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Remedial Precalculus32 chapters | 253 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will be able to understand and use the words associated with polynomial functions like a pro. You will see words, such as leading coefficient, and you will immediately know what it is referring to.

This video lesson is all about the keywords associated with **polynomial** functions. A polynomial is made up of several combinations of constants, variables, and exponents. Recall that a constant is simply a number, a variable is a letter, and exponents are the powers that the variable is raised to. When you graph a polynomial, you will get a line that looks like a roller coaster.

This graph is showing the polynomial function *f*(*x*) = 4*x*^5 - *x*^4 - *x*^3 + 2*x*^2 - 3x. See how it looks like a roller coaster? I personally wouldn't get on this roller coaster. It looks a bit scary to me. But I'll talk about the polynomial function any day.

This polynomial function is written in standard form because the exponents linked to our variables are written from large to small. Our first exponent is 5, followed by a 4, then the 3, then the 2, then a 1.

Why should you be interested in learning about keywords associated with polynomials? Well, because you will come across these terms more and more often the more you learn about polynomials. Also, the more math you do, the more polynomials you will come across. You will see polynomials in real life, and if you understand these terms, you will be in a better position to talk about them. Are you ready to dig in?

If you look at the polynomial function we just graphed, you will see numbers in front of the variables. These numbers are called our **coefficients**. If we don't see a number, then we have an invisible 1. So our coefficients going from left to right are 4, -1, -1, 2, and -3. Because our polynomial is written in standard form, the very first coefficient is our **leading coefficient**. It is always the coefficient linked to the variable with the highest exponent. For our polynomial, the leading coefficient is 4.

Next, we have our **terms**. The terms are products of constants and variables with their exponents. For polynomials, the exponents of our terms cannot be negative nor can they be fractions; the exponents must be positive whole numbers. For example, 4*x* and 2*x*^2 are both valid terms. But what if you have 4*x*^-1 or *x*^(1/2)? Are these valid terms for polynomials? No, they are not.

Why is this? The first isn't valid because it has a negative exponent. The second isn't valid because it has a fractional exponent. Look back at our polynomial that we graphed. This polynomial has a total of five terms. We have 4*x*^5, -*x*^4, -*x*^3, 2*x*^2, and -3*x.* Do you see how all our terms have exponents that are positive whole numbers? Do you also see how our polynomial is made up of separate terms separated by either addition or subtraction? If we have subtraction, then it means the term is negative. If we have addition, then our term is positive. We can also have terms that are just variables or even just numbers. If it is just a number, we sometimes refer to it as a **constant term**.

When our polynomial is written in standard form, the very first term, the one with the highest exponent, is the **leading term**. For our polynomial, it is 4*x*^5. If you can picture the roller coaster, then the terms are the individual cars of the roller coaster. The beginning car with the fancy front is then the leading term.

Let's look at our leading term of our roller coaster polynomial. Our leading term is 4*x*^5. Do you see the exponent? It is the highest exponent of our polynomial. In math, we call this the **degree**.

We can also describe our polynomial by its degree. For example, our roller coaster polynomial can also be called a polynomial of degree 5 because it has a degree of 5. What about this polynomial: 2*x*^3 - 4*x* + 6? What do you think the degree of this polynomial is? Is it 2, 3, 4, or 6? It's 3 because 3 is our highest exponent.

Let's analyze a polynomial using all the keywords we've learned now. Look at this polynomial: 5*x*^6 - 3*x* + 4. What are the coefficients? We remember that coefficients are the numbers in front of our variables. So our coefficients are 5, -3, and 4. What about our leading coefficient? What is that? Our leading coefficient is the very first number, so it is 5.

What about the terms? We have three terms. The first term is 5*x*^6, and it is positive. The second term is -3*x*. Our last term is 4. Our leading term is 5*x*^6. The degree of our polynomial is 6 because that is our highest exponent.

We did pretty good. Let's review everything we've learned now. We've covered the most important keywords related to polynomial functions. **Polynomials** are made up of several combinations of constants, variables, and exponents. Recall that a constant is simply a number, a variable is a letter, and exponents are the powers that the variable is raised to. When graphed, you will get a roller coaster-looking kind of line.

Now, for the keywords. **Coefficients** are the numbers in front of our variables. The **leading coefficient** is the very first number. A **term** is the product of a constant and a variable with its exponent. The **leading term** is the very first term. And finally, the **degree** is the highest exponent of the polynomial.

After reviewing this lesson, you should have the ability to:

- Define polynomial
- Identify the graph of a polynomial function
- Describe the keywords for polynomials: coefficients, leading coefficient, term, leading term and degree

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Remedial Precalculus32 chapters | 253 lessons | 1 flashcard set

- Terminology of Polynomial Functions 5:57
- Understanding Basic Polynomial Graphs 9:15
- Finding Intervals of Polynomial Functions 7:16
- Short Run & Long Run Behavior of Polynomials: Definition & Examples 6:05
- How to Graph Cubics, Quartics, Quintics and Beyond 11:14
- Pascal's Triangle: Definition and Use with Polynomials 7:26
- The Binomial Theorem: Defining Expressions 13:35
- Go to Polynomial Functions Basics

- Go to Continuity

- Go to Limits

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