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
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
Okay, so let's just look at some polynomial functions here. So looking at this first one I have
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
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.