Hi, and welcome. My name is Anthony Varela. And today, we're going to talk about monomials and polynomials. So we're going to look at polynomials and the terms that make up polynomials. We'll talk about degrees in terms and polynomials. And we'll wrap up by ordering terms in polynomial expressions.
So first, what is a monomial? Well, it's an expression containing a single term. So remember, terms are coefficients, variables, and powers combined through multiplication. We don't see addition or subtraction in a term. So monomial we see that word, or that prefix mono, and that means one. So monomial has one term.
Here's an example of a monomial, 2x cubed. I can write this as 2 times x times x times x. So we see how that's all combined through multiplication. I don't see any addition or subtraction. This is also a monomial. Even though we see two variables, 3x squared y-- I can write this as 3 times x times x times y, still all combined through multiplication, no addition or subtraction.
Well, there are a couple of restrictions to what a monomial can and cannot be. Variable powers must not be negative or fractional. So here are some non-examples of monomial, a to the negative second power and a to the 1/2 power. Now, notice that we can have the coefficients that are negative or fractional, just not variable powers. And if we think about this, a to the negative squared can be written-- or a to the negative 2 can be written as 1 over a squared. And a to the 1/2 can be written as the square root of a.
And a special thing about my monomials and polynomials is that variables are allowed to take on any value. We notice, in these cases, our variable is not allowed to be 0, because we can't divide by 0. And here, our variable is not allowed to be negative, because we can't have negative numbers underneath the square root. So that's a reason why our variable powers must not be negative or fractional when we're talking about monomial and polynomials.
So now let's talk about polynomials and a couple of different types of polynomials. Well, this is an expression containing several terms. So we already know that a monomial contains one term. If we have two terms, this is a binomial. If we have three terms, this is a trinomial. And we typically don't go much farther than that as far as special names. So we have polynomial, poly, meaning many. And you notice that all of these terms are separated by addition or subtraction.
So now let's talk about degrees in polynomials. And we're actually going to start by talking about the degree of terms first. So the degree of a term is the sum of all variable exponent powers in the term. So let's consider this polynomial. I see 1, 2, 3, 4 terms. So let's identify the degree of each term. So we're looking for the variable exponent powers.
So we see x to the third power here, so this has a degree of 3. 2x cubed has a degree of 3. Looking at x squared, this has a degree of 2. Now, what about 3x. I don't see a variable power. And if I have a variable that I don't see a variable power to, that's an implied 1, so 3x is a degree of 1. How about this constant term, negative 5? I don't see a variable at all. So this would have a degree of 0. This is an implied x to the zero power.
Now let's consider a more complicated polynomial. And let's identify, now, the degrees of all of these terms. Well, this is not a degree 2, and it's not a degree 3. Remember, it's the sum of all the variable exponent powers. So this is a fifth-degree term because I'm adding 2 and 3. Now let's look at this term, xy cubed. Remember this is an implied 1. So 1 plus 3 is 4. So this term has a degree of 4. And finally, this term, remember, this is an implied 1 here, so we have a degree of 3, 2 plus 1.
So when we're determining the degree of a term, we add the variable exponent powers. Now, this is different from the degree of the polynomial itself. Now, this is also called order, and it's the highest degree of the terms in a polynomial expression. So we've already identified the individual degrees of these terms.
But now let's identify the degree of the polynomial as a whole. It's the highest degree of all of the terms. So looking at this polynomial here, the highest degree is 3, so this is a third-degree polynomial. In this polynomial, our highest degree was 5, so this is a fifth-degree polynomial. So when we're determining the degree of a polynomial itself, it's the highest degree of all its terms.
Lastly, I'd like to talk about how to order terms in a polynomial according to standard procedure. And we order this from highest degree to lowest degree. So let's take this polynomial, for example. It's not written in our standard form. The terms are out of proper order. So we'd like to rewrite this, ordering our terms from highest degree to lowest degree.
So taking a look at our terms here, this has a degree of 1. This has a degree of 0. This has a degree of 3. And this has a degree of 2. So the highest degree would be this term right here, 2x cubed. So we're going to write that first. And I know it's positive because it's plus 2x cubed and not minus 2 x cubed.
So now, in descending order, the next highest would have a degree of 2. And I see that here with my minus x squared. So that's going to be the next term that I write. Now, in descending order, next up would be a degree of 1, which would be my 3x. And I know that's positive 3x. And lastly, that constant term has a degree of 0, so I would have minus 4 at the end. So this is the same polynomial. It's just written in our standard form.
So when we're ordering terms of a polynomial, identify the degree of each term, and then write them in descending order. And notice, one of the great things about ordering your polynomial is I can look at this first term and identify right away that this is a third-degree polynomial. So let's review my monomials and polynomials.
Polynomials have many terms, which are coefficients, variables, and powers separated by addition and subtraction. We talked about a monomial having one term, binomials having two terms, trinomials having three terms, and then polynomial is the general term for an expression that has many terms.
We talked about the degree of a term being the sum of all of those variable powers. And the degree of a polynomial is the highest degree of all of the terms. And lastly, we talked about ordering the terms in a polynomial expression, identify the degree of each term, and write them in descending order.
Thanks for watching this tutorial on monomial and polynomials. Hope to catch you next time.