in this video you're going to learn how to do polinomial long division using the box method so there's different methods to do polinomial long division I demonstrate them in some of my other videos where it's just like how you learn how to divide numbers back you know in elementary school but this method is a different way that I recently was introduced to through one of my students because their teacher insisted that they do it this way so I want to share it with you here in this video and the key here is you want to set up this as a box and I'm going to show you some of the uh backstory behind this as well as we're going to go through four examples so you can get some experience and practice with it so what you do is you say I'm dividing by X plus 2 we're going to put X and positive2 right here along one side of the Box okay and then what we're going to do is we're trying to get this quantity here inside the box to be equal to 3x Cub + 11 x^2 + 11 x + 2 so what we do is we start off with uh this first quantity here 3x Cub this first term and we say what * x = 3X Cub that's 3x^2 then we distribute 3x2 * X is 3x Cub that matches 3x^2 * 2 gives us 6 x^2 okay and now we look at our polinomial here and we say well I have 11 x^2 this is 6x^2 that means I need an additional 5x^2 now we've got our 11 x^2 so I say what * X X gives me 5x^2 that's going to be 5x so I distribute 5x * X is 5x^2 5x * 2 is 10 x but you can see here I have 11x that means that I need another X right 1x + 10 x is 11 x what * X is X that's going to be positive 1 1 * X is x 1 * 2 is 2 and there's our two right there and you can see now this is our quotient this is our answer answer 3x^2 + 5x + 1 it's basically just like finding the area of a rectangle you take the length time the width equals the area right or base times height equals the area but now what we're doing is we're kind of working backwards it's like we've got the area and maybe we have the width but we don't have the length you see make another analogy like say if you had uh what's 24 / 6 well we know that that's four right cuz 4 * 6 is 24 Z remainder if we want to get back to to the 24 we say oh 4 * 6 is 24 see this times this so this is our quotient now what happens when you have a remainder what do you do in that situation let's look at that situation next okay example number two now we've got this fraction here which remember when you see this fraction bar this is really like a division sign so we have the numerator being divided by the denominator and so we're going to do the polinomial long division using the box method we're going to take this quantity that we're uh dividing in into the polinomial we're going to put it on the side of our box here so 2X and -3 and we're going to start with this highest powered term this uh leading term 10 x cubed and we're going to say what * 2x = 10 x cubed that's going to be 5 x^2 then we're going to distribute that gives us the 10 x Cub 5x^2 * -3 gives us -15 x^2 okay now when we look at our polinomial here we we've got -2 3x^2 here we have -15 x^2 that means we need another 8x^2 so what * 2x = 8x^2 that's -4x we distribute that gives us A8 x^2 -4x * -3x gives us a POS 12x well we have 16x that means we need another 4X see there's our 16x so now we say what * 2x is 4X so that's just going to be 2 if I distribute I get the 4X 2 * -3 gives me -6 and now we say well we have -9 here this is -6 that means we need another -3 but we've already reached the end of the line here this is a constant we really can't go lower than a constant and so we've already gone in descending order so what do we do with thisg -3 well we take the -3 and we put it over the divisor what we're dividing by that's 2x - 3 now you can write this as plus a -3 over 2x - 3 or you could just write this as -3 over 2x - 3 and this is going to be your quotient now if you wanted to check your work you take 2x - 3 and multiply it by this and you're going to get the this quantity here which represents like our area now to make an analogy say if you had how many times does 6 go into 25 so previously we had 24 right so 4 * 6 is 2 4 remainder one so you can see you're taking this quantity here * 2x - 3 gives you this area but then we have to add on thisg -3 same thing here 4 * 6 is 24 then we have to add on this remainder here one to get back to the 25 okay the other way to look at it is when you multiply these together you're going to get this area when you multiply 2x - 3times this fraction the 2x - 3 is are going to cancel numerator and denominator going to be left with that -3 so we're getting this polinomial back let's take a look at another example this is an interesting one here we've got this polom / x squ + 3 notice we've got a quadratic term a constant term but no x to the first term no linear term so it's almost like there's a term missing there but we're going to take this x^2 + 3 put it along this side of the box and we're going to try to get this polinomial back here so we're going to start with our highest power term which is 2x 4th and we're going to say what * x^2 = = 2x 4 that's 2x^2 now when we distribute we get 2x 4 2x^2 * 3 is 6x^2 but working our way down in descending order here you can see we've got a Negative X cubed let's put a negative X cubed here what * x^2 ISX Cub that's X we distribute we get back thex cubed we get -3x okay now we have 11 x^2 here we have 6 x^2 so we need an additional 5x^2 okay so that's our 11 x^2 what * x^2 is 5x^2 that's going to be five distribute that's 5 x^2 5 * 3 gives us 15 but we need to get n uh 19 so we need an additional four here okay so when we write our final answer it's going to be this plus 4 over our divisor which is x^2 + 3 and that's our quotient that's our answer let's take a look at another example okay for example number four this is an interesting one because we're dividing by a trinomial this time three terms so we're going to do is we're going to take what we're dividing by put it along this side of the box here so x^2 - 2x + 5 and so you can see we've got three rows here now and we're going to take that highest powered term let's put it up in the upper leftand corner what time x^2 = 3x 4th that's going to be 3x^2 now if I distribute get 3x 4th I get -6x cubed and I get 15 x^2 okay now I want -4x Cub I already have a -6x cub so I'm going to add 2x Cub so that this adds up to the -4x Cub but what * x^2 is 2x Cub that's going to be 2x when I distribute I get the 2x cubed I get - 4x^2 and I get 10 x okay now let's see we've got 15 x^2 + -4 x^2 that's 11 x^2 I need 12 X2 so I need another 1 x^2 here so what * x^2 is 1x2 that's going to be 1 if I distribute I get -2X I get 1 * 5 is 5 okay and now let's look what we have here so so this you can diagonal here you can see added up to the 12x s now we want 8X but 10x - 2x oh that is 8x that's great okay and then we've got uh five but we want three so that means we have a remainder here of -2 5 + -2 gives us back to three so our final result is going to be this - 2 over x^2 - 2x + 5 you just put the remainder over the divisor which you're dividing by and that's your final result so let me know in the comments below what is your preferred method of dividing pols and if you want to see uh another way of doing this besides the box method follow me over to that video right there where I demonstrate polinomial long division I'll see you in that video