Dividing polynomials by x
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- [Instructor] We're already familiar with the idea of a polynomial and we've 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 three x plus two, and I wanted to divide it by x plus one. Pause this video and think about what would that be? What would I have to multiply x plus one by to get x squared plus three x plus two? Well, one way to approach it is we could try to factor x squared plus three x plus two and we've done that multiple times in our lives. We think about well, what two numbers add up to three and if I were to multiply them I get two. And the one that might jump, or the ones that might jump out at you are two and one. And so we could express x squared plus three x plus two as x plus two times x plus one, and then all of that is going to be over x plus one. And so, if you were to take x plus two times x plus one, and then divide that by x plus one, what is that going to be? Well, you're just going to be left with an x plus two. This is going, you don't have to put parentheses, this is going to be an x plus two. 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 one, because if x equals negative one in this expression or this expression we're gonna be dividing by zero. 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 zero here, this expression is going to be the same thing as x plus two, and that's because x plus two times x plus one 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 as easy to do just purely through factoring. And that's where we're gonna have a 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 in 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 gonna do a very quick example right over here, but we're gonna 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, I have a second degree term here. How many times is x going to 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, three x minus x. We are left with a two x. And then you bring down that two. So two x plus two. And you say how many times does x go into two x? Well, it goes two times. So you have a plus two here. Two times x plus one. Two times x is two x. Two times one is two. You can subtract these and then you are going to be left with nothing. Two minus two is zero. Two x minus two x is zero. So in this situation it divided cleanly into it and we got x plus two, which is exactly what we had over there. Now, an interesting scenario that we are also going to approach in the next few videos, is what if things don't divide cleanly? For example, if I were to add one to x squared plus three x plus two, I would get x squared plus three x plus three. And if I were to try to divide that by x plus one, well, it's not going to divide cleanly anymore. You could it either approach. One way to think about it, if we know we can factor x squared plus three x plus two is say hey, this is the same thing as x squared plus three x 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 were able to divide x plus one into, so we're just left with the one over x plus one. 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 part of what we have in the numerator? And as we do our polynomial long division we'll see that the remainder will show up at the end when we are done dividing. We'll see those examples in future videos.