# Rational Roots Theorem Lesson Plan

Instructor: Bret Sikkink

Bret has a Master's degree in Education and taught Economics for college credit in Mexico for six years.

In this lesson, students will learn the Rational Roots Theorem method for generating a list of possible solutions to a polynomial. They will apply their understanding to create an infographic with a description, example, and history of the Theorem.

## Learning Objectives

After completing this lesson, students will be able to

• Generate a list of possible solutions to polynomials using the Rational Roots Theorem.
• Identify which of the possible solutions are in fact solutions to the polynomial.

60-80 minutes

## Curriculum Standards

• CCSS.Math.Content.HSA.APR.C.4

Prove polynomial identities and use them to describe numerical relationships.

## Instructions

### Access Prior Knowledge

Begin the lesson by reminding students of what they have been learning. You may want to solve a simple example together, by factoring a polynomial such as 2x^2 + 9x + 4.

Before watching the video, try factoring a more difficult polynomial together. Alternatively, you could allow students to work with partners or teams on the solution. You could project or write on the board or poster paper a polynomial with a larger coefficient on the leading term or a polynomial with a greater number of terms, such as a polynomial with a leading term that is cubed.

As you are preparing to project the video, ask students if they would appreciate an easier way to generate a list of solutions to polynomials. How could you generate that solution list? You can remind them of the term 'factoring' and ask whether they could imagine a relationship amongst the factors of the coefficients in a polynomial.

### Use the Lesson

#### Video Segment 1

Watch the first 20 seconds of the video How to Use the Rational Roots Theorem: Process & Examples, and pause the video with the definition still visible on the screen.

Ask students to generate a list of possible solutions to the polynomial presented previously, 2x^2 + 9x + 4. Remind them that while they already know the correct answers, right now they are to use the Rational Roots Theorem as projected to generate a list of possibilities, including what they know already to be the correct solutions.

#### Video Segment 2

After giving students time to generate their own list of solutions, play the video up through 2:40, enough to work through the polynomial 2x^2 + 9x + 4 along with the narration. Stop the video and ask if there are any questions or clarifications.

If no one brings up the fact that it would have been easier to simply factor that polynomial, ask students if there would be any advantages to using the Rational Roots Theorem in that first case. Ask students if they can imagine any polynomials when it would make more sense to generate a list of possibilities rather than trying to factor the polynomial.

Before watching the third and final segment of the video, project or write the polynomial that the narrator will discuss: x^3 - 6x^2 + 11x - 6. For a collaborative alternative, ask students to pair up and identify a Partner A and a Partner B. Have all A partners try to solve the polynomial by factoring, and all B partners start by generating a list of possible solutions using the Rational Roots Theorem. See which method seems to work fastest and allow students to discuss what they find. If working with an individual student, have them use a timer to determine which method is faster.

#### Video Segment 3

Now play the video through to the end. Students can watch how the narrator generates possibilities and solutions to the polynomial. Make sure they are checking their work, and clear up any misunderstandings or common errors that may have arisen.

### Activity

This is a short activity intended to give students a chance to dig into the Rational Roots Theorem a bit before checking for understanding through a problem set. You may use it as a jigsaw activity by having students break into teams of 4. Alternatively, you can assign each of the four topics to be completed independently. Let students know that they will be making an infographic for the Rational Roots Theorem. They can produce this by hand on paper or using a digital tool such as Piktochart.

#### Research for the Infographic

Provide a time for gathering responses to the following four topics. For each topic, students should generate a short paragraph summary of their work time.

Topic 1: The basics of the Theorem. Students should write a definition of the theorem in their own words (or mathematical symbols). They should also show their understanding of key terms in the definition. Key vocabulary could include leading term, coefficient, polynomial, or factor.

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