Lesson 14

This warm-up asks students to consider what they know about a polynomial when given a few limited facts about the polynomial. In the following activity, which is an Information Gap, students will use some of the thinking they do here to ask precise questions in order to graph a polynomial with a known factor.

Arrange students in groups of 2. Tell students there are many possible answers for the question. After 2 minutes of quiet work time, ask students to briefly compare their responses to their partner’s and see what they have in common and what is different. Follow with a whole-class discussion. 

Invite students to state some things they know must be true about the polynomial, recording these for all to see. In particular, highlight any potential sketches of \(G\), making clear that there are 2 main options depending on which factor has a multiplicity of 2.

This Info Gap activity gives students an opportunity to determine and request the information needed to make a sketch of a polynomial. The goal of this activity is to give students additional practice dividing polynomials when one factor is already known and sketching polynomials from known factors. Since the focus is on working through the logic of the polynomial division to identify other factors and then making a sketch from the linear factors, graphing technology is not an appropriate tool.

Here is the text of the cards for reference and planning:

Tell students they will continue to work with polynomials with known factors. Explain the Info Gap structure, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “What are some things that are helpful to know when sketching a polynomial function?” (the linear factors of the expressions, end behavior, actual points)
• “How did you use the known factor and the equation for the polynomial?” (I used long division to find \((x^3-3x^2-61x+63) \div (x-9)\), and then factored the result, \(x^2+6x-7\) mentally to find the three linear factors to use to make my sketch.)

Highlight for students that the only piece of information they needed starting from a known factor is the equation for the polynomial. From there, they can use division (possibly more than once) to identify the other linear factors and make the sketch.

This activity is optional. Use this activity to give students extra practice rewriting polynomial expressions in different forms and graphing polynomials from the factored form.

Arrange students in groups of 2. Tell students that they are going to write their own polynomial and then their partner is going to factor and graph it. This activity is meant to be flexible based on student needs. While the task statement is written broadly, here are some modifications to consider:

• Restrict all horizontal intercepts to being between -10 and 10.
• Require that at least 1 factor have a multiplicity of 2.
• Allow the students to use graphing technology when they are writing their polynomials.

Once students have completed graphing their partner’s polynomial, discuss the following:

• “Which parts of graphing your partner’s polynomial were tricky? Explain why.”
• “Which parts of graphing your partner’s polynomial went well? What advice would you give to other students?”