Back To Course

Remedial Precalculus32 chapters | 253 lessons | 1 flashcard set

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

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.

Watch this video lesson to learn how you can find the intervals of a polynomial function. Also learn how to find out whether each interval is negative or positive.

In math, we come across all kinds of **polynomial functions**; they are the functions that are made up of constants, variables, and exponents where each term has a different exponent for the function's variable. Our terms are combinations of constants, variables, and exponents all multiplied together, and a polynomial function is our terms added together.

We can have simple polynomial functions, such as *f*(*x*) = *x* or *f*(*x*) = *x* + 1. We can also have more complicated functions, such as *f*(*x*) = *x*^2 + 4*x* + 3 or *f*(*x*) = 3*x*^4 + 2*x*^2 - *x*. All of these are polynomial functions. Do you see how all of these polynomials are made up of terms that are added together? Where we have subtraction is when our term is negative.

What do we do with all these polynomials? Why, we want to solve them and we want to find out how they curve or behave when graphed out. When we solve them, we find where the function equals zero. This is the beginning of our lesson. The rest of our lesson is going to teach us about finding intervals from our solutions and how the function behaves in those intervals.

This is useful information because knowing how our function behaves between intervals can give us a glimpse of what kinds of answers we can expect from our function. We will be able to answer questions such as 'Will we get a positive or negative answer when we give our function a value such as three?' Are you ready to begin? Let's go.

Let's go through and look at solving this polynomial: *f*(*x*) = (*x* - 7)(*x* + 1)(*x* - 2). This polynomial is already in factored form, so finding our solutions is fairly straightforward. We set our function equal to 0 and then find all the *x* values that will give us a true statement. Setting our function equal to 0, we get (*x* - 7)(*x* + 1)(*x* - 2) = 0.

We look at this and we remember from what we learned about solving polynomials that, since our polynomial is already in factored form, all we have to do is to set each factor equal to 0 to find our solutions: we have *x* - 7 = 0, which gives us *x* = 7; we have *x* + 1 = 0, which gives us *x* = -1; and we have *x* - 2 = 0, which gives us *x* = 2. So our solutions are *x* = 7, *x* = -1, and *x* = 2.

Now that we have our solutions, we can now find our intervals. We list our solutions in order from least to greatest. We have -1, 2, and 7. We have three solutions, so that means we will have four intervals. We have one interval from negative infinity to our first solution. Then, we have an interval between our first and second solution and another interval between our second and third solution. Lastly, we have an interval between our last solution and positive infinity.

You can easily find the number of intervals your function has by looking at the number of solutions. Your number of intervals will always be one more than the number of solutions. If you have two solutions, then you will have three intervals. If you have four solutions, then you will have five intervals, and so on.

In our polynomial, we have three solutions and four intervals. Our intervals are from negative infinity to -1, from -1 to 2, from 2 to 7, and from 7 to positive infinity. We can write our intervals using parentheses like this:

Now that we have our intervals, we can go ahead and determine the behavior of our polynomial function between these intervals. The way we can go about doing this is to choose a number between each of our intervals and plug that number into our polynomial function to see if we get a positive or negative answer.

Since we have four intervals, we will pick four numbers to plug in and evaluate, one for each interval. Our first interval is between negative infinity and -1. We can pick -2 since that number is inside this interval. Plugging in -2 into our polynomial, we get *f*(-2) = (-2 - 7)(-2 + 1)(-2 - 2) = (-9)(-1)(-4) = -36. Our answer is negative, so our function is negative in the first interval.

Let's look at the second interval. Our interval is between -1 and 2. We can pick an easy number like 0 inside this interval. Plugging in 0 into our polynomial function, we get *f*(0) = (0 - 7)(0 + 1)(0 - 2) = (-7)(1)(-2) = 14. Our answer is positive, so our function is positive in this second interval.

What about the third interval? Which number would you pick? I'm going to pick 3 because 3 is a smaller number, and I find it easier to work with smaller numbers. Plugging in 3, we get *f*(3) = (3 - 7)(3 + 1)(3 - 2) = (-4)(4)(1) = -16. We have a negative answer, so this third interval is negative.

For the last interval, I'm going to pick the number 10 because, of the larger digits, I find 10 easier to work with. You can pick whatever number you find easy to work with. Plugging in 10, we get *f*(10) = (10 - 7)(10 + 1)(10 - 2) = (3)(11)(8) = 264. Our answer is positive, so our function is a positive in this interval. We can make a table of this information:

Interval | Function Behavior |
---|---|

(negative infinity, -1) | - |

(-1, 2) | + |

(2, 7) | - |

(7, positive infinity) | + |

From this information, we see that our function first rises and passes through our first solution, then dips passing through the second solution, then rises again to pass through the third and final solution. We see that the *x*-axis cuts our function into four distinct pieces, one for each interval, which we can see more clearly if we graph out our function:

Let's review what we've learned now. We learned that **polynomial functions** are the functions that are made up of constants, variables, and exponents where each term has a different exponent for the function's variable. In math, we want to solve them and find out how they behave or curve.

To find out a function's behavior, we can find the intervals of the function and then the function's behavior in those intervals. What we look for in these intervals is whether the function is positive or negative.

The intervals are separated by the solutions of the polynomial function. There will always be one more interval than the number of solutions. If there are three solutions, then there will be four intervals. To check if the function is positive or negative within an interval, we plug in a number from that interval into our function and then evaluate.

Following this video lesson, you should be able to:

- Define polynomial functions
- Explain how to find the intervals of a polynomial function
- Describe how to use the intervals to find out the function's behavior

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
4 in chapter 15 of the course:

Back To Course

Remedial Precalculus32 chapters | 253 lessons | 1 flashcard set

- Terminology of Polynomial Functions 5:57
- How to Evaluate a Polynomial in Function Notation 8:22
- Understanding Basic Polynomial Graphs 9:15
- Finding Intervals of Polynomial Functions 7:16
- 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

- Computer Science 335: Mobile Forensics
- Electricity, Physics & Engineering Lesson Plans
- Teaching Economics Lesson Plans
- U.S. Politics & Civics Lesson Plans
- US History - Civil War: Lesson Plans & Resources
- HESI Admission Assessment Exam: Factors & Multiples
- HESI Admission Assessment Exam: Probability, Ratios & Proportions
- HESI Admission Assessment Exam: 3D Shapes
- HESI Admission Assessment Exam: Punctuation
- HESI Admission Assessment Exam: Linear Equations, Inequalities & Functions
- CPCE Prep Product Comparison
- CCXP Prep Product Comparison
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison
- TACHS Prep Product Comparison
- Top 50 Blended Learning High Schools
- EPPP Prep Product Comparison

- History of Sparta
- Realistic vs Optimistic Thinking
- How Language Reflects Culture & Affects Meaning
- Logical Thinking & Reasoning Questions: Lesson for Kids
- Middle School Choir Activities
- Holocaust Teaching Activities
- Shades of Meaning Activities
- Quiz & Worksheet - Dolphin Mating & Reproduction
- Octopus Diet: Quiz & Worksheet for Kids
- Quiz & Worksheet - Frontalis Muscle
- Quiz & Worksheet - Fezziwig in A Christmas Carol
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- Teaching Strategies | Instructional Strategies & Resources
- Assessment in Schools | A Guide to Assessment Types

- Soft Skills for Engineers
- Introduction to Organizational Behavior: Certificate Program
- Earth Science: Homework Help Resource
- Things Fall Apart Study Guide
- MTLE Life Science: Practice & Study Guide
- Graphs of Motion
- Congenital Cardiovascular Defects: Help and Review
- Quiz & Worksheet - Determining the Earth's Age
- Quiz & Worksheet - Succession in Freshwater and Terrestrial Ecosystems
- Quiz & Worksheet - Mass, Weight & the Effect of Gravity
- Quiz & Worksheet - Types of Tectonic Plate Boundaries
- Quiz & Worksheet - Types of Land Ownership & Use in the US

- Cycles of Matter: The Nitrogen Cycle and the Carbon Cycle
- User Interface Design: Patterns & Best Practices
- Certification to Teach English Abroad
- How to Study for Biology in College
- How to Prepare for College
- Creative Writing Prompts
- Science Topics for Middle School Students
- How to Become a User Experience Designer
- Study.com Grant for Teachers
- Alternative Teacher Certification in Maryland
- AP Calculus Exam Calculator: What's Allowed?
- Chicago Adult Education

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject