What Do You Know About Polynomials?
14.1: What Else is True? (5 minutes)
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.
\(G(x)\) is a polynomial. Here are some things we know about it:
- It has degree 3.
- Both \(x\) and \((x+4)\) are factors of \(G\).
- It has 2 horizontal intercepts, but only 1 is negative.
- Its leading coefficient is negative.
What else do we know is true about \(G(x)\)?
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.
14.2: Info Gap: More Polynomials (25 minutes)
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.
Design Principle(s): Cultivate Conversation
Supports accessibility for: Memory; Organization
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the data card:
- Silently read the information on your card.
- Ask your partner, “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
- Before telling your partner the information, ask, “Why do you need to know (that piece of information)?”
- Read the problem card, and solve the problem independently.
- Share the data card, and discuss your reasoning.
If your teacher gives you the problem card:
- Silently read your card and think about what information you need to answer the question.
- Ask your partner for the specific information that you need.
- Explain to your partner how you are using the information to solve the problem.
- When you have enough information, share the problem card with your partner, and solve the problem independently.
- Read the data card, and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
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.
14.3: Even More Polynomials (10 minutes)
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.
Design Principle(s): Support sense-making; Cultivate conversation
Supports accessibility for: Language; Social-emotional skills
- Without letting your partner see, do the following:
- write a polynomial of degree 3 or 4 in factored form
- sketch the graph of your polynomial
- rewrite its expression in standard form
- On a separate slip of paper, write the standard form of your polynomial along with 1 of the factors (or 2 factors, if the polynomial has degree 4). Trade slips with your partner.
- Use the information your partner gave you about their polynomial to:
- rewrite their polynomial in factored form
- sketch a graph of their polynomial showing all horizontal intercepts
- Once you and your partner have finished graphing, check your factored form and graph with your partner and discuss any differences.
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?”
Display the writing prompt “How would you explain to a student who isn’t here today how to solve a problem like the one in the Information Gap? Include what questions you think they should ask and any other solution steps needed to graph the polynomial.” Give students 2–3 minutes to respond, then invite as many students as time allows to share their explanations. Highlight students with particularly efficient strategies, such as recognizing that the only required piece of information was the expression for the polynomial.
14.4: Cool-down - How Would You Factor? (5 minutes)
Student Lesson Summary
We can look at the same polynomial in many different ways. Let’s think about \(P(x) = x^3 - 7x + 6\). It’s written in standard form, but we could also write it in factored form as \((x-2)(x+3)(x-1)\). If we graph \(P(x)\), we get this:
Depending on what we know about \(P(x)\) and what we want to do, different forms of it will be more useful. If we want to quickly estimate the value of \(P(x)\) for some value of \(x\), the graph might be most helpful. If we don’t know what the graph of \(P(x)\) looks like, the factored form can help us find the zeros and sketch it. If we want to know the general shape of the graph, we can use the standard form to find the end behavior. If we want to know the factors of \(P(x)\) and we only know the standard form, we can guess some possible factors and divide \(P(x)\) by them. If we have the factored form and we want to know the standard form, we can multiply all the factors together.