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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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Lesson Transcript

Instructor:
*David Liano*

After completing this lesson, you will know what the rational zeros theorem says. You will also know how to apply this theorem to find zeros of polynomial functions.

Factoring polynomial functions and finding zeros of polynomial functions can be challenging. This lesson will explain a method for finding real zeros of a polynomial function. Please note that this lesson expects that students know how to divide a polynomial using synthetic division. You can watch our lessons on dividing polynomials using synthetic division if you need to brush up on your skills.

Let's first state some definitions just in case you forgot some terms that will be used in this lesson. A **zero** of a polynomial function is a number that solves the equation *f*(*x*) = 0. These numbers are also sometimes referred to as roots or solutions. A **rational zero** is a rational number, which is a number that can be written as a fraction of two integers. An **irrational zero** is a number that is not rational, so it has an infinitely non-repeating decimal.

The **rational zeros theorem** helps us find the rational zeros of a polynomial function. Once you find some of the rational zeros of a function, even just one, the other zeros can often be found through traditional factoring methods. Let's state the theorem:

'If we have a polynomial function of degree *n*, where (*n* > 0) and all of the coefficients are integers, then the rational zeros of the function must be in the form of *p*/*q*, where *p* is an integer factor of the constant term *a*0, and *q* is an integer factor of the lead coefficient *a**n*.'

Just to be clear, let's state the form of the rational zeros again. The rational zeros of the function must be in the form of *p*/*q*. The number *p* is a factor of the constant term *a*0. The number *q* is a factor of the lead coefficient *a**n*. Let's look at how the theorem works through an example: *f*(*x*) = 2*x*^3 + 3*x*^2 - 8*x* + 3.

In this function, the lead coefficient is 2; in this function, the constant term is 3; in factored form, the function is as follows: *f*(*x*) = (*x* - 1)(*x* + 3)(*x* - 1/2).

The zero product property tells us that all the zeros are rational: 1, -3, and 1/2. If *x* - 1 = 0, then *x* = 1; if *x* + 3 = 0, then *x* = -3; if *x* - 1/2 = 0, then *x* = 1/2. Zeros are 1, -3, and 1/2.

Let's write these zeros as fractions as follows: 1/1, -3/1, and 1/2. Notice that each numerator, 1, -3, and 1, is a factor of 3. Also notice that each denominator, 1, 1, and 2, is a factor of 2.

List the possible rational zeros of the following function: *f*(*x*) = 2*x*^3 + 5*x*^2 - 4*x* - 3.

The constant term is -3, so all the factors of -3 are possible numerators for the rational zeros. The lead coefficient is 2, so all the factors of 2 are possible denominators for the rational zeros.

All possible combinations of numerators and denominators are possible rational zeros of the function. The possible rational zeros are as follows: +/- 1, +/- 3, +/- 1/2, and +/- 3/2.

Find the rational zeros for the following function: *f*(*x*) = 2*x*^3 + 5*x*^2 - 4*x* - 3.

This is the same function from example 1. The rational zeros theorem showed that this function has many candidates for rational zeros. Therefore, we need to use some methods to determine the actual, if any, rational zeros. One good method is synthetic division. Let's try synthetic division.

This method will let us know if a candidate is a rational zero. Let's show the possible rational zeros again for this function:

There are eight candidates for the rational zeros of this function. The number -1 is one of these candidates. To determine if -1 is a rational zero, we will use synthetic division.

The synthetic division problem shows that we are determining if -1 is a zero. The first row of numbers shows the coefficients of the function. If -1 is a zero of the function, then we will get a remainder of 0; however, synthetic division reveals a remainder of 4. Therefore, -1 is not a rational zero.

We could select another candidate from our list of possible rational zeros; however, let's use technology to help us. If we graph the function, we will be able to narrow the list of candidates.

The graph of our function crosses the *x*-axis three times. It certainly looks like the graph crosses the *x*-axis at *x* = 1. If you recall, the number 1 was also among our candidates for rational zeros. To determine if 1 is a rational zero, we will use synthetic division.

The synthetic division problem shows that we are determining if 1 is a zero. Synthetic division reveals a remainder of 0. Therefore, 1 is a rational zero. In other words, *x* - 1 is a factor of the polynomial function.

Those numbers in the bottom row are coefficients of the polynomial expression that we would get after dividing the original function by *x* - 1.

We started with a polynomial function of degree 3, so this leftover polynomial expression is of degree 2. In other words, it is a quadratic expression. We can now rewrite the original function.

First, let's show the factor (*x* - 1). Next, let's add the quadratic expression: (*x* - 1)(2*x*^2 + 7*x* + 3).

We could continue to use synthetic division to find any other rational zeros. However, it might be easier to just factor the quadratic expression, which we can as follows: 2*x*^2 + 7*x* + 3 = (2*x* + 1)(*x* + 3).

Let's add back the factor (*x* - 1). Using the zero product property, we can see that our function has two more rational zeros: -1/2 and -3.

2*x* + 1 = 0

2*x* = -1*x* = -1/2

*x* + 3 = 0*x* = -3

Find the rational zeros of the following function: *f*(*x*) = *x*^4 - 4*x*^2 + 1.

The only possible rational zeros are 1 and -1. Let's use synthetic division again. Both synthetic division problems reveal a remainder of -2. Therefore, neither 1 nor -1 is a rational zero.

This function has no rational zeros. Let's look at the graph of this function. The graph clearly crosses the *x*-axis four times. Therefore, all the zeros of this function must be irrational zeros.

The **rational zeros theorem** is a method for finding the **zeros** of a polynomial function. The theorem tells us all the possible **rational zeros** of a function. We showed the following image at the beginning of the lesson:

The rational zeros of a polynomial function are in the form of *p*/*q*. The term *a*0 is the constant term of the function, and the term *a**n* is the lead coefficient of the function.

Using synthetic division and graphing in conjunction with this theorem will save us some time. The rational zeros theorem will not tell us all the possible zeros, such as **irrational zeros**, of some polynomial functions, but it is a good starting point.

Following this lesson, you'll have the ability to:

- Describe the rational zeros theorem
- Identify the form of the rational zeros of a polynomial function
- Explain how to use synthetic division and graphing to find possible zeros

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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

- How to Add, Subtract and Multiply Polynomials 6:53
- Factoring Polynomials Using Quadratic Form: Steps, Rules & Examples 8:38
- How to Divide Polynomials with Long Division 8:05
- How to Use Synthetic Division to Divide Polynomials 6:51
- Remainder Theorem & Factor Theorem: Definition & Examples 7:00
- Dividing Polynomials with Long and Synthetic Division: Practice Problems 10:11
- Finding Rational Zeros Using the Rational Zeros Theorem & Synthetic Division 8:45
- Using Rational & Complex Zeros to Write Polynomial Equations 8:59
- Go to Polynomial Functions of a Higher Degree

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