# Ch 16: Higher-Degree Polynomial Functions

### About This Chapter

## Higher-Degree Polynomial Functions - Chapter Summary and Learning Objectives

Instructors teaching this chapter on higher-degree polynomials, such as cubics, quartics and quintics, introduce you to methods for factoring these types of functions and solving polynomial equations. Practice problems are also included to help you get the hang of determining which technique to use and when, along with lessons showing you the practical applications of the fundamental theorem of algebra. Once you've worked your way through this chapter, you should be able to:

- Perform basic arithmetic operations on polynomials
- Convert higher-degree polynomials into quadratic form
- Use synthetic and long division to factor higher-degree polynomials
- Apply the remainder, factor and rational zero theorems
- Determine the number of solutions to a polynomial function

Video | Objectives |
---|---|

Solving Polynomials of Degree Greater Than or Equal to Three | Learn how to solve polynomial equations of the third degree and higher. |

How to Add, Subtract and Multiply Polynomials | Understand how to perform arithmetic operations on polynomials. |

Factoring Polynomials Using Quadratic Form: Steps, Rules & Examples | Identify higher-degree polynomials that can be written in quadratic form. Learn how to factor them. |

How to Divide Polynomials with Long Division | Explore techniques for performing long division on higher-degree polynomials. |

How to Use Synthetic Division to Divide Polynomials | Use synthetic division to divide a polynomial by a monomial. |

Remainder Theorem & Factor Theorem: Definition & Examples | Examine how the remainder and factor theorems can be used as an alternative to synthetic or long division. |

Dividing Polynomials with Long and Synthetic Division: Practice Problems | Get extra practice determining when and how to use synthetic or long division. |

Finding Rational Zeros Using the Rational Zeros Theorem & Synthetic Division | Explore the uses of the rational zero theorem and synthetic division to factor polynomials. |

Fundamental Theorem of Algebra: Explanation and Example | Learn how a polynomial's degree can be used to determine the number of solutions it has. |

### 1. Solving Higher Degree Polynomials

After watching this video lesson, you will be able to solve polynomials where the degree is three or higher. Learn the technique that you can use to solve any one of these polynomials.

### 2. How to Add, Subtract and Multiply Polynomials

Adding, subtracting and multiplying polynomials are, basically, the same as adding, subtracting and multiplying numbers. They only difference is that we have a pesky variable to worry about, but this video will show you that's no problem, so no worries! This method has worked for many of my students, and I think it will work for you, too!

### 3. Factoring Polynomials Using Quadratic Form: Steps, Rules & Examples

Factoring a polynomial of degree 4 or higher can be a difficult task. However, some polynomials of higher degree can be written in quadratic form, and the techniques used to factor quadratic functions can be utilized.

### 4. How to Divide Polynomials with Long Division

Arithmetic long division and polynomial long division are very similar. Yes, it's a long process, but once you have the rhythm you will get every problem correct!

### 5. How to Use Synthetic Division to Divide Polynomials

Synthetic division is a 'short-cut' way of dividing a polynomial by a monomial. You still need to know long division, sorry, but this method is way fun when you're dividing by a monomial!

### 6. Remainder Theorem & Factor Theorem: Definition & Examples

In this lesson, you will learn about the remainder theorem and the factor theorem. You will also learn how to use these theorems to find remainders and factors of polynomials.

### 7. Dividing Polynomials with Long and Synthetic Division: Practice Problems

Let's look at some more polynomial division problems. We will use long division and synthetic division, but this time we will have a couple of more involved problems. So, get out some paper and a pencil and let's begin!

### 8. Finding Rational Zeros Using the Rational Zeros Theorem & Synthetic Division

After completing this lesson, you will know what the rational zeros theorem says. You will also know how to apply this theorem to find zeros of polynomial functions.

### 9. Fundamental Theorem of Algebra: Explanation and Example

In this lesson, you will learn what the Fundamental Theorem of Algebra says. You will also learn how to apply this theorem in determining solutions of polynomial functions.

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### Other Chapters

Other chapters within the Remedial Precalculus course

- Working with Linear Equations
- Working With Inequalities
- Absolute Value Equations
- Working with Complex Numbers
- Systems of Linear Equations
- Mathematical Modeling
- Introduction to Quadratics
- Working with Quadratic Functions
- Geometry Basics for Precalculus
- Functions: Basics for Precalculus
- Understanding Function Operations
- Graph Symmetry
- Graphing with Functions
- Rate of Change
- Polynomial Functions Basics
- Rational Functions & Difference Quotients
- Rational Expressions and Function Graphs
- Exponential Functions & Logarithmic Functions
- Using Trigonometric Functions
- Trigonometric Graphs
- Solving Trigonometric Equations
- Trigonometric Identities
- Trigonometric Applications
- Graphing Piecewise Functions
- Vectors, Matrices and Determinants
- Mathematical Sequences and Series
- Analytic Geometry and Conic Sections
- Polar Coordinates and Parameterizations
- Continuity
- Limits
- Sets in Algebra