Back To Course

Math 101: College Algebra12 chapters | 94 lessons | 11 flashcard sets

Watch short & fun videos
**Start Your Free Trial Today**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 55,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Kathryn Maloney*

Kathryn teaches college math. She holds a master's degree in Learning and Technology.

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!

Polynomial long division is very similar to the long division we did as kids, except now we have numbers and *x*s. It takes only two steps that are repeated until you're done.

- Divide the first terms.
- Multiply the result from the divisor, then subtract it from the dividend.
- Repeat!

Let's review the parts of a division problem. We have the divisor divided into the dividend, and of course our answer is the quotient.

Here's our first example: (*x*^2 + 7*x* + 12) / (*x* + 3).

To fill in our long division, *x*^2 + 7*x* + 12 is the dividend, so it goes under the long division symbol. *x*+3 is the divisor, so it goes to the outside, in front of the long division symbol. Now we're ready to start the division. The steps are the same every time. It's easier to show and talk about the steps than listing them, so let's take a look. To find the first term of the quotient, we take the first terms from the divisor and the dividend and divide them. I like to write them as a fraction; it's easier to divide or reduce: *x*^2 / *x* = *x*. *x* is the first term of the quotient; we write it above the long division symbol. To figure out what we will subtract from the dividend, we multiply *x* times the divisor, *x*+3: *x*(*x*+3) = *x*^2 + 3*x*. *x*^2 + 3*x* is written under the dividend, matching like terms. Here's where I like to make life easy. To subtract the polynomials, I simply change their signs and add! So *x*^2 -*x*^2 = 0, and 7*x* - 3*x* = 4*x*. Just like we did in long division, we bring down the next term, which is 12.

Guess what? We do the exact same step! To find the next term of the quotient, we take the new first terms and divide them, so we'll have 4*x* / *x*. I like to write them as a fraction; it's easier: 4*x* / *x* = *4*. 4, or positive 4, is the next term of the quotient. Remember, that gets written next to the *x* above the long division symbol. To figure out what we will subtract from the dividend, we multiply 4 times the divisor, (*x* + 3). (4)(*x* + 3) = 4*x* + 12. 4*x* + 12 is written under the dividend, matching like terms. Next, I like to change the signs and add down: 4*x* + 12 - 4*x* - 12 = 0. It turns out our remainder is zero. What does our answer look like? Just look above the division symbol: *x* + 4. Are you thinking to yourself 'That wasn't too bad'? Let's try one a little longer. Don't panic. Follow the exact same steps, repeat and we'll have an answer.

Divide (15*x*^2 + 26*x* + 8) / (5*x* + 2).

To fill in our long division, 15*x*^2 + 26*x* + 8 is in the dividend, so it goes under the long division symbol. 5*x* + 2 is the divisor, so it goes to the outside in the front of the long division symbol. Now we're ready to start the division. To find the first term of the quotient, we take the first terms from the divisor and dividend and divide them: 15*x*^2 / 5*x* = 3*x*. 3*x* is the first term of the quotient; we write it above the long division symbol. To figure out what we will subtract from the dividend, we multiply 3*x* times the divisor, 5*x* + 2: 3*x*(5*x* + 2) = 15*x*^2 + 6*x*. 15*x*^2 + 6*x* is written under the dividend, matching like terms. I like to change the signs and add straight down: 15*x*^2 - 15*x*^2 = 0, and 26*x* + (-6*x*) = 20*x*. Just like we did in long division, we're going to bring down the next term, which is 8.

And we do exactly the same step! To find the next term of the quotient, we take the new first terms and divide them: 20*x* / 5*x* = *4*. 4, or +4, is the next term of the quotient. Remember, that gets written next to the 3*x* above the long division symbol. To figure out what we will subtract from the dividend, we multiply 4 times the divisor, 5*x* + 2: (4)(5*x* + 2) = 20*x* + 8. 20*x* + 8 is written under the dividend, matching like terms. I'm going to change their signs and add straight down: 20*x* + (-20*x*) = 0; 8 + (-8) = 0. It turns out our remainder is zero. To find your answer, or the quotient, just look above the division symbol: 3*x* + 4. Okay, I know what you're thinking: 'What happens when I have a remainder?' Let's try a long division problem that has a remainder. No magic here! We're going to follow the same steps, except our final answer will look a little different.

(4*x*^2 + 8*x* - 5) / (2*x* +1).

To fill in our long division, 4*x*^2 + 8*x* - 5 is the dividend, so it goes under the long division symbol. 2*x* + 1 is the divisor, so it goes to the outside in front of the long division symbol. Now we're ready to start division. To find the first term of the quotient, we take the first terms from the divisor and dividend and divide them: 4*x*^2 / 2*x* = 2*x*. 2*x* is the first term of the quotient; we write it above the long division symbol. To figure out what we will subtract from the dividend, we multiply 2*x* times the divisor, 2*x*+1: 2*x*(2*x* + 1) = 4*x*^2 + 2*x*. 4*x*^2 + 2*x* is written under the dividend, matching like terms. I like to change their signs and add straight down: 4*x*^2 + (-4*x*^2) = 0, and 8*x* + (-2*x*) = 6*x*. Just like we've done before, we're going to bring down the next term, which is -5.

We're going to do the exact same step again! To find the next term of the quotient, we take the new first terms and divide them: 6*x* / 2*x* = *3*. 3, or +3, is the next term of the quotient. Remember, that gets written next to the 2*x* above the long division symbol. To figure out what we will subtract from the dividend, we multiply 3 times the divisor, 2*x*+1. (3)(2*x* + 1) = 6*x* + 3. 6*x* + 3 is written under the dividend, matching like terms. Once again, I like to change the signs and add straight down, so 6*x* + (-6*x*) = 0; -5 + (-3) = -8. It turns out our remainder is -8. We're going to write the remainder as a fraction: -8 / (2*x* + 1). So our final answer is going to look like (2*x* + 3) + -8/(2*x* + 1).

Dividing polynomials using long division takes only two steps that are repeated until you're done!

- Divide the first terms.
- Multiply that quotient by the divisor and subtract it from the dividend.
- And repeat!

Remember how much work you did with long division? How much paper you used? The same is true for polynomial long division. Don't skip steps, use your paper and don't try this in your head. Multiple-choice answers look very similar for a reason! One small mistake and the whole problem is wrong. There isn't partial credit on a multiple-choice test.

After you complete this lesson you should have no problems on dividing polynomials.

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

Create
your account

Already a member? Log In

BackDid you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 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
8 in chapter 6 of the course:

Back To Course

Math 101: College Algebra12 chapters | 94 lessons | 11 flashcard sets

- What Are the Five Main Exponent Properties? 5:26
- How to Define a Zero and Negative Exponent 3:13
- How to Simplify Expressions with Exponents 4:52
- Rational Exponents 3:22
- Simplifying Expressions with Rational Exponents 7:41
- How to Graph Cubics, Quartics, Quintics and Beyond 11:14
- How to Add, Subtract and Multiply Polynomials 6:53
- How to Divide Polynomials with Long Division 8:05
- Dividing Polynomials with Long and Synthetic Division: Practice Problems 10:11
- Go to Exponents and Polynomials

- Go to Functions

- English 310: Short Stories
- Nurse Entrance Test (NET): Exam Prep & Study Guide
- CCMA Basic Exam: Study Guide & Test Prep
- Personalized Learning in the Classroom
- Certified Emergency Nurse (CEN): Study Guide & Exam Prep
- 19th-Century British Short Stories
- 19th & Early 20th-Century American Naturalist Short Stories
- 19th-Century Russian Realism in Short Stories
- Early 20th-Century Feminist Short Stories
- Medical Records & HIPAA
- Professional Publications in Literacy
- Dyslexia Programs in Texas
- Study.com's Teacher Edition
- Study.com School Plans
- Study.com's Virtual Classrooms
- How to Set Up a Class and Invite Students in Your Study.com Virtual Classroom
- How to View Grades and Export CSVs in Your Study.com Virtual Classroom

- Tree Diagrams in Math: Definition & Examples
- How to Square a Trinomial
- How to Find the Least Common Multiple of Expressions
- Converting 1 Radian to Degrees
- Rip Van Winkle Literary Criticism
- Working Papers in the Audit Process: Definition & Development
- German Genitive Pronouns
- Treaty of Paris Lesson Plan
- Quiz & Worksheet - The Advancement of Learning by Francis Bacon
- Quiz & Worksheet - Practice Graphing Radical Functions
- Quiz & Worksheet - Finding the Major Axis of an Ellipse
- Quiz & Worksheet - Types of Triangles & Their Properties
- Quiz & Worksheet - Point of Care Technology in Healthcare
- Developing Presentation Skills Flashcards
- Hypothesis Testing in Statistics Flashcards

- 9th Grade English: Tutoring Solution
- Financial Accounting: Homework Help Resource
- CLEP Human Growth and Development: Study Guide & Test Prep
- Effective Communication in the Workplace: Help and Review
- Science 102: Principles of Physical Science
- Integration & Integration Techniques - AP Calculus: Homeschool Curriculum
- Inflows, Outflows and Restrictions: Homeschool Curriculum
- Quiz & Worksheet - Using Sharp Objects in a Veterinary Office
- Quiz & Worksheet - Veterinary Hospitals & Large Animal Facilities
- Quiz & Worksheet - Types of Hyperparathyroidism
- Quiz & Worksheet - Sudden Infant Death Syndrome Risks & Prevention
- Quiz & Worksheet - Characteristics of Tissue Swelling

- Biotechnology Applications in Medicine & Genetics
- The Muse Terpsichore: Greek Mythology, Definition
- What are Passing Scores for the Praxis Tests?
- Buoyancy Experiments for Kids
- What Are the Pass Rates for the CLEP?
- What is the Center for Change in Utah?
- How Long Should I Study For the GMAT?
- Is Studying Computer Science Hard?
- Probability Lesson Plan
- How to Get Tuition Reimbursement
- What is the Center for Deployment Psychology?
- Albert Einstein Experiments for Kids

Browse by subject