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Current time:0:00Total duration:5:27

Polynomial division introduction

Video transcript

we're already familiar with the idea of a polynomial and we spent some time adding polynomials subtracting polynomials and multiplying polynomials and factoring polynomials and what we're going to think about in this video and really start to think about in this video is the idea of polynomial division so for example if I had the polynomial and this would be a quadratic polynomial let's say x squared plus 3x plus 2 and I wanted to divide it by X plus 1 pause this video and think about what would that be what would I have to multiply X plus 1 by to get x squared plus 3x plus 2 well one way to approach it is we could try to factor x squared plus 3x plus 2 and we've done that multiple times in our lives we think about well what two numbers add up to 3 you know if I were to multiply them I get 2 and the one that might jump or the ones that might jump out at you are 2 and 1 and so we could express x squared plus 3x plus 2 as X plus 2 times X plus 1 and then all of that is going to be over X plus 1 and so if you were to take X plus 2 times X plus 1 and then divide that by X plus 1 what is that going to be well you're just going to be left with and X plus 2 this is going out you have to put parentheses this is going to be an X plus 2 and if we want to be really mathematically precise we would say hey this would be true as long as X does not equal X does not equal negative 1 because if x equals negative 1 in this expression or this expression we're going to be dividing by 0 and we know that leads to all sorts of mathematical problems but as we see for any other X as long as we're not dividing by 0 here this expression is going to be the same thing as X plus 2 and that's because X plus 2 times X plus 1 is equal to what we have in this numerator here now as we go deeper into polynomial division we're going to approach things that aren't is easy to do just purely through factoring and that's where we're going to have technique called polynomial long division polynomial long division sometimes known as algebraic long division and if it sounds familiar because you first learned about long division and fourth or fifth grade it's because it's a very similar process where you would take your X plus one and you would try to divide it into your x squared plus three X plus two and you do something very and I'm going to do a very quick example right over here but we're going to do much more detailed examples in future videos but you look at the highest degree terms you say okay I have a first degree term and a second degree term here how many times does X go into x squared well it goes x times so you put the X in the first degree column and then you multiply your x times X plus one x times X is x squared X times one is X and then you subtract this from that so you might already start to see some parallels with the long division that you first learned in school many years ago so when you do that these cancel out 3x minus X we are left with a 2x and then you bring down that 2 so 2x plus 2 and they say how many times does X go into 2x well it goes two times so you have a plus 2 here 2 times X plus 1 2 times X is 2x 2 times 1 is 2 you can subtract these and then you are going to be left with nothing 2 minus 2 is 0 2x minus 2x is 0 so in this situation it divided cleanly into it and we got X plus 2 which is exactly what we had over there now an interesting scenario that we're also going to approach in the next few videos is what if things don't divide cleanly for example if I were to add 1/2 x squared plus 3x plus 2 I would get x squared plus 3x plus 3 and if I were to try to divide that by X plus 1 well it's not going to divide cleanly anymore you could do it either approach one way to think about it if we know we know can factor x squared plus 3x plus 2 is say hey this is the same thing as x squared plus 3x plus two plus one and then all of that's going to be over X plus one and then you could say hey this is the same thing as x squared plus three X plus two over X plus one over X plus one plus one over X plus one plus one over X plus one and we already figured out that this expression on the left as long as X does not equal negative one this is going to be equal to X plus two so this is going to be equal to X plus two but then we have that one that we weren't able to divide X plus 1 into so we're just left with a 1 over X plus 1 and we'll study that in a lot more detail in other videos what does this remainder mean and how do we calculate it if we can't factor a part of what we have in the numerator and as we do our polynomial long division will see that the remainder will show up at the end when we are done dividing we'll see those examples in future videos