# 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

Higher degree polynomials include those with a degree of 3 and higher, and they require a slightly different technique than those with lower degrees. Use the rational roots theorem to explore possible solutions to example problems.

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

Polynomials can be added, subtracted, and multiplied similarly to regular numbers once the variables have been organized. Learn the steps required to apply the principles of addition, subtraction, and multiplication to polynomials through a series of example problems.

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

One of the best methods for factoring polynomials of a higher degree is using the quadratic form. Discover the steps and rules of how to change expressions into the quadratic form through examples provided in this lesson.

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

Polynomials can be divided using long division by dividing the first terms, multiplying the quotient by the divisor, subtracting it from the dividend, and continuously repeating the steps. Learn more about the steps for dividing polynomials with long division and why it is necessary to repeat the steps to complete the operation.

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

Synthetic division is a shortened form of dividing polynomials by monomials that is shorter than using long division. Explore several examples of the synthetic division of polynomials and follow the correct order of steps to arrive at much simpler expressions.

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

In algebra, the remainder theorem is a formula used to find the remainder when dividing a polynomial by a linear polynomial, while the factor theorem links a polynomial's zeros and factors. Learn what this means by exploring the definitions and examples of both theorems.

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

Polynomials can be divided using both long and synthetic division, and so it is important to be comfortable using both. Learn the steps of both long and synthetic division of polynomials and use them to solve practice problems.

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

A rational zero is a number that can be expressed as a fraction of two numbers, while an irrational zero has a decimal that is infinite and non-repeating. Learn how to use the rational zeros theorem and synthetic division, and explore the definitions and work examples to recognize rational zeros when they appear in polynomial functions.

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

The Fundamental Theorem of Algebra states that every polynomial function includes a minimum of one complex zero. Using an analogy based on bank fees, explore an explanation and examples of how to apply the theorem, and learn about imaginary, repeated, and complex solutions.

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### Other Chapters

Other chapters within the Remedial Precalculus course

- Working with Linear Equations
- Working With Inequalities Review
- Absolute Value Equations Review
- Working with Complex Numbers Review
- 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 Review
- 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 & Conic Sections Review
- Polar Coordinates and Parameterizations
- Continuity
- Limits
- Sets in Algebra