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Monomials & Polynomials

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Today we're going to talk about monomials and polynomials. So we'll start by defining and giving examples of both monomials and polynomials. And then we'll talk about how to find the degree of a polynomial, as well as how to order the terms in a polynomial.

So let's look at some examples of monomials. A monomial is just an expression with only one term. The mono in monomial means one. And a term is just a collection of numbers, variables, and powers or exponents. So this first example, 3x to the sixth, is combined together with a number, a variable, and an exponential or power. And its combined together only through multiplication. Similarly, this example, negative 12 x to the third y to the seventh, is a combination of numbers, variables, and powers. And it's combined only through multiplication. So these are examples of monomials. They have only one term. And if we had an expression with an addition or subtraction sign, that would give us more than one term, which would make it not a monomial.

We also have two examples that are not monomials. And these examples are not monomials because they have either a fractional exponent or a negative exponent. So fractional exponents in an expression will make the expression not a monomial and a negative exponent in an expression will make the expression not a monomial.

So let's look at some examples of polynomials. A polynomial is just an expression with two or more terms. So this first example, 8x squared plus 4x, is considered a polynomial because it has at least two terms. But it also has a more specific name, binomial. Bi meaning two, because this has two unlike terms. This next example has three terms, so it's considered a polynomial. But it also has a more specific name, trinomial. The tri in trinomial meaning three, so we call this a trinomial because it has three unlike terms. This last example has four terms and so we consider this another example of a polynomial. This does not have a more specific name other than polynomial.

So let's look at how to find the degree of a polynomial. The degree is also referred to the order. And when we're finding the degree of a polynomial or the order of a polynomial, we're going to start by finding the degree of the terms in the polynomial. The degree of a term in a polynomial is just the sum of the variable exponents in that term. So for my first example of my polynomial, I'm going to look at my first term and find the sum of the variable exponents. And so I only have one exponent so the degree of this term is 5. Similarly, the degree of this term is going to be 3 and the degree of this term is going to be 1 because there's a 1 exponent here even though we typically don't write it. So to find the degree of my polynomial, I simply need to find the highest degree in all of the terms in the polynomial. So the highest degree is 5 so the degree of my polynomial is going to be 5.

For my second example, I have different variables in my polynomial, but I'll find the degree in the same way. So I'll start by finding the degree of each term. The degree of this first term is 5. The degree of my second term is 2. And for my third term, I have two variable exponents, so I need to sum or add them together to find the degree of this term. 6 plus 2 is 8 so the degree of this term is going to give me 8. So now to find the degree of the polynomial, I'm simply going to find the highest degree in all of the terms. And I see that that is 8, so the degree of this polynomial is 8.

So finally, let's look at how we can order the terms in a polynomial. The standard way to write a polynomial is to order the terms by their degrees from highest to lowest. So we want to rearrange these terms so that the term with the highest degree comes first and the term with the lowest degree comes last. So I'm going to start by identifying the degree of each of these terms. This first term has a degree of 1 because there's no exponent, which indicates an exponent of 1. The second term has a degree of 4. This has a degree of 5. This constant, 6, has a degree of 0 and this last term has a degree of 2.

So the term with my highest degree is positive 2x to the fifth. So that will come first. The next highest degree is found in the term negative 3x to the fourth. So I'm going to bring my minus sign along with this term. So this will become minus 3x to the fourth. Then I have another minus in front of 7x squared, which will be my term with the next highest degrees. So I'll bring the minus sign along with the term. Then I have a positive 11x. And lastly, my constant of positive 6. So you might notice that when you have a polynomial written properly, then the degree of the entire polynomial is just going to be the degree of your first term.

So let's go over our key points from today. A term is a collection of numbers, variables, and powers. A monomial is an expression with one term. A binomial has two terms. And a trinomial has three terms. These are also all polynomials. The degree of a term is a sum of the powers in the term. The degree of a polynomial is the highest degree in all of the terms in the polynomial. It's also called the order of the polynomial. And a polynomial is in standard form when the terms are in order of a degree from highest to lowest.

So I hope that these key points and examples helped you understand a little bit more about monomials and polynomials. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.