Lesson 10

Multiplicity

10.1: Notice and Wonder: Duplicate Factors (10 minutes)

Warm-up

The purpose of this warm-up is to elicit the idea that the number of times a linear factor occurs in an equation is reflected in the shape of the graph near the intercept related to that factor, which will be useful when students sketch graphs of polynomial functions in the following activity. While students may notice and wonder many things about these images, the connections between factors, zeros, and the look of the graph near the related horizontal intercept are the important discussion points.

Launch

Display the graphs for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

$y=(x-3)^2$

Graph of a polynomial function on xy-plane.

$y=(x+1)(x-3)^2$

Graph of a polynomial function on xy-plane.

$y=(x-3)^3$

Graph of a polynomial function on xy-plane.

$y=(x-6)(x-3)^2$

Graph of a function, curved increase (non-linear) until -3 comma 0, where there is a turning point, decreases, then increases again for remainer of graph. crosses x axis at 6 comma 0.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the different look of the horizontal intercepts of the graphs does not come up during the conversation, ask students to discuss this idea and why the parts of each graph near \(x=3\) look the way they do in the different graphs. Tell students that this is an example of multiplicity. The multiplicity of a factor is the number of times the factor occurs when a polynomial is written in factored form. Three of the equations all have \((x-3)^2\), and so the graph of the polynomial near \(x=3\) looks quadratic, and we say that the factor \((x-3)\) has a multiplicity of 2. The other equation has \((x-3)^3\), so the graph looks cubic near \(x=3\), and we say that the factor \((x-3)\) has a multiplicity of 3.

10.2: Sketching Polynomials (25 minutes)

Activity

The goal of this activity is for students to use what they have learned about the end behavior of polynomials and the relationship between factors, zeros, and horizontal intercepts in order to sketch a rough graph from an equation. Since these are only sketches, the vertical scale has been purposely left off the axes to help students focus on features they can identify from the factored equations without computation.

Launch

Arrange students in groups of 2. Tell students that they are going to sketch the graphs of polynomials from factored equations. Display the equation \(f(x)=(x+9)(x+3)(x-4)^2\) and a blank set of axes like those in the activity statement for all to see. Give students quiet think time and ask them to give a signal when they have identified one thing they know is true about the graph of \(f\). Select several students to share and record their responses as points on the graph or notes next to the graph as appropriate (for example, information about the end behavior).

Once all horizontal intercepts have been marked as points on the axis, ask students to tell their partner how they would complete the sketch. Invite partners to share ideas with the class and record these for all to see. Finish the sketch of the polynomial using one of the suggested strategies.

Before students begin the activity, tell them not to worry about the vertical scale since they are only making a sketch. The important thing is to show where the output of the function is positive and where it is negative. Provide access to devices that can run Desmos or other graphing technology.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their verbal and written responses to the question, “After marking the horizontal intercepts, explain how to complete the sketch of the graph of \(f(x)=(x+9)(x+3)(x-4)^2\).” Give students time to meet with 2–3 partners to share and get feedback on their responses. The listener should press for detail and precision in language by asking "Why did you decide to do that?", "How did that help you sketch the graph?", or “Can you say that a different way?” Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students use mathematically correct language when explaining how to sketch the graph of a polynomial.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 4 of the examples to complete. Check to make sure students include at least one of either B or D to gain familiarity with what happens when the leading term is negative. Chunking this task into more manageable parts may also benefit students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

  1. For polynomials \(A\)\(F\):
    1. Write the degree, all zeros, and complete the sentence about the end behavior.
    2. Sketch a possible graph.
    3. Check your sketch using graphing technology.

      Pause here for your teacher to check your work.

  2. Create your own polynomial for your partner to figure out.
    1. Create a polynomial with degree greater than 2 and less than 8 and write the equation in the space given.
    2. Trade papers with a partner, then fill out the information about their polynomial and complete a sketch.
    3. Trade papers back. Check your partner’s sketch using graphing technology.

  

\(A(x)=(x+2)(x-2)(x-8)\)

Degree:                     Zeros: 
End behavior: As \(x\) gets larger and larger in the negative direction,

  

A set of x and y axes are drawn, with arrows on each indicating positive direction. The x-axis is labeled negative 10 to 8 by twos. The y axis is unlabeled.

\(B(x)=\text-(x+2)(x-2)^2\)

Degree:                     Zeros: 
End behavior: As \(x\) gets larger and larger in the negative direction,

  

A set of x and y axes are drawn, with arrows on each indicating positive direction. The x-axis is labeled negative 10 to 8 by twos. The y axis is unlabeled.

\(C(x)=(x+6)(x+2)^2\)

Degree:                     Zeros: 
End behavior: As \(x\) gets larger and larger in the negative direction,

  

A set of x and y axes are drawn, with arrows on each indicating positive direction. The x-axis is labeled negative 10 to 8 by twos. The y axis is unlabeled.

\(D(x)=\text-(x+6)^2(x+2)\)

Degree:                     Zeros: 
End behavior: As \(x\) gets larger and larger in the negative direction,

  

A set of x and y axes are drawn, with arrows on each indicating positive direction. The x-axis is labeled negative 10 to 8 by twos. The y axis is unlabeled.

​​​​​​

\(E(x)=(x+4)(x-2)^3\)

Degree:                     Zeros: 
End behavior: As \(x\) gets larger and larger in the negative direction,

  

A set of x and y axes are drawn, with arrows on each indicating positive direction. The x-axis is labeled negative 10 to 8 by twos. The y axis is unlabeled.

\(F(x)=x^3(x+4)(x-3)^2\)

Degree:                     Zeros: 
End behavior: As \(x\) gets larger and larger in the negative direction,

  

A set of x and y axes are drawn, with arrows on each indicating positive direction. The x-axis is labeled negative 10 to 8 by twos. The y axis is unlabeled.

Your polynomial:

Degree:                     Zeros: 
End behavior: As \(x\) gets larger and larger in the negative direction,

A set of x and y axes are drawn, with arrows on each indicating positive direction. The x-axis is labeled negative 10 to 8 by twos. The y axis is unlabeled.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are unsure how to start a sketch because they’re not used to seeing factors raised to powers greater than 1, remind them to use what they know about end behavior to decide what the ends of the graph should look like, and that they can use their knowledge of multiplicity from the warm-up to see what happens at each zero.

Students can also test individual \(x\)-values to see what happens to the graph in any areas that they’re unsure about. If they get bogged down in calculating exact outputs of the function, it may help them to remember that they only need to know whether the output is positive or negative. For example, if a student wants to know what happens to the graph of \(f(x) = (x - 2)(x + 1)(x + 5)\) between \(x = \text-1\) and \(x = \text-5\), and chooses \(x = \text-3\) to test, they only need to know that the first two factors are negative and the third is positive, and that multiplying them will therefore result in a positive number.

Activity Synthesis

Begin the discussion by revisiting the list of strategies students suggested at the start of the activity. Invite students to share which ones they found useful or to add on a new strategy they figured out while working. For example, a student may have thought they would need to identify \((x,y)\) pairs in between each horizontal intercept in order to figure out if the curve was above or below the \(x\)-axis, but then realized that if they can identify the end behavior and the multiplicity of the factors, they can work from left to right to sketch the curve without figuring out individual points.

At the end of the discussion, select 2–3 partners to share the polynomials they wrote for one another and the resulting sketches.

10.3: Using Knowledge of Zeros (10 minutes)

Optional activity

This activity is optional because it goes beyond the depth of understanding required to address the standard. The goal of this activity is for students to use what they have learned about multiplicity to reason about a polynomial function whose graph is never negative. Provide access to devices that can run Desmos or other graphing technology.

Student Facing

  1. Sketch a graph for a polynomial function \(y=f(x)\) that has 3 different zeros and \(f(x)\ge0\) for all values of \(x\).

    Blank coordinate plane. X axis from -10 to 8, by 2’s. Y axis is blank.
  2. What is the smallest degree the polynomial could have?
  3. What is a possible equation for the polynomial? Use graphing technology to see if your equation matches your sketch.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

What is a possible equation of a polynomial function that has degree 5 but whose graph has exactly three horizontal intercepts and crosses the \(x\)-axis at all three intercepts? Explain why it is not possible to have a polynomial function that has degree 4 with this property.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask 2–3 students to share their thinking process for making the graph and finding a possible equation.

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: “I agree because . . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. For example, a statement such as, “The degree is 6,” can be restated as a question such as, “Does the polynomial have a degree of 6?”
Design Principle(s): Support sense-making
Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, invite students who are stuck to sketch a graph with the same conditions, except for only having a single zero. Once they have drawn a quadratic, ask them what it would look like with a second zero and then a third. Encourage students to arrive at the degrees and multiplicity required for the equation by reflecting on the degrees and multiplicity required for each iteration of their graph.
Supports accessibility for: Social-emotional skills; Conceptual processing

Lesson Synthesis

Lesson Synthesis

Tell students to write an equation for a 4th degree polynomial in factored form that has 3 unique factors of the form \((x-a)\), and then make a sketch of the graph of the polynomial. Here are some questions for discussion:

  • “If you have a 4th degree polynomial with only 3 unique factors, what must be true about the multiplicity of the factors?” (One of the factors has a multiplicity of 2.)
  • “If you changed which of your factors had a multiplicity of 2, how would the end behavior of your graph change?” (The end behavior of my graph would not change since the degree of the polynomial did not change.)
  • “What is one number you could change in your equation that would result in a graph with different end behavior?” (If the leading coefficient was the opposite sign, the end behavior of the new graph would be the opposite of the old graph. Or, increasing the multiplicity of one of the factors from 1 to 2 and changing the overall degree of the polynomial would also change the end behavior.)

10.4: Cool-down - One Last Sketch (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Earlier, we learned to identify the zeros of a polynomial function from the factored expression. These factors let us figure out the points where the graph of the polynomial intersects the horizontal axis. The number of times a factor is repeated also gives us important information: it tells us the shape of the graph at that point on the horizontal axis.

For example, \(y=(x+3)(x-1)(x-4)\) has three factors with no duplicates. This results in a graph that looks a bit like a linear function near \(x=\text-3\), \(x=1\), and \(x=4\) when we zoom in on each of those places.

$y=(x+3)(x-1)(x-4)$

Graph of a polynomial function on xy-plane.

near $x=\text-3$

Graph of a polynomial function on x-axis, zoomed view near x=3.

near $x=1$

Graph of a polynomial function on x-axis, zoomed view near x=1.

near $x=4$

Graph of a polynomial function on x-axis, zoomed view near x=4.

We say that each factor, \((x+3)\), \((x-1)\), and \((x-4)\), has a multiplicity of 1.

For \(y=(x+3)^2(x-4)\), there are still three factors, but two of them are \((x+3)\). This results in a graph that looks a bit like a quadratic near \(x=\text-3\) and a bit like a linear function near \(x=4\). We say that the factor \((x+3)\) has a multiplicity of 2 while the factor \((x-4)\) has a multiplicity of 1.

$y=(x+3)^2(x-4)$

Graph of a polynomial function on xy-plane.

near $x=\text-3$

Graph of a polynomial function on x-axis, zoomed view near x=-3.

near $x=4$

Graph of a polynomial function on x-axis, zoomed view near x=4.

Combining what we know about factors, degree, end behavior, the sign of the leading coefficient, and multiplicity gives us the ability to sketch polynomials written in factored form.

For example, consider what the graph of \(y=(x+3)(x-4)^3\) would look like. The factors help us identify that the function has zeros at -3 and 4. We also know that since \((x+3)\) has a multiplicity of 1 and \((x-4)\) has a multiplicity of 3, the graph looks a bit like a linear polynomial crossing the \(x\)-axis at -3 and a bit like a cubic polynomial crossing the \(x\)-axis at 4. Since this is a 4th degree polynomial with a positive leading coefficient, we know that as \(x\) gets larger and larger in either the negative or positive direction, \(y\) gets larger and larger in the positive direction.

$y=(x+3)(x-4)^3$

Graph of a polynomial function on xy-plane.