Back To Course

Precalculus: Help and Review11 chapters | 88 lessons

Instructor:
*DaQuita Hester*

DaQuita has taught high school mathematics for six years and has a master's degree in secondary mathematics education.

Trinomials can have different leading coefficients and therefore, have to be factored differently. Learn how to factor and solve any trinomial. Then, test your knowledge with a quiz.

A **trinomial** is an equation that consists of three terms. For this lesson, we will examine trinomials written in the form *ax^2 + bx + c*, where *a*, the leading coefficient, does not equal zero.

To **factor** a trinomial means to rewrite it as a product of two binomials. This means that we are going to rewrite the trinomial in the form *(x + m) (x + n)*. Your task is to determine the value of *m* and *n*.

For factoring, trinomials are divided into two groups: those with a leading coefficient of 1 and those with a leading coefficient not equal to 1. Let's examine both.

Use the following steps to factor the trinomial *x^2 + 7x + 12*.

Step 1: **Determine the factor pairs of c that will add to get b**. For *x^2 + 7x + 12*, *a* = 1, *b* = 7, and *c* = 12. So to complete this step, we have to figure out which factor pairs of 12 will add together to equal 7. Let's list them. The factor pairs are 1 & 12, 2 & 6, and 3 & 4. The third pair is what we need, because the sum of these two numbers is 7, which is our *b*.

Step 2: **In separate parentheses, add each number to x**. Here, we are simply going to take our factor pair and add each one to

To ensure that we've factored correctly, let's multiply (x + 3) by (x + 4) to see if we get our original trinomial. When we distribute, we see that (x)(x) = x^2, (x)(4) = 4x, (3)(x) = 3x, and (3)(4) = 12. By combining our like terms, we get x^2 + 7x +12, which was our original trinomial. Therefore, we can conclude that our factoring was done correctly.

To factor trinomials with *a* not equal to 1, the process will be a little different. Let's walk through the steps below and use them to factor *2x^2 - 5x - 3*.

Step 1: **Multiply a and c together**. For this trinomial,

Step 2: **Identify the factor pairs of this product that will add together to equal b**. To complete this step, we must list the factor pairs of -6. They are -1 & 6, 1 & -6, 2 & -3, and -2 & 3. The pair that will add together to get

Step 3: **Re-write the original equation, but replace b with the correct factor pair**. Since -5 = 1 - 6, we will replace -5x with 1x - 6x. This gives us

Step 4: ** Group the equation into two parentheses, each with two terms. Then, factor each one**. Our two parentheses will be (2x^2 + x) and (-6x - 3). Let's factor them. From the first set of parentheses, we can factor out an *x* to get x*(2x + 1). In the second parenthesis, let's factor out -3. This leaves us with -3*(2x + 1). Now we can see that *2x^2 - 5x - 3* = (2x^2 + x)+(-6x - 3) = x*(2x + 1) + -3*(2x + 1). Notice that the equations left inside of both parentheses are the same. If we are factoring correctly, this should happen every time.

Step 5: ** Place the factored terms into a separate parenthesis**. The terms that we factored out of the parenthesis were *x* and *-3*. By placing them into their own parenthesis, we end up with (x - 3) and (2x + 1) as the factors of *2x^2 - 5x - 3*.

Let's see if we did this correctly and multiply (x - 3) by (2x + 1). From distributing, we see that (x)(2x) = 2x^2, (x)(1) = x, (-3)(2x) = -6x, and (-3)(1) = -3. After combining like terms, we see that the product is 2x^2 - 5x - 3, which is what we started with. We can now be sure that we have factored correctly.

Now that we are able to factor trinomials, let's practice solving them. To do so, we want *ax^2 + bx + c* to equal zero.

Take a look at the equation 3x^2 - 5x - 5 = -3.

Before we can factor, we must get our trinomial to equal zero. To achieve this, we should add 3 to both sides. This leaves us with 3x^2 - 5x - 2 = 0, and now we are ready to factor.

Our leading coefficient is not equal to 1, so let's make sure that we follow the appropriate steps from above and begin by multiplying *a* and *c*. From this, we get (3)(-2) = -6. Now it's time to list the factor pairs. They are -1 and 6, 1 and -6, 2 and -3, & -2 and 3. The pair that will add together to get - 5 is 1 and -6, and our equation becomes 3x^2 + x - 6x - 2 = 0.

After grouping, we get (3x^2 + x)(-6x - 2) = 0. Then, once we factor each set of parentheses, we have x(3x + 1) and -2(3x + 1). At this point, we give those factored terms a new sets of parentheses, and conclude that 3x^2 - 5x - 2 factored is (x - 2)(3x + 1).

By substitution, (3x^2 + x)(-6x - 2) = 0 becomes (x - 2)(3x + 1) = 0.

To have a product of zero, one of the parentheses in our equation must be equal to zero. Truth is, it could be either one of them. Therefore, let's set each parenthesis equal to zero and solve for x. For (x - 2) = 0, we have a solution of x = 2. For (3x + 1) = 0, we have a solution of -1/3.

We have solved the trinomial 3x^2 - 5x - 5 = -3 and determined that the two solutions are 2 and -1/3.

The steps followed to factor a trinomial are determined by whether the leading coefficient is equal to 1 or not equal to 1. To solve a trinomial by factoring, the trinomial must be equal to zero.

Once you are finished, you should be able to identify, factor, and solve a trinomial.

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

BackDid 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
15 in chapter 6 of the course:

Back To Course

Precalculus: Help and Review11 chapters | 88 lessons

- What is a Parabola? 4:36
- Parabolas in Standard, Intercept, and Vertex Form 6:15
- What is a Function? - Applying the Vertical Line Test 5:42
- Multiplying Binomials Using FOIL and the Area Method 7:26
- How to Factor Quadratic Equations: FOIL in Reverse 8:50
- Factoring Quadratic Equations: Polynomial Problems with a Non-1 Leading Coefficient 7:35
- How to Complete the Square 8:43
- Completing the Square Practice Problems 7:31
- How to Solve a Quadratic Equation by Factoring 7:53
- How to Use the Quadratic Formula to Solve a Quadratic Equation 9:20
- How to Solve Quadratics That Are Not in Standard Form 6:14
- Graphing Circles: Identifying the Formula, Center and Radius 8:32
- Factoring Quadratic Expressions: Examples & Concepts
- Reflectional Symmetry: Definition & Examples 3:35
- Trinomials: Factoring, Solving & Examples
- Go to Factoring and Graphing Quadratic Equations: Help and Review

- GRE Information Guide
- Computer Science 310: Current Trends in Computer Science & IT
- Earth Science 105: Introduction to Oceanography
- Computer Science 331: Cybersecurity Risk Analysis Management
- Computer Science 336: Network Forensics
- Practical Applications for Excel
- Practical Applications in Business Law
- Practical Applications for How to Network
- Practical Application for Technical Writing
- Practical Applications for Workplace Communications with Computers
- MEGA Test Accomodations
- Study.com Grant for Teachers
- What are the MEGA Tests?
- MOGEA Test Score Information
- ASWB Prep Product Comparison
- What is the MOGEA Test?
- TASC Exam Registration Information

- Promoting Motor Learning With Activities, Sports & Games
- How to Teach Conflict Resolution to Kids
- Witchcraft in the Jacobean Era
- Best Practices for Teaching Gifted Students
- Practical Application: Principles of Composition in Graphic Design Infographic
- Recursion & Recursive Algorithms in Python: Definition & Examples
- Practical Application: Five Major Consumer Rights Infographic
- Food & Beverage Service & Operations: Assignment 2 - Research Paper
- Quiz & Worksheet - Aristotelian Virtue Ethics
- Quiz & Worksheet - Assessing Diverse Learners
- Quiz & Worksheet - Citizen Journalism Overview
- Quiz & Worksheet - Neolithic Period Overview
- Quiz & Worksheet - When to Intercede in Conflict as a Manager
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies

- Collaborating Effectively as a Team at Work
- MTTC Earth/Space Science (020): Practice & Study Guide
- FTCE School Psychologist PK-12 (036): Test Practice & Study Guide
- Foundations of Education: Help and Review
- College Earth Science: Help and Review
- The Roaring 20s: Homework Help
- Working with Quadratic Functions in Trigonometry: Help and Review
- Quiz & Worksheet - Characteristics of Protozoa
- Quiz & Worksheet - Antimicrobial Peptides
- Quiz & Worksheet - Word Problems with Multi-Step Algebra Equations
- Quiz & Worksheet - Prisoner's Dilemma
- Quiz & Worksheet - Spanish Vocabulary: Childhood Activities

- Dividing Exponential Expressions
- L-DOPA: Benefits, Side Effects & Toxicity
- Jobs for Retired Teachers
- Cover Letter Lesson Plan
- English Language Learning Programs in California
- Dividing Fractions Lesson Plan
- Best Free Online English Courses
- How to Prep for the NYS Geometry Regents Exam
- Creative Writing Lesson Plan
- Telling Time Games & Activities
- Kindergarten Book List
- How to Pass Intermediate Algebra

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

Browse by subject