 we have here for in basic math terms, polynomials is defined as f of x, a n, x to the power of n plus a n minus one, which means it's another constant, it's a coefficient, x to the power of n minus one, which basically means this x has to be one, and the power has to be one less, at least one less than the previous one, right? Basically, so this is saying that the powers have to be in descending order, right? And the a's just basically mean just toggling the n minus one, minus two, minus three, it's just basically saying the constants could be the same or could be different, right? So the a's could be anything from the real number set, and x, n, n has to be, again, one of the whole numbers, it can only be a whole number or positive integers if you wanna think about it, but those are whole numbers, right? So this toggles all the way down to, you could have just a constant, and just a constant number, like if you had f of x is equal to a number, even zero, that's considered to be a polynomial function, right? So polynomial functions are anything that has a combination of these, or a combination of these, right? Or a zero. So it could be a zero or anything with x to a positive power. So for example, you couldn't have x to a power of a half, that's not a polynomial function. You couldn't have x to a power of a negative number, that's not a polynomial function, right? So you couldn't have only numbers from the rational number set, all the numbers from the rational number set, because that doesn't work. Rational numbers includes negative numbers and fractions, so you can't have any powers that are fractions. So for example, let me just go up here or write down some of the ones you can't have, right? So for example, anything in the orange, x to the power of negative two, x to a power of one over two, or three over two, or five over two, square root of, or any root of x, or one over x, x can't be in the denominator. Those are not considered to be polynomial functions or polynomial equations. Now, in some of the previous videos, we've solved some equations that weren't polynomial equations, specifically when it came to dealing with GCF, taking out the greatest common factor, because all we had to do to be able to solve equations with the greatest common factor was, just combine all the x's to one x term to any power and just get it by itself, right? And that was easy, right? You just did the opposite and you just had a number or other variables on the other set. So anything that has this type of x variable in it is not considered to be a polynomial function or polynomial equation. It has to be x to a power that's a whole number, including zero, so it could be x to the power of zero. Anything to the power of zero is just one. Now, one thing to keep in mind is polynomial functions can be based on one variable where we just have x here, or there could be multivariable functions or multivariable equations. So you could have an x or y, a z, a w or whatever it is. So you could have multiple variables in the function, okay? As long as the power is part of the whole number set, right? Now, there is another terminology that we use, which is called the degree of a polynomial, the degree of an equation. And the degree refers to the highest power in the function. Highest power in the function, right? So if you had f of x is equal to x to the power of three plus two x squared or something like this, the highest power obviously you write first because they have to be in descending order. So the highest power decides what a degree of a function is. So if you have an x cubed as the highest power, that's called a third degree function. If you have x to the power of eight as the highest power, that's called a polynomial function or polynomial equation to the eighth degree, okay? Now, that's when it comes to, you know, when polynomials have one variable, right? So with one variable, you take the highest power and that's the degree of a function. If you have multivariable polynomials, multivariable functions, right, or equations, you add up the powers in the top with the, you add up the powers with the variables and that becomes your degree. So for example, if you had x squared and y cubed as your highest powers, your first term, then you add the two and the three and it becomes degree five. So that would be a polynomial function to degree five, okay? So with functions or equations, polynomials, when you have one variable, the highest power decides what a degree of a function is. If you have multivariable functions, polynomials, which means multiple letters in the equation, you add up the highest powers, you add up the powers for all the variables and the one that gives you the highest number is the degree of the function, right? And you order things. Well, with multivariable functions, you can order things based on different criteria but we're gonna stay away from those for now anyway. And what we're gonna do is specifically just talk about one variable functions, one variable equations. And we've already solved some equations like this, we've already talked about some of this stuff and graphed them. What we're gonna do is delve a little bit deeper into this stuff and graph higher power functions such as cubes and extra power five, six, or something like this, okay? So we're gonna learn some new rules, some new properties of polynomial functions which help us graph them on a Cartesian coordinate system. It's solving it, which basically means finding the X intercepts. It's just one property of a polynomial function for a polynomial equation. There are other properties that these functions have which help us graph them. One thing which is quite important to remember, okay, is that all this terminology that we're learning, all these terms and us defining a function or a polynomial or anything in mathematics, the words that we use are specifically in English, right? So I doubt it if an equation is called an equation in any other language, right? So all this terminology is good to learn them, it's good to understand what it is, you know, to define what it is that we're talking about. But keep in mind that when you see something like this, right, down here, that's a polynomial function, but that's us calling it a polynomial function in mathematics. This is what it is in any other language, you know, I'm not sure if all languages call them polynomials or not. I doubt it if every language in the world calls these types of things equations, right? Equation is an English word. So keep this in mind. What we're doing right now when we're learning terminology is giving names, English names to things, to words, to functions, to equations, to things in mathematics that appear in the language of mathematics. So we're using one language to describe another language. In mathematics, when you see an equation, you don't sit there and go, aha, that's a polynomial equation. So, you know, write down polynomial, look at what a polynomial is and deal with it. When you start using these a lot, you automatically know what a polynomial is. You just notice it, you recognize it. You know that nothing here, those guys are not polynomial, are not part of a polynomial function.