 # K12MATH011: Algebra II

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.

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.

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.

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):

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Attribution-NonCommercial-ShareAlike 3.0 United States
is attributed to Khan Academy.
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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):

• 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):

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) = 2x3 + 5x2 – 7x + 1 behaves just like ƒ(x) = 2xas 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):

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):