that means that we change all the signs and we can add these cancel and this is two X cubed minus five X squared and we're going to bring down the next term minus eight X and then from there we are going to go back and look at the leading terms again, two X cubed divided by the X squared and that leaves me with two and X, two cancels two of those so we have plus two X and when we multiply, we get two X cubed plus zero X squared plus ten X. We're going to come in here and change the signs and from subtraction then it becomes an addition. These cancel, this is negative five X squared, this is minus 18 X and then bring down the last term, one more time leading terms, negative five X squared over X squared will just be negative five and when we multiply, we're going to get negative five X squared and zero X and negative 25 and then we're going to change the signs and then because we changed the signs, we went from subtraction into addition, these cancel out minus 18 X plus nine and we would check by saying, so this is my dividend, this is my quotient and this is my remainder and this is P and you can go ahead and multiply that all out and check for yourself but you should come up with X to the fourth plus two X cubed minus eight X minus 16.