Back To Course

Intermediate Algebra for College Students23 chapters | 151 lessons

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Melanie Olczak*

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will explain how to find complex zeros of a polynomial. You will be shown examples to apply factoring and the quadratic formula or square roots to solve for complex zeros.

Have you ever been on a roller coaster? Did you know that the path of a roller coaster can be modeled by a mathematical equation called a polynomial? The up and down motion of a roller coaster can be modeled on the coordinate plane by graphing a polynomial. Imagine that you want to find the points in which the roller coaster touches the ground. These points are called the zeros of the polynomial. The **zeros of a polynomial** are also called solutions or roots of the equation.

A **polynomial** is a function that has multiple terms. Each term is made up of variables, exponents, and coefficients. **Variables** are letters that represent numbers. For polynomial functions, we'll use *x* as the variable. **Coefficients** are numbers that are multiplied by the variables. The **degree** of the polynomial is the highest exponent of the variable.

As we mentioned a moment ago, the **solutions** or **zeros** of a polynomial are the values of *x* when the *y*-value equals zero. Polynomials can have real zeros or complex zeros. **Real zeros** to a polynomial are points where the graph crosses the *x*-axis when *y* = 0. When we graph each function, we can see these points. **Complex zeros** are the solutions of the equation that are not visible on the graph. Complex solutions contain imaginary numbers. An **imaginary number** is a number *i* that equals the square root of negative one. So complex solutions arise when we try to take the square root of a negative number.

The **Fundamental Theorem of Algebra** states that the degree of the polynomial is equal to the number of zeros the polynomial contains. So if the largest exponent is four, then there will be four solutions to the polynomial. If the largest exponent is a three, then there will be three solutions to the polynomial, and so on. Thinking in terms of the roller coaster, if it reaches the ground five times, the polynomial degree is five.

We can graph polynomial equations using a graphing calculator to produce a graph like the one below.

Looking at this graph, we can see where the function crosses the *x*-axis. This graph has an *x*-intercept of -2, which means that -2 is a real solution to the equation.

Looking at the equation, we see that the largest exponent is three. This means the polynomial has three solutions. When we look at the graph, we only see one solution. How do we find the other two solutions?

Since the graph only intersects the *x*-axis at one point, there must be two complex zeros. In order to find the complex solutions, we must use the equation and factor.

We will find the complex solutions of the previous problem by factoring. Since this polynomial has four terms, we will use **factor by grouping**, which groups the terms in a way to write the polynomial as a product of its factors.

1. First, we replace the *y* with a zero since we want to find *x* when *y* = 0. Then we group the first two terms and the last two terms.

2. Next, we look at the first two terms and find the greatest common factor. What numbers or variables can we take out of both terms? In the first set of parentheses, we can remove two *x*'s. In the second set of parentheses, we can remove a 3.

3. Now, we group our two GCFs (greatest common factors) and we write (*x* + 2) only once.

4. Now, we can set each factor equal to zero. If we know that the entire equation equals zero, we know that either the first factor is equal to zero or the second factor is equal to zero.

5. Now we solve each equation.

On the right side of the equation, we get -2. We already knew this was our real solution since we saw it on the graph. On left side of the equation, we need to take the square root of both sides to solve for *x*.

When we take the square root, we get the square root of negative 3. We cannot solve the square root of a negative number; therefore, we need to change it to a complex number. To do this, we replace the negative with an *i* on the outside of the square root. We now have two answers since the solution can be positive or negative.

We now have both a positive and negative complex solution and a third real solution of -2. We have successfully found all three solutions of our polynomial.

Let's review what we've learned about finding complex zeros of a polynomial function. First off, **polynomials** are equations with multiple terms, made up of numbers, variables, and exponents. **Variables** are letters that represent numbers, in this case *x* and *y*. **Coefficients** are the numbers that are multiplied by the variables. The **degree** of the polynomial is the highest exponent of the variable. The number of zeros is equal to the degree of the exponent. **Zeros** are the solutions of the polynomial; in other words, the *x* values when *y* equals zero.

**Real zeros** are the values of *x* when *y* equals zero, and they represent the *x*-intercepts of the graphs. **Complex zeros** are values of *x* when *y* equals zero, but they can't be seen on the graph. Complex zeros consist of imaginary numbers. An **imaginary number**, *i*, is equal to the square root of negative one.

The **Fundamental Theorem of Algebra** states that the degree of the polynomial is equal to the number of zeros the polynomial contains. We can tell by looking at the largest exponent of a polynomial how many solutions it will have.

To solve polynomials to find the complex zeros, we can factor them by grouping by following these steps.

- Group the first two terms and the last two terms.
- Find the greatest common factor (GCF) of each group.
- Group the GCFs together in a set of parentheses and write the leftover terms in a single set of parentheses.
- Set each factor equal to zero.
- Solve for
*x*.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
2 in chapter 19 of the course:

Back To Course

Intermediate Algebra for College Students23 chapters | 151 lessons

- AFOQT Information Guide
- ACT Information Guide
- Computer Science 335: Mobile Forensics
- Electricity, Physics & Engineering Lesson Plans
- Teaching Economics Lesson Plans
- FTCE Middle Grades Math: Connecting Math Concepts
- Social Justice Goals in Social Work
- Developmental Abnormalities
- Overview of Human Growth & Development
- ACT Informational Resources
- AFOQT Prep Product Comparison
- ACT Prep Product Comparison
- CGAP Prep Product Comparison
- CPCE Prep Product Comparison
- CCXP Prep Product Comparison
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison

- What Are the 5 Ws in Writing? - Uses & Examples
- Phenol: Preparation & Reactions
- What is a Color Wheel? - Definition & Types
- What Are Abbreviations? - Meaning, Types & Examples
- Zentangle Lesson Plan for High School
- West Side Story Discussion Questions
- Fireboat: The Heroic Adventures of the John J. Harvey Activities
- Quiz & Worksheet - Solvay Process
- Quiz & Worksheet - Acetone Reactions
- Quiz & Worksheet - Themes in A Raisin in the Sun
- Quiz & Worksheet - Act & Rule Utilitarianism Comparison
- Analytical & Non-Euclidean Geometry Flashcards
- Flashcards - Measurement & Experimental Design
- Common Core Math Worksheets & Printables
- Bullying in Schools | Types & Effects of Bullying

- Counseling 101: Fundamentals of Counseling
- Microbiology: Help and Review
- High School World History: Tutoring Solution
- Writing Review for Teachers: Study Guide & Help
- Introduction to Organizational Behavior: Certificate Program
- Data Visualization & Programming Languages
- Introduction to Chemistry: Help and Review
- Quiz & Worksheet - The Phoenicians
- Quiz & Worksheet - Polyphonic Texture in Music
- Quiz & Worksheet - 45-45-90 Triangles
- Quiz & Worksheet - Heart of Darkness Themes & Analysis
- Quiz & Worksheet - King Agamemnon & the Trojan War

- Author Willa Cather: Biography & Works
- Listening to Someone Give Directions in Spanish
- Shape Games for Kids
- Chinese New Year Lesson Plan
- Finding Travel Grants for Teachers
- English Conversation Topics
- What is the International Baccalaureate Primary Years Program?
- When Do You Apply for Community College?
- How to Pass the Series 63
- How to Pass the ATI Exit Exam
- Corporate Team Building Activity Ideas
- 5th Grade Common Core Math Standards

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject