 Welcome back to our lecture series Math 1050, College Asper for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misledine. In this lecture, lecture 24, we're going to talk a lot about graphing polynomial functions, and in the previous lecture, we talked a lot about this, some things that will affect the shape of the graph of a polynomial function, things like in behavior, x-intercepts, turning points, just to name some of the things we talked about. Before we start graphing polynomials, there's one more topic related to the graph of a polynomial we want to introduce, and this is the idea of multiplicity. When you have a factored polynomial, we need to factor the polynomial in order to find the x-intercepts of the graph. We see this from the factor theorem, which we've seen previously, but the number of times that a factor shows up affects the shape of the graph. This is what we mean by multiplicity. Suppose that the factor x-minus c to the m is a factor of the polynomial. We learn by the factor theorem that if x-minus c is a factor that corresponds to c being a root of the polynomial. But in particular, we're looking at x-minus c. So x-minus c shows up m minus time, but it does not show up in plus one many times. So m here is the number of times you see this factor in the factorization of the polynomial. This number m is called the multiplicity of the corresponding root. So a quick example of that, we look at the following polynomial, f of x equals five times x-minus two times x-plus three squared times x-minus one-half to the fourth. This polynomial is factored. We can very quickly see from it the roots of the polynomial. We say that x-equals two is a root. We get that from the factor x-minus two. We also get the root negative three, which shows up here from the factor x-plus three. We're also going to get the factor of one-half, I should say the root. These are the roots of the polynomial. We're now going to describe the multiplicities of these roots. So when we look at x-minus two, notice it shows up exactly once. So we say the multiplicity of the root two is the number one. And I'm going to often denote this as a superscript to our root, although so we don't confuse this with an actual exponent, like we're not actually taking two to the first per se or two squared or anything. I'm going to use a different color. So we're going to write two to the one here. So you see that the multiplicity of two is the number one. When we look at the factor x-plus three, this corresponds to the root negative three, and this factor shows up two times. So we say the multiplicity of negative three is two. And then finally, x-minus one to the one-half tells us that one-half is a root of the polynomial. It shows up four times in the factorization. So the multiplicity of one-half is equal to four. And that's all that comes down to for computing the root or the multiplicities of roots of the polynomial. Now, why does this matter? So what we see here is that the multiplicity of a polynomial's root affects the shape of the polynomial. And the reason is the following, that if you have a polynomial f of x and you have that f of c is equal to zero, so it's a root, then we're gonna see that as x approaches the number c, we're gonna see that our polynomial f of x is gonna be approximately the same thing as a times x-c raised to its multiplicity m. So there might be some coefficient here, but when f of x gets close to c, the function f of x will behave very much like the monomial x minus c to the m where some coefficient a comes into play here. And so if we think about that, this really has to do with the multiplicity there, whether it's an even number or an odd number. So for example, if m is an even number, what this means is our function will look like an even monomial when you're close to x equals c. So it'll look something like this or something like this. In particular, if you were to draw the x-axis, because after all, x equals c is a root of this thing, what we're gonna see is that x equals c, our polynomial's gonna come, it's either gonna touch it from above or it's gonna touch it from below. In particular, the sign of the function does not change signs when you go from one side of c to the other side. You're either both positive or you're both negative. So the function won't cross the x-axis, it'll just touch the x-axis. And that's the terminology we're gonna use in this situation, that when your multiplicity is even, we say that the function will touch the x-axis either from above or from below. Now when your multiplicity is odd, you're gonna get something that looks like an odd monomial. Your function will either look like this near x equals c or it'll look like this when you're near x equals c. So in either case, your function will either do something like this, it'll go from a negative to a positive, it'll switch its sign or the other option is that it'll switch its sign from plus to negative. And so when m is an odd number, the sign of f does change from one side of c to the others. And in other words, you're gonna cross the x-axis. And so this is the thing you wanna remember when you have a, when you look at the multiplicities of these factors here, when you have an even multiplicity, you're gonna touch the x-axis, you don't cross it. When you have an odd multiplicity, you will cross the x-axis. And this is all about switching the sign here. So let's look at a specific example. We can see what one right here, f of x equals x squared times x minus two. So if we record the roots of the polynomial real quick, we're gonna get x equals zero and two as the roots. And the multiplicities of said roots because x shows up twice x squared, we see the multiplicity of zero is gonna be two and x minus two shows up once. So we're gonna get a multiplicity of one right here. In particular, x equals zero is gonna have an even multiplicity and x equals two is gonna have an odd multiplicity. Odd multiplicity, meaning that the function will cross at x equals two and at x equals zero, it's only gonna touch the x-axis. And so if I were graphing this function, I might think of something like the following, that okay, some important points to pay attention to at x equals zero, we have x-intercept and at x equals two, we have an x-intercept. At x equals zero, it's going to touch the x-axis but not cross. At x equals two, it's gonna cross, but not it will cross the x-axis going from one side to the other. And we can then use information about like in behavior, the y-intercept to determine what's gonna happen. We'll see that a little bit more detail in the next example actually. So what we can do in the following is say, what we can do right now is say the following. So when the function, when you are, when x approaches zero, our function is gonna look like f of x and it's gonna be approximately equal to x squared but then we have to determine the coefficient. The coefficient will be determined by plugging in zero for every factor except the factor that makes the whole thing go to zero. So x equals zero, let me say that again. x equals zero came about as a root because x squared is a factor. We're gonna plug in x equals zero into every factor of x of f of x except for its corresponding one. So let me write this to the side right here. So as x approaches zero, f of x will be approximately the same thing as x squared times zero minus two. And this tells us that we'll be approximately the same thing as negative two x squared. So this will look like an even monomial that's pointing downward. So our graph is gonna look something like the following at x equals zero. It'll look like a parabola pointing down. Now as x approaches two, we see that f of x will be approximately the same thing as two squared times x minus two. That is it'll look like four times x minus two. This will look like a linear function with a positive slope. So it'll be increasing, it'll go from negative to positive. And so we connect those, when we look at that, when you're near zero, it looks like a concave down parabola. When you're at two, when you're close to two, it'll look like an increasing line, a slope approximately four. And that's what we see right here. Now, so we could actually plug in zero, plug in two into the function to determine these things. And that's how we got these little pictures right here. But we also could have used a test value like going to what we had before, erasing this for a second. I had mentioned earlier that it's gonna touch right here and it's gonna cross right here. What if I picked a number like say x equals one that sits between the two points there, zero and two. If I look at x equals one, this is gonna be one squared times one minus two, which ends up being negative one in the end. So that means the point one comma negative one's on the graph. Well, how can I connect this information together if I go to the right, if I'm below the x-axis crossing means I'm gonna switch to the positive side, then it's gonna have to cross like that, give me the picture. And then if you're touching, that means go into the right, I'd have, or go into the left, I'd have to touch the graph right here. Now, this picture might be a little bit misleading because by no means am I assuming that one comma negative one is the local minimum, but it does help us fill in the information. So we can determine how the graph behaves based upon the multiplicity. So we get a picture like this right here. Now, if you couple this information multiplicity with the end behavior and the intercepts, which we already know the x-axis intercepts here, but if we combine it with the y-intercepts, we would actually have all the information we need to actually start putting together the picture of our polynomial graphs.