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. 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. 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 in 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. 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
If I subtract 1 from that, I get 2, which is my very next power here. And then all of these
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
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
Let's go ahead and move to our next one. Here we have
Let's go ahead and move on to our last example here. We have
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
So my highest power on that first term