 Welcome back to our lecture series Math 1050, College Algebra for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Mistledine. In this first video for lecture 25, we're gonna talk about the idea of polynomial division. And there's two types of division we're gonna talk about in this lecture. For this video, we're gonna get into the idea of what's typically referred to as long division of polynomials. And in the next video, we'll talk about synthetic division, which is an abbreviation of it. And it turns out that dividing polynomials is essentially the same way that we divide integers. And I wanna try to convince us of that before we proceed to do this example. If we were trying to divide integers, you might take something like 107 and divide that by say five. The first thing we're gonna do is we're gonna look for a multiple of five that goes into 10. Because one's too small, so we didn't have the two digits there. So we need to look for a multiple and we can take two times five, right? Two times five would be 10 itself. We subtract. The difference turns out to be zero. We then bring down what's left over. We get a seven. Then we repeat the process. Five goes into seven exactly one time. We subtract from it five. Seven minus five is a two. And five can't go into two, two is too small. So we end up with this remainder of two. In other words, we have the following observation that 107 is exactly 21 times five plus two. So we have this quotient and we have this remainder. You know, we could do division with decimals here, but when we first learned division, we probably learned it with quotient and remainder. And we can write this with like mixed numbers versus improper fractions, right? We take the number 107 divided by five. As a mixed number, we write this as 21 and two-fifths, which be aware that 21 and two-fifths really just means we're taking 21 plus two-fifths. There's no multiplication going on there. We just often suppress the plus sign right there. And so again, we have our quotient and our remainder right here. When it comes to division of polynomials, the same principle is gonna come into play. We're going to first take the numerator. So if we take 6x squared minus 26 plus 12, this we wanna think of as the numerator of the fraction that's gonna be divided by x minus five, the denominator. So we write our numerator or our dividend, same thing. Six minus x squared minus 26x plus 12. You're gonna write this thing in descending order for organizational purposes. Your dividend, your numerator's gonna go inside this block and then to the left, we're gonna write our divisor, aka the denominator here, x minus four. And so then we have to ask ourselves, how many times does x minus four divide into this? But we don't have to use all of the digits, so to speak, right? We only use the 10 when we divide it by five right here. And so when it comes to the divisor, we're only gonna look at the leading terms right here. So we ask ourselves, because this is a much easier question, how many times does x divide into 6x squared? Well, that actually turns out to be some type of monomial expression. We're gonna take 6x squared and divide it by x, which that gives us just a 6x. And we're then gonna record that on top, but we're not gonna record it on the top digit. We're gonna record it based upon its place value. Since this is a linear term, have x to the first, we're gonna record this over the x to the first portion right there. So we get a 6x. This is like when we wrote the two over here with the integers, okay? And so next, we're gonna take the divisor x minus four and we're gonna times it by this partial quotient. So we're gonna get 6x times x minus four, and this gives us a 6x squared minus 24x. And then we're gonna record this below 6x squared minus 24x, much like we wrote the 10 over here. But then we subtracted the 10, so we have to subtract the 6x squared and the negative 24x right here. You're gonna see that the 6x squared minus 6x squared, those are gonna cancel each other. Now, we're gonna have a negative 24x, we're gonna subtract from that a negative 24x, which actually makes it a positive. So you get negative 26x plus 24x, which gives us a negative 2x. And then we're gonna bring down the next term right here, which is a 12 and this process then repeats itself. We ask ourselves how many times does the divisor x minus four go into this? But we can ask that question by just looking at leading terms. How many times does x go into negative 2x? So if we ask ourselves that, we take negative 2x and divide it by x, that is going to give us just a negative two. Negative two. We're gonna record that on top, negative two right here. Then we're gonna take the negative two and we're gonna times it by our divisor. So we get negative two times x minus four. This gives us negative 2x plus eight, for which we record that here, negative 2x plus eight. But then we have to subtract it from above, for which case then the leading terms will cancel out. If we chose our coefficients correctly, those will cancel out. And now we're left with 12 minus eight will be a positive four. But four is much too small to divide by x here because the degree is too small. So this actually turns out to be our remainder. We get this remainder of four. And so keeping track of things, our quotient is then 6x minus two and our remainder is then the number four. The remainder should always have, as its degree, something smaller than the divisor. Our divisor was a linear polynomial. It has a degree one. So the only thing smaller than that would be a degree zero polynomial, which is a constant like we see right here. And so then gathering this information together, we can say a couple of things. So for example, we can say that 6x squared minus 26x plus 12. This factors in the following way, we're gonna have x minus four times 6x minus two and then plus four right here. And so what do we see? We see the original dividend, a.k.a. the numerator. We see the divisor, a.k.a. the denominator. Here we see the quotient 6x minus two and then we see the remainder here. Our goal whenever we have these polynomials is you're gonna write, we're gonna write f of x, the numerator as our divisor g of x times q of x, the quotient plus r of x, the remainder. This is our goal. We can also think of this as a mixed number, right? So in a mixed number, we start off with this improper fraction, 6x squared minus 6x plus 12. This sits above x minus four. We can think of this right here as this improper fraction. What am I in my improper fraction? Improper fractions like we saw above 107 over five. This would suggest that the numerator is either bigger than or equal to the denominator and so we could kind of reduce it down a little bit. The same thing going on here. Our numerator, it has a degree two, our denominator has a degree one. So since the denominator is smaller degree than the numerator, we say this isn't proper. Well, we can write this as a mixed number, right? What does a mixed number mean? A mixed number means you have a whole number part and then you have your remainder sitting above the divisor. When it comes to polynomials, the whole number part, that is the part with no fraction would be a polynomial and so we're gonna get 6x minus two, that's our quotient. And then we add to it a proper fraction, which is gonna be four over x minus four. So we could argue that this right here is our so-called mixed number, but we shouldn't call it a mixed number. It's like a mixed polynomial because we have a whole number part, a polynomial part right here, and then we have this rational function part for which it's a proper fraction. The numerator, which has degree zero, is smaller than the denominator, which has degree one. And this demonstrates how we can perform polynomial division or so-called long division and it mimics the exact same long division formula that we learned for dividing whole numbers back in grade school. And in the next video, I'm gonna do some more examples of this long division of polynomials. And so take a look at that if we wanna see some more practice.