Hey everyone. We've worked with polynomial expressions and we've even worked with a specific type of polynomial function, a quadratic function where my highest power is 2. Now I want to take a broader look at all the possibilities of polynomial functions, which can really just be any polynomial, but now \( f(x) \) is equal to that polynomial, making it a polynomial function. So that might sound like a lot, any polynomial, and now it can be a function too. But don't worry, we're going to rely on a lot of what we already know about polynomials and some of what we just learned about quadratic functions in order to learn more about polynomial functions and their graphs. So let's go ahead and jump right in.

So looking at the polynomial function that I have right here, we want to remind ourselves of a couple of things that we learned with polynomials, the first of which is that polynomials can only have positive whole number exponents. So that means no negatives and no fractions in those exponents. The other thing is that whenever we write our polynomials in standard form, all of our like terms need to be combined and it needs to be written in descending order of power. So if I start with a power of 3 my next power is going to be 1 lower and then 1 lower, 1 lower until I get to the last one descending order. Now, looking at these polynomials that I have here, you might notice this one looks similar to what you've seen in your textbook. And this can look a little bit intimidating when you first see it, but don't worry, this is just showing us exactly how to write any polynomial in standard form. So this \( a_n \) right here represents my leading coefficient. So in the polynomial that I have here, it's simply 6, and \( n \) represents the degree of the polynomial. So in my polynomial here, it's just 3. Now when I look at my next term, I see that that this power goes down to \( n-1 \), and that's just showing us that it's in descending order of power. So here I started with 3. If I subtract 1 from that, I get 2, which is my very next power here. And then all of these \( a \) terms down to my \( a_0 \) just represent all of my coefficients and then my constant. So don't let that form freak you out, it's just showing you how to write in standard form.

Okay, so let's just look at some polynomial functions here. So looking at this first one I have \( f(x) = -x^2 + 5x^3 - 6x + 4 \) and I want to determine if this even is a polynomial function. And if it is, write it in standard form and state both the degree and leading coefficient. So let's first determine if this even is a polynomial function. And since I only have positive whole number exponents, I don't have any fractions or any negatives in those exponents, it looks like, yes, this is a polynomial function. So let's go ahead and write it in standard form. Now here it looks like I just need to switch these two terms in order for it to be in standard form. So I end up with \( f(x) = 5x^3 - x^2 - 6x + 4 \), and we're good; we're in standard form. So let's go ahead and identify that degree. So the degree of this polynomial, looking at that first term, my highest power, is simply 3. And then my leading coefficient is the one that's attached to that x^3. It is 5. So we're good on that one.

Let's go ahead and move to our next one. Here we have \( f(x) = 2x^{1/2} + 3 \). Now is this a polynomial function? Well, I have a fraction in my exponent there. So no, this is not a polynomial function. I cannot have fractions as exponents, so I don't have to worry about anything else because it's not a polynomial function at all.

Let's go ahead and move on to our last example here. We have \( f(x) = -\frac{2}{3}x^4 + 1 + 3 \). Now looking at this, there is a fraction as a coefficient. Does that mean that it's not a polynomial? Well, no, because it's just a coefficient. It's not in my exponent and my exponents here are positive whole numbers. So it looks like yes, this is a polynomial function. You can have a fraction as a coefficient just not as an exponent. So let's go ahead and write this in standard form, which I can do by simply combining these like terms that I have here. So I end up with \( f(x) = -\frac{2}{3}x^4 + 4 \), combining that \( 1 + 3 \). So let's identify our degree here. So my highest power on that first term is 4. And then lastly, my leading coefficient, looking here, I have -2/3 as my leading coefficient on that term with the highest power.

Okay, so we've taken a look at some polynomial functions, let's look at what their graphs may look like. So there are 2 things that we want to think about with the graphs of polynomial functions, and that is that they are both smooth and continuous. This will be true of the graph of any polynomial function. And what that means is that there will be no corners in our graph and there will be no breaks. So looking at these polynomial functions on the left side, I have this curve here that is smooth. It is a smooth curve. It is continuous. It never breaks off. And then here you might recognize this as a quadratic function, which we know is definitely a polynomial function and it is both smooth and continuous as well. Now looking over to this right side here and this graph, I have a really harsh corner right here so that tells me that I am not dealing with a polynomial function. It is not a polynomial function at all. It is not smooth and continuous. So let's look at one more here. And looking at this graph, it breaks off and then it keeps going on that side. Now this is not okay. It is not a polynomial function because it has that break. So these 2 are not polynomial functions at all and we want to look for things that look more similar to these 2 that are both smooth and continuous. So one more thing that I want to mention about the graphs of polynomial functions is the domain, which is always for any polynomial function going to go from negative infinity to infinity, which is something you may remember from working with quadratic functions. All quadratic functions had this domain, and all polynomial functions have it as well. All real numbers are included in that domain. So that's a little of the basics of polynomial functions. Thanks for watching, and I'll see you in the next video.