Extended Graphing of polynomial Functions and Using Polynomials to Represent a Melody

In this lesson, you will extend your knowledge of graphing polynomial functions by considering the multiplicity of roots.

Finally, you will see an example of how you can represent a musical melody using a polynomial function.

Later, in Module 8, you will build on the application of polynomial functions in context.

Let’s review your work from the previous lesson.

A. Factors, zeros and turning points of polynomial functions

Recall that in the previous lesson, we explored how algebraic expressions of polynomials can help us identify the factors, zeros and maximum number of turning points of their graphs.

Worked Example
Self Check

Worked Example:

Given the polynomial,

$LaTeX: f\left(x\right)=2\left(x-2\right)^3\left(x+1\right)^{}\left(x-5\right)$

Part A: What are its factors?

Part B: What are its zeros?

Part C: What is the maximum number of turning points for the polynomial?

Solution

Part A: The factors are $LaTeX: \left(x-2\right),\left(x+1\right)$ and $LaTeX: \left(x-5\right)$

Part B: The zeros are LaTeX: x=2,x=-1 and LaTeX: x=5

Part C: Since the degree of the polynomial is 5, the maximum number of turning points is 4.

Self Check

Given the polynomial in factored form,

$LaTeX: f\left(x\right)=2\left(x-1\right)\left(x+3\right)^2$

Part A: Identify the factors of the polynomial

Part B: Identify the maximum number of turning points

Part C: Identify the zeros of the polynomial

Solution

Part A: The factors are $LaTeX: \left(x-1\right)$ and $LaTeX: \left(x+3\right)$

Part B: Maximum number of turning points is 2

Part C: Zeros are 1 and -3

Instructional Content:

Part I. Multiplicity of Roots or Zeros of Polynomial Functions

We will define what is meant by the multiplicity of a root or zero of a polynomial. Let's consider some examples of polynomials in their factored forms:

Polynomial A:

$LaTeX: 5\left(x+1\right)^2\left(x-2\right)^3$

Polynomial B:

$LaTeX: x^4\left(x-3\right)^3\left(x+5\right)$

We know from our previous work that the factors of polynomial A are (x+1) and (x-2) and that the roots or zeros are x = -1 and x = 2. Since the factor (x+1) is squared, we say that the root or zero x = -1 has multiplicity 2. Since (x-2) is raised to the third power, we say that the root or zero x = 2 has multiplicity 3.

In general, we have

is a zero of a polynomial and the exponent on the term that produced the root is LaTeX: k

then we say that LaTeX: r

has multiplicity LaTeX: k

. Zeros with a multiplicity of 1 are often called simple zeroes. $LaTeX: \left(x-r\right)^k$ would be a term in the factored form of the polynomial. So LaTeX: r

is the zero and LaTeX: k

is exponent on the term $LaTeX: \left(x-r\right)$ .

Polynomial B $LaTeX: x^4\left(x-3\right)^3\left(x+5\right)$ has 3 roots, they are LaTeX: x=0,x=3, and LaTeX: x=-5 . The multiplicity of is 4. The multiplicity of LaTeX: x=3 is 3 and the multiplicity of is 1. We say that a root is repeated if it has a multiplicity larger than 1. So and are repeated roots.

Related to this idea, an important theorem is given below:

Fundamental Theorem of Algebra:

If $LaTeX: P\left(x\right)$ is a polynomial of degree LaTeX: n then $LaTeX: P\left(x\right)$ will have exactly zeroes, some of which may repeat.

The degree of Polynomial B is 8. The Fundamental Theorem of Algebra states that Polynomial B has 8 zeroes. The zero x = 0 is repeated 4 times, the zero x = 3 is repeated 3 times and the zero x = -5 is a simple zero. So 4 + 3 +1 = 8 is the number of zeros, with some of those zeros being repeated zeros.

Worked Example
Self-Check

Worked Example:

Given the polynomial in factored form, identify the zeros of the polynomial and the multiplicities of each zero.

$LaTeX: x\left(x-3\right)^2\left(x+1\right)^3$

The zeroes/roots of this polynomial are: LaTeX: x=0,x=3 and LaTeX: x=-1 .

For the multiplicities just remember that the multiplicity of the zero/root is simply the exponent on the term that produces the zero/root. Therefore, the multiplicities of each zero/root is,

LaTeX: x=0 : multiplicity 1

LaTeX: x=3 : multiplicity 2

LaTeX: x=-1 : multiplicity 3

Self-Check

Part A: Identify the zeros or roots of the given polynomial

$LaTeX: \left(x+1\right)^2\left(x-4\right)^2\left(x-1\right)\left(x+3\right)^3$

Part B: Give the multiplicity of each root.

Solution:

Part A: The zeroes/roots of this polynomial are: LaTeX: x=-1,x=4,x=1 and LaTeX: x=-3

Part B: For the multiplicities just remember that the multiplicity of the zero/root is simply the exponent on the term that produces the zero/root. Therefore, the multiplicities of each zero/root is,

LaTeX: x=-1 : multiplicity 2

LaTeX: x=4 : multiplicity 2

LaTeX: x=1 : multiplicity 1

LaTeX: x=-3 : multiplicity 3

Part II. Connecting the Multiplicity of the Roots to the Graphical Behavior of a Polynomial Function

The multiplicity of a root of a polynomial can give us important information about the behavior of the graph of the polynomial at that root.

In the examples below, you can see how the multiplicity of each zero/root affects the graph of the polynomial.

Worked Example
Worked Example
Self-Check

Worked Example:

Polynomial:

$LaTeX: f\left(x\right)=x\left(x-3\right)^3\left(x+2\right)$

Roots: LaTeX: x=0,x=3 and LaTeX: x=-2 .

Multiplicity of each root:

LaTeX: x=0 has a multiplicity of 1 so the graph will pass through the x-axis at since the multiplicity is odd

LaTeX: x=3 has a multiplicity of 2 so the graph will bounce off the x-axis at since the multiplicity is even

This root is called a "touch root".

LaTeX: x=-2 has a multiplicity of 1 so the graph will pass through the x-axis at since the multiplicity is odd

Graph:

Screenshot 2024-11-12 at 10.18.58 AM.png

Worked Example:

Polynomial:

$LaTeX: g\left(x\right)=\left(x-1\right)^2\left(x+2\right)^3$

Roots: LaTeX: x=1 and LaTeX: x=-2 .

Multiplicity of each root:

LaTeX: x=1 has a multiplicity of 2 so the graph will bounce off the x-axis at because the multiplicity is even. is called a “touch” root.

LaTeX: x=-2 has a multiplicity of 3 so the graph will pass through the x-axis at since the multiplicity is odd

Graph:

Screenshot 2024-11-12 at 10.18.33 AM.png

Self-Check

Given the polynomial in factored form,

$LaTeX: g\left(x\right)=\left(x+2\right)^{}\left(x-1\right)^2\left(x-3\right)^2$

Part A: Identify the roots of the polynomial

Part B: Identify the multiplicity of each root.

Part C: Identify the behavior of the graph of the polynomial at each root.

Solution:

Part A: The roots of this polynomial are: LaTeX: x=-2,x=1 and LaTeX: x=3

Part B: The multiplicities of each zero/root is,

LaTeX: x=-2 : multiplicity 1

LaTeX: x=1 : multiplicity 2

LaTeX: x=3 : multiplicity 2

Part C:

At x = -2, the graph of f passes through the x-axis.

At x = 1 and at x = 3 the graph of f bounces off of the x-axis. x = 1 is called a “touch” root.

Part III. Putting it all together: Graphing Polynomial Functions & Writing an Equation from the Graph

Now that you know how the degree, the leading coefficient and the roots and their multiplicities of a polynomial affect the graph of a polynomial, you can use this information to graph polynomials by hand.

Worked Example:

Consider the polynomial:

$LaTeX: f\left(x\right)=-\left(x-2\right)^2\left(x+3\right)^2$

Identify thew following:

Part A: The Degree of the Polynomial

Part B: The leading coefficient of the Polynomial

Part C: The End Behavior of the Polynomial

Part D: The Roots of the Polynomial

Part E: The Multiplicity of Each Root of the Polynomial

Part F: The y-intercept of the Polynomial

Part G: The maximum number of turning points of the Polynomial

Use the information in Parts A-G to sketch a graph of the polynomial:

Screenshot 2024-11-12 at 10.31.38 AM.png

In the next example, we show a graph of a polynomial and then use the information on the graph to write a possible equation for the function.

Writing the Equation from the graph

Watch the Video to see how you can write a possible equation of a polynomial function from a given graph.

Self Check:

Given the graph of the polynomial below, write a possible equation for the function:

Screenshot 2024-11-14 at 5.16.39 PM.png

Solution

$LaTeX: f\left(x\right)=\frac{1}{4}\left(x-2\right)^2\left(x+2\right)\left(x+3\right)$

The roots are x = 2, x = -2 and x = -3. Since x = 2 is a touch root, we square the (x-2) factor.

To find the leading coefficient, we use the y-intercept (0,6) as follows:

$LaTeX: f\left(x\right)=a\left(x-2\right)^2\left(x+2\right)\left(x+3\right)$

Substitute 6 in for y and 0 in for x:

$LaTeX: 6=a\left(0-2\right)^2\left(0+2\right)\left(0+3\right)$

$LaTeX: 6=a\left(4\right)\left(2\right)\left(3\right)$

LaTeX: 6=24a

$LaTeX: \frac{6}{24}=a$

$LaTeX: \frac{1}{4}=a$

Part IV Using a Polynomial to Represent a Musical Melody

In this part of the lesson, we will demonstrate an example of how we can create a polynomial representation of a musical melody from a given data set that represents the melody. Consider the melody for the Children’s Song, “The Wheels on the Bus”.

A portion of the sheet music is shown below. Don’t worry if you don’t know how to read music. We will provide the data that represents the music. We have created a data set that roughly represents the first part of the melody.

Screenshot 2024-11-14 at 5.33.02 PM.png

A graph of a Data Set that represents the Melody is shown below:

Screenshot 2024-11-14 at 6.00.11 PM.png

We can use the curve fitting tool in Desmos to create a polynomial function that represents the data set.

The graph below shows a graph of the polynomial along with a graph of the data.

Screenshot 2024-11-14 at 6.01.01 PM.png

Here is the algebraic representation for the polynomial shown above with the domain restricted to $LaTeX: 0\le x\le7$

$LaTeX: f\left(x\right)=0.02341x^5-0.448984x^4+2.99x^3-8.18x^2+8.932x-1.05137\left\lbrace0\le x\le7\right\rbrace$

You will have an opportunity to use Desmos to create a polynomial representation of the Melody for the state song of New Mexico in the Module Assessment.

Extended Graphing of polynomial Functions and Using Polynomials to Represent a Melody

Worked Example

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Worked Example

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Worked Example

Worked Example

Self-Check

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