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Current time:0:00Total duration:12:09

Intro to long division of polynomials

Video transcript

in this video we're going to learn to divide polynomials and sometimes this is called algebraic long division but you'll see what I'm talking about when we do a few examples so let's say I just wanted to divide 2x plus 4 and divide it by 2 and we're not really changing the value we're just changing how we're going to express the value so we already know how to simplify this we've done this in the past we could divide the numerator and the denominator by 2 and then this would be equal to what this would be equal to X plus 2 let me write it this way it would be equal to if you divide this by 2 it becomes an X you divide the 4 by 2 it becomes a 2 if you divide the 2 by 2 you get a 1 so this is equal to X plus 2 which is pretty straightforward I think the other way is you could have factored a 2 out of here and then those would have cancelled out but I'll also show you how to do it using algebraic long division which is a bit of overkill for this problem but I just want to show you that it's not fundamentally anything new it's just a different way of doing things but it's useful for more complicated problems so you could have also written this as 2 goes into 2x plus 4 how many times and you would perform this the same way you would do traditional long division you would say to you always start with the highest degree term 2 goes into the highest degree term you would ignore the 4 2 goes into 2x how many times well it goes into 2x x times and you put the X in the x place x times 2 is 2x and just like traditional long division you now subtract you now subtract so 2x plus 4 minus 2x is what it's for right and then 2 goes into 4 how many times it goes into 2 times a positive 2 times put that in the constants place 2 times 2 is 4 you subtract remainder 0 so this might seem overkill for what was probably a problem that you already knew how to do and do it in a few steps we're not going to see that this is a very generalizable process you can do this really for any degree polynomial dividing into any their degree polynomial let me show you what I'm talking about so let's say we wanted to divide we wanted to divide X plus 1 into X plus 1 into x squared plus 3x plus 3 X plus 6 so what do we do here so you look at the highest degree term here which is an X and you look at the highest degree term here which is an x squared so you can ignore everything else and that really simplifies the process and you say X goes into x squared how many times well x squared divided by X is just X right X goes into x squared x times and you put it in the x place this is the x place right here or the X to the first power place so x times X plus 1 is what x times X is x squared X times 1 is X so it's x squared plus X and just like we did over here we now subtract and what do we get x squared plus 3x plus 6 minus x squared let me be very careful this is minus x squared plus X don't want to make sure that negative sign only you know it applies to this whole thing so x squared minus x squared those cancel out 3x there's going to be a minus X let me let me put that sign there so this is minus x squared minus X just to be clear we're subtracting the whole thing 3x minus X is 2x and then you bring down the 6 or 6 minus 0 is nothing so 2x plus 6 now you look at the highest degree term in X and a 2x how many times does X go into 2x let's go into 2 times 2 times X is 2x 2 times 1 2 times 1 is 2 2 times 1 is 2 so we get 2 times X plus 1 is 2x plus 2 we're going to subtract this from this up here so we're going to subtract it instead of writing 2x plus 2 we could just write negative 2x minus 2 and then add them these guys cancel out 6 minus 2 is 4 and how many times does X go into 4 well we could just say that 0 times we could say that 4 is the remainder 4 is the remainder remainder so if we were to rewrite so if we wanted to rewrite x squared plus 3x plus 6 over X plus 1 notice this is the same thing as x squared plus 3x plus 6 divided by X plus 1 this thing divided by this we can now say that this is equal to X plus 2 it is equal to X plus 2 plus the remainder divided by X plus 1 plus 4 over X plus 1 these this right here and this right here are equivalent and if you wanted to check that if you wanted to go from this back to that what you could do is multiply this by you could multiply it by X plus 1 over X plus 1 and then add the two so this should this is the same thing as as X plus 2 and I'm just going to multiply that times X plus 1 over X plus 1 right that's just multiplying it by 1 and then to that add 4 over X plus 1 I did that so I have the same common denominator and when you perform this addition right here when you multiply these two binomials and then add the 4 up here you should get x squared plus 3x plus 6 let's do another one of these the kind of fun so let's say that we have let's say that we have we want to simplify x squared plus 5x plus 4 over over X plus 4 so once again we can do our algebraic long division we can divide X plus 4 sorry X plus 4 into into x squared plus 5x plus four and once again same exact process look at the highest degree terms in both of them X goes into x squared how many times it goes into it x times put in the x place this is our x place right here X times X is x squared x times 4 is 4x and then of course we're going to want to subtract these from there so let me just put a negative sign there and then these cancel out 5x minus 4x is X 4 minus 0 is plus 4 X plus 4 and then you can even see this coming you could say X plus 4 because an X plus 4 obviously one time or you could just look at if you were you know not looking at the constant terms you would completely just say well X goes into X how many times well one time plus one one times X is x one times four is four we're going to subtract them from up here so it cancels out so we have no remainder so this right here simplifies to this is equal to X plus one and there's other ways you could have done this we could have tried to factor this numerator right x squared plus 5x plus 4 over X plus 4 this is the same thing as what we could have factored this numerator as X plus 4 times X plus 1 4 times 1 is 4 4 plus 1 is 5 all of that over X plus 4 that cancels out and you're left just with X plus 1 either way would have worked but the algebraic long division will always work even if it's not a perfect if you can't cancel out factors like that even if you did have a remainder in this situation you didn't so this was equal to X plus 1 let's do let's do another one of these let's do another one of these just to make sure that you're really hit because this is actually a very very useful skill to have in your toolkit so let's say we have x squared we have let me let's see let's say we had let me just change it up let's say we had 2 2x squared I could really make these numbers up on the fly 2 X square - 20 X + 12 / actually let's make it really interesting just to show you that it'll always work I want to go above quadratic so let's say we have 3 X to the 3rd minus 2x squared plus 7x minus 4 and we want to divide that we want to divide that by x squared plus 1 I just made this up but we can just do the algebraic long division to figure out what this is going to be or what this is we simplify it when it'll be x squared plus 1 divided into this thing up here 3x to the third minus 2 x squared plus 7x minus 4 once again look at the highest degree term through x squared goes into 3x to the third how many times well it's going to go into it 3x times right you multiply 3x times this you get 3x to the third so it's going to go into 3x times so we have to write the 3x over here in the X term so it's going to go into it 3x times just like that now let's multiply 3x times x squared is 3x to the third right 3x times x squared plus 3x times 1 so we have a 3x over here I'm making sure to put it in the X place and we're going to want to subtract them and what do we have what do we have when we do that these cancel out we have a minus 2x squared and then 7x minus I because I just subtracted 0 from there 7x minus 3x is plus 4x and we have a minus 4 once again look at the highest degree term x squared and a negative 2x squared so x squared goes into negative 2 x squared negative 2 times negative 2 put it in the constants place negative 2 times x squared is negative 2 x squared negative 2 times 1 is negative 2 negative of to now we're going to want to subtract these from there so let's multiply them by negative one or those become a positive these two guys cancel out 4x minus 0 is 4x minus let me switch colors 4x minus 0 is 4x negative 4 minus negative 2 or negative 4 plus 2 is equal to negative 2 and then x squared now it has a higher degree than 4x and the highest degree here so we view this as the remainder we view this as the remainder remainder so this expression we could rewrite it as being equal to 3x minus 2 that's the 3x minus 2 plus our remainder plus our remainder 4x minus 2 all of that over x squared plus 1 x squared plus 1 hopefully you found that as fun as I did