**Unit 4: General Polynomial Functions**
*Having been introduced to first-power (linear) and second-power
(quadratic) forms, you are now ready to approach general polynomial
equations and functions. Polynomials are expressions that are made up of
sums and differences of terms, such that the coefficients are all real
numbers and the exponents on the variables are all whole numbers. This
unit begins with an introduction to polynomials in one variable,
stressing the degree (highest power) and what the degree may imply about
a polynomial function’s solutions. You will learn to perform basic
operations (adding, subtracting, multiplying, and dividing) on
polynomial expressions and how to apply these operations to the
composition of functions. You will learn how to graph polynomial
functions and end the unit with a fairly detailed method for finding all
the roots (solutions) of a polynomial.*

**Unit 4 Time Advisory**

Completing this unit should take approximately 7 hours and 45 minutes.

☐ Subunit 4.1: 45 minutes

☐ Subunit 4.2: 2 hours and 30 minutes

☐ Subunit 4.3: 1 hour

☐ Subunit 4.4: 1 hour

☐ Subunit 4.5: 2 hours and 30 minutes

**Unit4 Learning Outcomes**

Upon successful completion of this unit, you will be able to:
- Add, subtract, divide, and multiply polynomial functions.
- Find the compositions of functions.
- Use the composition of two functions to determine whether they are
inverses.
- Determine the basic graph of a polynomial function.
- Find the roots of a polynomial function using Descartes’s method.

Standards Addressed (*Common Core*):
- CCSS.Math.Content.HSA-APR.A.1
- CCSS.Math.Content.HSA-APR.B.2
- CCSS.Math.Content.HSA-REI.D.10
- CCSS.ELA-Literacy.RST.11-12.1
- CCSS.ELA-Literacy.RST.11-12.2
- CCSS.ELA-Literacy.WHST.11-12.1

**4.1 Introduction to Polynomials**
*The most common of all functions that are used in the scientific and
business communities today are polynomials. This is because they are
easy to use and have centuries of study upon them. Polynomials are
single terms or sums and differences of terms, such that the powers on
all possible variables are whole numbers. That is, if there is a
variable (and there doesn’t have to be), the variable must have a whole
number power or else it’s not a polynomial.*

**Explanation: Carl Stitz and Jeff Zeager’s**Link: Carl Stitz and Jeff Zeager’s*Precalculus*: “Chapter 3: Polynomial Functions”*Precalculus*: “Chapter 3: Polynomial Functions” (PDF)

Instructions: Read pages 235–245 for a detailed discussion of polynomial basics, such as definition, recognition, degree, and evaluation. There is also discussion of graphing.

Reading the material and writing notes should take approximately 45 minutes.

Standards Addressed (*Common Core*):

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. It is attributed to Carl Stitz and Jeff Zeager.**Explanation: Khan Academy’s “Terms, Coefficients and Exponents in a Polynomial”**Link: Khan Academy’s “Terms, Coefficients and Exponents in a Polynomial” (YouTube)

Instructions: This video discusses terms, coefficients, and exponents in a polynomial.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (*Common Core*):

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to Khan Academy.

**4.2 Operations with Polynomials**
*Since polynomials may represent real numbers, polynomial functions may
be added, subtracted, multiplied, and divided. Polynomial functions play
an important role in business and the sciences, as many processes can be
precisely modeled using polynomial functions. For this reason, we need
to know as much about working with polynomial equations as possible.
Also, techniques learned now will be useful later, as you will see these
techniques again and again.*

**Explanation: Khan Academy’s “Addition and Subtraction of Polynomials”**Link: Khan Academy’s “Addition and Subtraction of Polynomials” (YouTube)

Instructions: This video discusses the process of adding and subtracting polynomials.

Watching the video and writing notes should take approximately 30 minutes.

Standards Addressed (*Common Core*):Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to Khan Academy.

**Explanation: Khan Academy’s “Multiplication of Polynomials”**Link: Khan Academy’s “Multiplication of Polynomials” (YouTube)

Instructions: This video discusses the process of multiplying polynomials.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (*Common Core*):

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to Khan Academy.**Did I Get This? Activity: Khan Academy’s “Adding and Subtracting Polynomials”**Link: Khan Academy’s “Adding and Subtracting Polynomials” (HTML)

Instructions: This is a self-quiz on adding and subtracting polynomials. Note that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (*Common Core*):

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to Khan Academy.**Did I Get This? Activity: Khan Academy’s “Multiplying Polynomials”**Link: Khan Academy’s “Multiplying Polynomials” (HTML)

Instructions: This is a self-quiz on multiplying polynomials. Note that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (*Common Core*):

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to Khan Academy.

**4.3 Dividing Polynomials**
- **Explanation: Khan Academy’s “Polynomial Division”**
Link: Khan Academy’s “Polynomial
Division”
(YouTube)

Instructions: This video discusses the general process of dividing
polynomials.

Watching the video and writing notes should take approximately 30
minutes.

Standards Addressed (*Common Core*):

```
- [CCSS.Math.Content.HSA-APR.B.2](http://www.corestandards.org/Math/Content/HSA/APR/B/2)
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial-ShareAlike 3.0 United States
License](http://creativecommons.org/licenses/by-nc-sa/3.0/us/). It
is attributed to Khan Academy.
```

**Explanation: Khan Academy’s “Synthetic Division”**Link: Khan Academy’s “Synthetic Division” (YouTube)

Instructions: This video discusses the process of synthetic division. Note that unlike general polynomial division, this only works for dividing a polynomial by a binomial. This is a fast way to determine if a particular value is a root of the polynomial (the remainder is zero for synthetic division, in that case).

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (*Common Core*):

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to Khan Academy.**Explanation: Khan Academy’s “Why Synthetic Division Works”**Link: Khan Academy’s “Why Synthetic Division Works” (YouTube)

Instructions: This video discusses the process of synthetic division, explaining how it works, in plain English.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (*Common Core*):

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to Khan Academy.

**4.4 Polynomial Equations and Functions**
*Polynomials form smooth curves with no breaks or skips. Their graphs
have no breaks in them and their extremes are dominated by the highest
power term. For example, the function* ƒ(x) = 2x^{3} +
5x^{2} – 7x + 1 *behaves just like* ƒ(x) = 2x^{3 }*as
the values of x become more and more positive or negative. The following
material will make it easy to understand and graph these equations and
functions.*

**Did I Get This? Activity: Khan Academy’s “Views of a Function”**Link: Khan Academy’s “Views of a Function” (HTML)

Instructions: This is a self-quiz on general functions, including polynomials (which include linear and quadratic functions). Note that there are hints available if needed.

Completing the practice quiz and writing notes should take approximately 1 hour.

Standards Addressed (*Common Core*):

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to Khan Academy.

**4.5 Finding Roots (Solutions) to Polynomial Functions**
*Finding the roots of any polynomial is possible, but it requires time
to master. In general, great minds came up with methods for solving
polynomials nearly 300 years ago. The methods are still useful today and
can be translated to computer programming.*

**Explanation: CK-12: “Finding all Solutions of Polynomial Functions”**Link: CK-12: “Finding all Solutions of Polynomial Functions” (HTML and YouTube)

Instructions: Review the examples, practice problems, and videos on the accumulated methods for finding all roots (solutions) for any given polynomial. The material article covers the rational root theorem as well as the use of synthetic division to quickly find all real roots. Note: Once all real roots are exhausted, what remains, in pairs, will be irrational roots. Methods for finding those are demonstrated as well. This is an accumulation of all techniques learned to this point. Now, consider conjugate pairs of complex numbers (like 2 -3i and 2 + 3i or 7 + 5i and 7 – 5i). Experiment with pairs of conjugate complex roots and single or odd numbers of complex roots. Write a one-page essay explaining why any complex roots of a polynomial with real coefficients*must*be in conjugate pairs.

Completing this activity should take approximately 1 hour and 30 minutes.

Standards Addressed (*Common Core*):- CCSS.Math.Content.HSA-APR.B.2
- CCSS.ELA-Literacy.RST.11-12.1
- CCSS.ELA-Literacy.RST.11-12.2
- CCSS.ELA-Literacy.WHST.11-12.1

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License. It is attributed to CK-12.**Checkpoint: MyOpenMath: “Beginning and Intermediate Algebra: Chapter 5 Review”**Link: MyOpenMath: “Beginning and Intermediate Algebra: Chapter 5 Review” (HTML)

Instructions: This is an assessment on understanding polynomials. Work all 26 problems. Answers are available for viewing by a button on each problem. There is also a link for a printed version at the bottom left of the page. Note: New versions of the 26 problems can be created by accessing the link in the upper right corner of the page. You may take a nearly unlimited number of attempts with different problems.

Completing this assessment should take approximately 1 hour.

Standards Addressed (*Common Core*):

Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License.