Finding Rational Zeros Using the Rational Zeros Theorem & Synthetic Division

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  • 0:01 Definitions
  • 0:58 Rational Zeros Theorem
  • 3:16 Example 1
  • 3:52 Example 2
  • 7:07 Example 3
  • 7:45 Lesson Summary
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Lesson Transcript
Instructor: David Liano

David has a Master of Business Administration, a BS in Marketing, and a BA in History.

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.

Rational Zeros Theorem

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 a0, and q is an integer factor of the lead coefficient an.'

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 a0. The number q is a factor of the lead coefficient an. Let's look at how the theorem works through an example: f(x) = 2x^3 + 3x^2 - 8x + 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.

Example 1

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

Example 2

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

Possible rational zeros for example 2
possible zeros for example problem

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.


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