Finding Intervals of Polynomial Functions

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  • 0:01 A Polynomial Function
  • 1:32 Finding the Solutions
  • 3:43 Is It Positive or Negative?
  • 6:18 Lesson Summary
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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.

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.

A Polynomial Function

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 + 4x + 3 or f(x) = 3x^4 + 2x^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.

Finding The Solutions

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:

polynomial intervals

Is It Positive or Negative?

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.

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