Creating Polynomial Functions as Models for Contextual Problems

In this lesson we will extend your knowledge of polynomial functions by considering how to factor more complex polynomial expressions to find the roots or zeros of a polynomial and using this information to help us graph the functions.

First we will review our work from the previous lesson.

A. Algebraic Operations with Polynomials Expressions

The examples below show how we can add, subtract and multiply polynomial expressions. In the 2nd semester of this course, you will learn how to divide polynomial expressions. We will also review factoring polynomials.

Worked Example
Worked Example
Worked Example

Example 1: Multiplying Polynomial Expressions

Multiply the following polynomial expressions:

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

Sure! Using the FOIL method (First, Outer, Inner, Last), we multiply each term of the polynomial by each term of the second polynomial:

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

$LaTeX: \left(2x^2\cdot x\right)+\left(2x^2\cdot3\right)+\left(-1\cdot x\right)+\left(-1\cdot3\right)$

Now, let's simplify:

LaTeX: 2x^3+6x^2-x-3

So, using the FOIL method, we. get the same result as before:

LaTeX: 2x^3+6x^2-x-3

Example 2: Adding polynomial Expressions

Add the following polynomial expressions:

LaTeX: 3x^3-x^2+9 and LaTeX: -4x^3+3x^2-20

To add the polynomial expressions LaTeX: 3x^3-x^2+9 and LaTeX: -4x^3+3x^2-20 , we simply combine like terms:

$LaTeX: \left(3x^3-x^2+9\right)+\left(-4x^3+3x^2-20\right)$

Combine the like terms:

$LaTeX: \left(3x^3-4x^3\right)+\left(-x^2+3x^2\right)+\left(9-20\right)$

$LaTeX: \left(-x^3+2x^2\right)+\left(-11\right)$

So, the sum of the polynomial expressions is:

LaTeX: -x^3+2x^2-11

Example 3: Factoring

Factor the following polynomial

LaTeX: x^2-5x+6

To factor the quadratic polynomial LaTeX: x^2-5x+6 , we need to find two numbers that multiply to 6 and add up to -5, which are -2 and -3. Then, we can express the quadratic as:

$LaTeX: x^2-5x+6=\left(x-2\right)\left(x-3\right)$

This is the factored form of the quadratic polynomial.

B. Predicting the Behavior of Graphs of Polynomial Functions

Recall that we can predict the behavior of the graphs of polynomial functions by considering the degree of the polynomial and the sign of its leading coefficient.

Here is a worked example

Worked Example
Shelf Check
Shelf Check

Worked Example:

Use words to explain the end behavior of the graph of the polynomial function:

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

Solution

Since the degree of the polynomial is odd and its leading coefficient is negative, the end behavior of the graph is that it extends up to the left and down to the right.

Problem 1:

Perform the given operation of the polynomials given:

$LaTeX: -3x^4+2x^{}-1-\left(2x^4-5x^2+x\right)$

Solution

To subtract the expression LaTeX: -3x^4+2x-1 from $LaTeX: -3x^4+2x-1-\left(2x^4-5x^2+x\right)$ , you can distribute the negative sign and then combine like terms:

LaTeX: -3x^4+2x-1-2x^4+5x^2-x

Combine like terms:

$LaTeX: \left(-3x^4-2x^4\right)+\left(2x-x\right)+\left(-1\right)+5x^2$

LaTeX: -5x^4+x-1+5x^2

So, the corrected expression is:

LaTeX: -5x^4+x+5x^2-1

Problem 2:

Match the graph of the polynomial to the polynomial expression.

Graph A:

Screenshot 2024-11-08 at 10.57.43 AM.png

Graph B:

Screenshot 2024-11-08 at 10.58.16 AM.png

Graph C:

Screenshot 2024-11-08 at 10.59.02 AM.png

Polynomial 1: $LaTeX: f\left(x\right)=5x^3-4x$

Polynomial 2: $LaTeX: g\left(x\right)=-x^4+3x^2+2$

Polynomial 3: $LaTeX: h\left(x\right)=x^4+3x^2+2$

Solution

Graph A is Polynomial 2

Graph B is Polynomial 3

Graph C is Polynomial 1

Instructional Content:

Part I. Factoring Polynomials and Connecting Factors to Roots

In Module 2, we saw that we can solve equations such as

LaTeX: x^2-3x-4=0

By factoring and setting each factor equal to zero as follows:

$LaTeX: \left(x-4\right)\left(x+1\right)=0$

LaTeX: x-4=0 or LaTeX: x+1=0

LaTeX: x=4 or LaTeX: x=-1

These solutions have a geometric interpretation. The zeros of this polynomial are the roots or x-intercepts of the graph of the polynomial, as shown in the graph below.

Screenshot 2024-11-08 at 11.38.22 AM.png

The factors of the polynomial are (x-4) and (x+1) and the zeros are 4 and -1.

In general, we know that if the factors of a polynomial are (x-a) and (x-b), then the zeros of the polynomial are a and b.

Worked Example
Self-Check
Self-Check

Worked Example:

A. Factor the following polynomial

B. List its factors

C. List the zeros of the polynomial

$LaTeX: f\left(x\right)=x^3-5x^2-6x$

Part A: To factor LaTeX: x^3-5x^2-6x , let's first look for common factors:

$LaTeX: x\left(x^2-5x-6\right)$

Now, let's factor the quadratic expression LaTeX: x^2-5x-6 . We need to find two numbers that multiply to LaTeX: 6 and add up to LaTeX: -5 , which are LaTeX: -6 and LaTeX: 1 .

$LaTeX: x\left(x-6\right)\left(x+1\right)$

So, the factored form of LaTeX: x^3-5x^2-6x is $LaTeX: x\left(x-6\right)\left(x+1\right)$

Part B: The factors of $LaTeX: f\left(x\right)$ are LaTeX: x , LaTeX: x-6 , and LaTeX: x+1

Part C: The zeros of $LaTeX: f\left(x\right)$ are LaTeX: 0,6,-1

Self-Check

INSTRUCTION: Identify the zeros of the given polynomial:

$LaTeX: f\left(x\right)=x\left(x+7\right)\left(x-3\right)$

Solution:

The zeros of $LaTeX: f\left(x\right)$ are LaTeX: 0,-7,3

Self-Check

A. Factor the following polynomial

B. Lists its factors

C. List the zeros of the polynomial

$LaTeX: f\left(x\right)=x^3-4x$

Solution:

Part A:

1. The given polynomial is LaTeX: x^3-4x

2. Both terms share a common factor of LaTeX: x .

So, we can factor out LaTeX: x :

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

Next, we notice that LaTeX: x^2-4 is a difference of squares, which can be further factored:

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

Thus, the fully factored form of LaTeX: x^3-4x is:

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

Part B:

The factors are LaTeX: x,x+2,x-2

Part C:

The zeros are LaTeX: 0,-2,2

Part II. Connecting the Degree of a Polynomial to the Number of Turning Points

We have seen that graphs of polynomial functions can vary depending on the degree of the polynomial, the leading coefficient and the number of roots or zeros.

Now we will consider how we can predict the possible number of turning points of the graph of a polynomial.

Definition of a Turning Point

A turning point is a point on a function's graph where the graph changes direction, from increasing to decreasing or vice versa.

For example:

The graph below has 2 turning points.

Screenshot 2024-11-08 at 5.46.54 PM.png

Below is an important fact about predicting the possible number of turning points of a polynomial.

The maximum number of turning points of a polynomial function is always one less than the degree of the function.

This means that if a polynomial has a degree of 5, it will have at most 4 turning points.

Below are some examples of graphs of polynomial of degree 5.

Example 1:

A polynomial of degree 5 with 4 turning points

Screenshot 2024-11-08 at 5.41.37 PM.png

Example 2:

A polynomial of degree 5 with 2 turning points

Screenshot 2024-11-08 at 5.40.18 PM.png

Example 3:

A polynomial of degree 5 with 3 turning points

Screenshot 2024-11-08 at 5.40.57 PM.png

Part III. Putting it all together: Graphing Polynomial Functions

Now that we know about how the degree, the leading coefficient and the factors of a polynomial affect the graph of a polynomial, we can use this information to help us graph polynomials by hand.

In the next lesson, we will learn even more about how the zeros of a polynomial can affect the graphs of a polynomial.

Worked Example:

Given the polynomial, we will identify its end behavior, its zeros, its zeros and the maximum number of turning points. Then we will graph the function.

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

The polynomial is of degree 4 and its leading coefficient is 2. This information leads us to the fact that its end behavior is up to the left and up to the right. Its factors are $LaTeX: \left(x-1\right),\left(2x+3\right),\left(x-3\right)$ . It's zeros are $LaTeX: 1,-\frac{3}{2},3$ . Because its degree is 4, it will have at most 3 turning point.

Let's start by marking the zeros on the axes as shown below:

Screenshot 2024-11-08 at 6.07.19 PM.png

Another point that is important to graph is the y-intercept. Recall that to find the y-intercept, we set LaTeX: x=0 as follows:

$LaTeX: \left(0-1\right)^2\left(2\left(0\right)+3\right)\left(0-3\right)=1\left(3\right)\left(-3\right)=-9$

The y-intercept is $LaTeX: \left(0,-9\right)$ . We have added that point to the graph and then marked the end behavior on the graph. We have also "filled in" a possible shape of the graph. We do not have enough information to know exactly where all of the turning points occur, so those are approximate values.

Screenshot 2024-11-08 at 6.14.12 PM.png

Creating Polynomial Functions as Models for Contextual Problems

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