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?
\(y=(x-3)^2\)
\(y=(x+1)(x-3)^2\)
\(y=(x-3)^3\)
$y=(x-6)(x-3)^2$
Student Response
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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.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Supports accessibility for: Organization; Attention; Social-emotional skills
Student Facing
- For polynomials \(A\)–\(F\):
- Write the degree, all zeros, and complete the sentence about the end behavior.
- Sketch a possible graph.
-
Check your sketch using graphing technology.
Pause here for your teacher to check your work.
- Create your own polynomial for your partner to figure out.
- Create a polynomial with degree greater than 2 and less than 8 and write the equation in the space given.
- Trade papers with a partner, then fill out the information about their polynomial and complete a sketch.
- 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,
\(B(x)=\text-(x+2)(x-2)^2\)
Degree: Zeros:
End behavior: As \(x\) gets larger and larger in the negative direction,
\(C(x)=(x+6)(x+2)^2\)
Degree: Zeros:
End behavior: As \(x\) gets larger and larger in the negative direction,
\(D(x)=\text-(x+6)^2(x+2)\)
Degree: Zeros:
End behavior: As \(x\) gets larger and larger in the negative direction,
\(E(x)=(x+4)(x-2)^3\)
Degree: Zeros:
End behavior: As \(x\) gets larger and larger in the negative direction,
\(F(x)=x^3(x+4)(x-3)^2\)
Degree: Zeros:
End behavior: As \(x\) gets larger and larger in the negative direction,
Your polynomial:
Degree: Zeros:
End behavior: As \(x\) gets larger and larger in the negative direction,
Student Response
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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
-
Sketch a graph for a polynomial function \(y=f(x)\) that has 3 different zeros and \(f(x)\ge0\) for all values of \(x\).
- What is the smallest degree the polynomial could have?
- What is a possible equation for the polynomial? Use graphing technology to see if your equation matches your sketch.
Student Response
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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
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Activity Synthesis
Ask 2–3 students to share their thinking process for making the graph and finding a possible equation.
Design Principle(s): Support sense-making
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
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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.
\(y=(x+3)(x-1)(x-4)\)
near \(x=\text-3\)
near \(x=1\)
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.
\(y=(x+3)^2(x-4)\)
near \(x=\text-3\)
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.
\(y=(x+3)(x-4)^3\)
