6.1 Factoring Polynomials with Common Factors. Copyright Cengage Learning. All rights reserved.

Size: px
Start display at page:

Download "6.1 Factoring Polynomials with Common Factors. Copyright Cengage Learning. All rights reserved."

Transcription

1 6.1 Factoring Polynomials with Common Factors Copyright Cengage Learning. All rights reserved. 1

2 What You Will Learn Find the greatest common factor of two or more expressions Factor out the greatest common monomials factor from polynomials Factor polynomials by grouping 2

3 Greatest Common Factor 3

4 Greatest Common Factor You have used the Distributive Property to multiply polynomials. In this chapter, you will study the reverse process, which is factoring. Multiplying Polynomials Factoring Polynomials 4

5 Greatest Common Factor To factor an expression efficiently, you need to understand the concept of the greatest common factor of two (or more) integers or terms. You have learned that the greatest common factor of two or more integers is the greatest integer that is a factor of each integer. For example, the greatest common factor of 12 = and 30 = is 2 3 = 6. 5

6 Example 1 Finding the Greatest Common Factor To find the greatest common factor of 5x 2 y 2 and 30x 3 y, first factor each term. 5x 2 y 2 = 5 x x y y = (5x 2 y)(y) 30x 3 y = x x x y = (5x 2 y)(6x) So, you can conclude that the greatest common factor is 5x 2 y. 6

7 Example 2 Finding the Greatest Common Factor To find the greatest common factor of 8x 5, 20x 3, and 16x 4 first factor each term. 8x 5 = x x x x x = (4x 3 )(4x 2 ) 20x 3 = x x x = (4x 3 )(5) 16x 4 = x x x x = (4x 3 )(4x) So, you can conclude that the greatest common factor is 4x 3. 7

8 Common Monomial Factor 8

9 Dividing a Polynomial by a Binomial Consider the polynomial 8x x x 3. The greatest common factor, 4x 3, of these terms is the greatest common monomial factor of the polynomial. When you use the Distributive Property to remove this factor from each term of the polynomial, you are factoring out the greatest common monomial factor. 8x x x 3 = 4x 3 (2x 2 ) + 4x 3 (4x) + 4x 3 (5) = 4x 3 (2x 2 + 4x + 5) Factor each term. Factor out common monomial factor. 9

10 Example 3 Greatest Common Monomial Factor Factor out the greatest common monomial factor from 6x 18. Solution: The greatest common integer factor of 6x and 18 is 6. There is no common variable factor. 6x 18 = 6(x) 6(3) Greatest common monomial factor is 6. = 6(x 3) Factor 6 out of each term. 10

11 Example 4 Greatest Common Monomial Factor Factor out the greatest common monomial factor from 10y 3 25y 2. Solution: For the terms 10y 3 25y 2, 5 is the greatest common integers factor and y 2 is the highest-power common variable factor. 10y 3 25y 2 = 5y 2 (2y) 5y 2 (5) = 5y 2 (2y 5) Greatest common factor is 5y 2. Factor 5y 2 out of each term. 11

12 Greatest Common Monomial The greatest common monomial factor of the terms of a polynomial is usually considered to have a positive coefficient. However, sometimes it is convenient to factor a negative number out of a polynomial. 12

13 Example 8 A Negative Common Monomial Factor Factor the polynomial 2x 2 + 8x 12 in two ways. a. Factor out a common monomial factor of 2. b. Factor out a common monomial factor of 2. Solution: a. To factor out the common monomial factor of 2, write the following. 2x 2 + 8x 12 = 2( x 2 ) + 2(4x) + 2( 6) Factor each term. = 2( x 2 + 4x 6) Factored form 13

14 Example 8 A Negative Common Monomial Factor b. To factor 2 out of the polynomial, write the following. 2x 2 + 8x 12 = 2(x 2 ) + ( 2)( 4x) + ( 2)(6) cont d Factor each term. = 2(x 2 4x + 6) Factored form Check this result by multiplying (x 2 4x + 6) by 2. When you do, you will obtain the original polynomial. 14

15 Greatest Common Monomial With experience, you should be able to omit writing the first step shown in Example 8. For instance, to factor 2 out of 2x 2 + 8x 12, you could simply write 2x 2 + 8x 12 = 2(x 2 4x + 6). 15

16 Factoring by Grouping 16

17 Factoring by Grouping There are occasions when the common factor of an expression is not simply a monomial. For instance, the expression x 2 (x 2) + 3(x 2) has the common binomial factor (x 2). Factoring out this common factor produces x 2 (x 2) + 3(x 2) = (x 2)(x 2 + 3). This type of factoring is part of a more general procedure called factoring by grouping. 17

18 Example 9 Common Binomial Factors Factor each expression. a. 5x 2 (7x 1) 3(7x 1) b. 2x(3x 4) + (3x 4) c. 3y 2 (y 3) + 4(3 y) Solution: a. Each of the terms of this expression has a binomial factor of (7x 1). 5x 2 (7x 1) 3(7x 1) = (7x 1)(5x 2 3) 18

19 Example 9 Common Binomial Factors b. Each of the terms of this expression has a binomial factor of (3x 4). cont d 2x(3x 4) + (3x 4) = (3x 4)(2x + 1) Be sure you see that when (3x 4) is factored out of itself, you are left with the factor 1. This follows from the fact that (3x 4)(1) = (3x 4). c. 3y 2 (y 3) + 4(3 y) = 3y 2 (y 3) 4(y 3) = (y 3)(3y 2 4) Write 4(3 y) as 4(y 3). Common factor is (y 3). 19

20 Factoring by Grouping In Example 9, the polynomials were already grouped so that it was easy to determine the common binomial factors. In practice, you will have to do the grouping as well as the factoring. To see how this works, consider the expression x 3 + 2x 2 + 3x + 6 and try to factor it. Note first that there is no common monomial factor to take out of all four terms. 20

21 Factoring by Grouping But suppose you group the first two terms together and the last two terms together. x 3 + 2x 2 + 3x + 6 = (x 3 + 2x 2 ) + (3x + 6) = x 2 (x + 2) + 3(x + 2) = (x + 2)(x 2 + 3) Group terms. Factor out common monomial factor in each group. Factored form When factoring by grouping, be sure to group terms that have a common monomial factor. For example, in the polynomial above, you should not group the first term x 3 with the fourth term 6. 21

22 Example 10 Factoring by Grouping Factor x 3 + 2x 2 + x + 2. Solution: x 3 + 2x 2 + x + 2 = (x 3 + 2x 2 ) + (x + 2) = x 2 (x + 2) + (x + 2) = (x + 2)(x 2 + 1) Group terms. Factor out common monomial factor in each group. Factored form 22

23 Factoring by Grouping Note that in Example 10 the polynomial is factored by grouping the first and second terms and the third and fourth terms. You could just as easily have grouped the first and third terms and the second and fourth terms, as follows. x 3 + 2x 2 + x + 2 = (x 3 + x) + (2x 2 + 2) = x(x 2 + 1) + 2(x 2 + 1) = (x 2 + 1)(x + 2) You can always check to see that you have factored an expression correctly by multiplying the factors and comparing the result with the original expression. 23

24 Example 12 Geometry: Area of a Rectangle The area of a rectangle of width (2x 1) feet is (2x 3 + 2x x 2 ) square feet, as shown below. Factor this expression to determine the length of the rectangle. 24

25 Example 12 Geometry: Area of a Rectangle Solution Verbal Model: cont d Labels: Area = 2x 3 + 2x x 2 (square feet) Width = 2x 1 Equation: 2x 3 + 4x x 2 2 = (2x 3 + 4x) + ( x 2 2) Group terms. (feet) = 2x(x 2 + 2) + (x 2 + 2) = (x 2 + 2)(2x 1) Factor out common monomial factor in each group. Factored form The length of the rectangle is (x 2 + 2) feet. 25

26 Homework: Page 272 # down Page 273 # 45, 47, 49 Page 274 # down

5.3 Multiplying Polynomials: Special Products. Copyright Cengage Learning. All rights reserved.

5.3 Multiplying Polynomials: Special Products. Copyright Cengage Learning. All rights reserved. 5.3 Multiplying Polynomials: Special Products Copyright Cengage Learning. All rights reserved. 1 What You Will Learn Find products with monomial multipliers Multiplying binomials using the Distributive

More information

When factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF.

When factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF. Factoring: reversing the distributive property. The distributive property allows us to do the following: When factoring, we look for greatest common factor of each term and reverse the distributive property

More information

6.2 Factoring Trinomials. Copyright Cengage Learning. All rights reserved.

6.2 Factoring Trinomials. Copyright Cengage Learning. All rights reserved. 6.2 Factoring Trinomials Copyright Cengage Learning. All rights reserved. 1 What You Will Learn Factor trinomials of the form x 2 + bx + c Factoring trinomials in two variables Factor trinomials completely

More information

Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).

Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32

More information

Factoring Polynomials

Factoring Polynomials Section P.5 Factoring Polynomials 51 P.5 Factoring Polynomials What you should learn: Factor polynomials with common factors Factor polynomials by grouping terms Factor the difference of two squares Factor

More information

Greatest Common Factor (GCF) Factoring

Greatest Common Factor (GCF) Factoring Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication

More information

Investigation: Multiplying Binomials

Investigation: Multiplying Binomials Investigation: Multiplying Binomials In this investigation, we will explore how to multiply two binomials together, using area of rectangles. After your comfortable with that method, we will use the distributive

More information

7-2 Factoring by GCF. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 1

7-2 Factoring by GCF. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 1 7-2 Factoring by GCF Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up Simplify. 1. 2(w + 1) 2. 3x(x 2 4) 2w + 2 3x 3 12x Find the GCF of each pair of monomials. 3. 4h 2 and 6h 2h 4. 13p and 26p

More information

Algebra Success. [OBJECTIVE] The student will learn how to multiply monomials and polynomials.

Algebra Success. [OBJECTIVE] The student will learn how to multiply monomials and polynomials. Algebra Success T697 [OBJECTIVE] The student will learn how to multiply monomials and polynomials. [MATERIALS] Student pages S269 S278 Transparencies T704, T705, T707, T709, T711, T713, T715 Red and yellow

More information

To find the greatest common factor among two or more numbers, write out the prime factorization of each number using the following procedure:

To find the greatest common factor among two or more numbers, write out the prime factorization of each number using the following procedure: Greatest Common Factors When factoring polynomials, the first thing to always check for is a greatest common factor (GCF) among all of the terms of the polynomial. To find the greatest common factor among

More information

MTH 098. Sections 4.1 & 4.2

MTH 098. Sections 4.1 & 4.2 MTH 098 Sections 4.1 & 4.2 4.1 The Product Rule and Power Rules for Exponents How do you write 2 2 2 in exponential form? Evaluate 2 2 2. Evaluate -2 Evaluate (-2) The Product Rule for Exponents 2 2 =

More information

The greatest common monomial factor of two or more monomials is the product of all integer and variable factors that are common to those monomials.

The greatest common monomial factor of two or more monomials is the product of all integer and variable factors that are common to those monomials. Chapter 8.1 Common Monomial Factors The greatest common monomial factor of two or more monomials is the product of all integer and variable factors that are common to those monomials. To find the greatest

More information

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial

More information

1.3 Polynomials and Factoring

1.3 Polynomials and Factoring 1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

More information

Factor Polynomials Completely

Factor Polynomials Completely 9.8 Factor Polynomials Completely Before You factored polynomials. Now You will factor polynomials completely. Why? So you can model the height of a projectile, as in Ex. 71. Key Vocabulary factor by grouping

More information

Name Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE

Name Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers

More information

MTH 098. Sections 5.1 & 5.2

MTH 098. Sections 5.1 & 5.2 MTH 098 Sections 5.1 & 5.2 5.1 The Greatest Common Factor: Factoring by Grouping What is the greatest common factor of two numbers? Find the greatest common factor (GCF) for each list of numbers. 1. 50,

More information

Factoring Quadratic Expressions VOCABULARY

Factoring Quadratic Expressions VOCABULARY 5-5 Factoring Quadratic Expressions TEKS FOCUS Foundational to TEKS (4)(F) Solve quadratic and square root equations. TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil,

More information

Integration Unit 8 Factoring and solving quadratics by Factoring

Integration Unit 8 Factoring and solving quadratics by Factoring Integration Unit 8 Factoring and solving quadratics by Factoring Name Period Objective 1: Factors and Greatest Common Factors Factors of a number: Numbers that divide evenly into a number are called factors.

More information

Factoring and Applications

Factoring and Applications Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

More information

Operations with Algebraic Expressions: Multiplication of Polynomials

Operations with Algebraic Expressions: Multiplication of Polynomials Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the

More information

Factoring is the process that reverses multiplication. Rewriting a polynomial as a product of factors is called factoring a polynomial.

Factoring is the process that reverses multiplication. Rewriting a polynomial as a product of factors is called factoring a polynomial. Factoring GCF s What is Factoring? Factoring is the process that reverses multiplication Rewriting a polynomial as a product of factors is called factoring a polynomial When we multiply 4 times 5 we obtain

More information

Chapter 4. Polynomials

Chapter 4. Polynomials 4.1. Add and Subtract Polynomials KYOTE Standards: CR 8; CA 2 Chapter 4. Polynomials Polynomials in one variable are algebraic expressions such as 3x 2 7x 4. In this example, the polynomial consists of

More information

Adding, Subtracting, Multiplying, and Factoring Polynomials Using Algebra Tiles. This Lesson is Designed for 8 th Grade

Adding, Subtracting, Multiplying, and Factoring Polynomials Using Algebra Tiles. This Lesson is Designed for 8 th Grade 1 Adding, Subtracting, Multiplying, and Factoring Polynomials Using Algebra Tiles This Lesson is Designed for 8 th Grade The Lesson will last 5 days and you will need: A Class Set of Algebra Tiles, Algebra

More information

Factoring - Everything you NEED to know

Factoring - Everything you NEED to know Factoring - Everything you NEED to know Topic 1: Greatest Common Factors and Factoring by Grouping Factoring is the opposite of multiplying; it is the process of expressing a polynomial as a product of

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

NSM100 Introduction to Algebra Chapter 5 Notes Factoring Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

More information

Note that the words divisor and factor are equivalent. They have the same meaning.

Note that the words divisor and factor are equivalent. They have the same meaning. 2 MODULE 5. FACTORING 5a Common Factors The Greatest Common Factor We begin this section with definitions of factors and divisors. Because 24 = 2 12, both 2 and 12 are factors of 24. However, note that

More information

Factors and Greatest Common Factors (Pages )

Factors and Greatest Common Factors (Pages ) A 10-1 Factors and Greatest Common Factors (Pages 558 563) S 11.0, 25.1 When two or more numbers are multiplied to form a product, each number is a factor of the product. Prime numbers are whole numbers

More information

MATH Fundamental Mathematics II.

MATH Fundamental Mathematics II. MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/fun-math-2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics

More information

Factoring Polynomials

Factoring Polynomials Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring

More information

Exponents, Polynomials and Functions. Copyright Cengage Learning. All rights reserved.

Exponents, Polynomials and Functions. Copyright Cengage Learning. All rights reserved. Exponents, Polynomials and Functions 3 Copyright Cengage Learning. All rights reserved. 3.1 Rules for Exponents Copyright Cengage Learning. All rights reserved. Rules for Exponents The basic concept of

More information

2x 2x 2 8x. Now, let s work backwards to FACTOR. We begin by placing the terms of the polynomial inside the cells of the box. 2x 2

2x 2x 2 8x. Now, let s work backwards to FACTOR. We begin by placing the terms of the polynomial inside the cells of the box. 2x 2 Activity 23 Math 40 Factoring using the BOX Team Name (optional): Your Name: Partner(s): 1. (2.) Task 1: Factoring out the greatest common factor Mini Lecture: Factoring polynomials is our focus now. Factoring

More information

State whether each sentence is true or false. If false, replace the underlined phrase or expression to make a true sentence.

State whether each sentence is true or false. If false, replace the underlined phrase or expression to make a true sentence. State whether each sentence is true or false. If false, replace the underlined phrase or expression to make a true sentence. 1. x + 5x + 6 is an example of a prime polynomial. The statement is false. A

More information

1.4 Review: Adding, Subtracting, and Multiplying Polynomials

1.4 Review: Adding, Subtracting, and Multiplying Polynomials 1.4 Review: Adding, Subtracting, and Multiplying Polynomials Key Concepts To add polynomials, collect like terms. To subtract polynomials, add the opposite. To multiply a polynomial by a monomial, use

More information

Math 154 :: Elementary Algebra

Math 154 :: Elementary Algebra Math 154 :: Elementary Algebra Section 7.1 Section 7. Section 7.3 Section 7.4 Section 7.5 Section 7.6 Section 7.7 Section 7.8 Section 7.1 Greatest Common Factor. This answer should be in your own words.

More information

Polynomials and Factoring

Polynomials and Factoring 7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of

More information

Factoring, Solving. Equations, and Problem Solving REVISED PAGES

Factoring, Solving. Equations, and Problem Solving REVISED PAGES 05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring

More information

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

More information

LESSON 6.2 POLYNOMIAL OPERATIONS I

LESSON 6.2 POLYNOMIAL OPERATIONS I LESSON 6.2 POLYNOMIAL OPERATIONS I Overview In business, people use algebra everyday to find unknown quantities. For example, a manufacturer may use algebra to determine a product s selling price in order

More information

6 1 Integer Exponents

6 1 Integer Exponents CHAPTER 6 EXPONENTS AND POLYNOMIALS LESSONS 6 1 6 2 6 2 II 6 3 6 4 6 5 6 5 II 6 6 WARM UPS 6 1 6 2 6 2 II 6 3 6 4 6 5 6 5 II 6 6 HOMEWORK 6 1 6 2 6 2 II 6 3 6 4 6 5 6 5 II 6 6 6 1 Integer Exponents Objectives

More information

Example: 2x 2 + 4x Notice that each term has a factor of 2x, so we can rewrite it as: 2x 2 + 4x = 2x(x + 2)

Example: 2x 2 + 4x Notice that each term has a factor of 2x, so we can rewrite it as: 2x 2 + 4x = 2x(x + 2) ALGEBRA 2: 6.4 SUPPLEMENT FACTORING POLYNOMIALS NAME Factoring a polynomial is the opposite process of multiplying polynomials. Recall that when we factor a number, we are looking for prime factors that

More information

Factors and Products

Factors and Products CHAPTER 3 Factors and Products What You ll Learn use different strategies to find factors and multiples of whole numbers identify prime factors and write the prime factorization of a number find square

More information

Factoring A Quadratic Polynomial

Factoring A Quadratic Polynomial Factoring A Quadratic Polynomial If we multiply two binomials together, the result is a quadratic polynomial: This multiplication is pretty straightforward, using the distributive property of multiplication

More information

4-3 Solving Quadratic Equations by Factoring. Write a quadratic equation in standard form with the given root(s) SOLUTION: Write the pattern.

4-3 Solving Quadratic Equations by Factoring. Write a quadratic equation in standard form with the given root(s) SOLUTION: Write the pattern. 17. 7 Write a quadratic equation in standard form with the given root(s). Write the pattern. Since there is only one root, it is a repeated root. Replace p and q with 7. Use the FOIL method to multiply.

More information

Lesson 8 Add Subt Polynomials.notebook

Lesson 8 Add Subt Polynomials.notebook Lesson 8 Adding and Subtracting Polynomials Students understand that the sum or difference of two polynomials produces another polynomial and relate polynomials to the system of integers; students add

More information

Math 002 Intermediate Algebra Spring 2012 Objectives & Assignments

Math 002 Intermediate Algebra Spring 2012 Objectives & Assignments Math 00 Intermediate Algebra Spring 01 Objectives & Assignments Unit 3 Exponents, Polynomial Operations, and Factoring I. Exponents & Scientific Notation 1. Use the properties of exponents to simplify

More information

8-3 Multiplying Polynomials. Find each product. 1. (x + 5)(x + 2) SOLUTION: 2. (y 2)(y + 4) SOLUTION: 3. (b 7)(b + 3) SOLUTION:

8-3 Multiplying Polynomials. Find each product. 1. (x + 5)(x + 2) SOLUTION: 2. (y 2)(y + 4) SOLUTION: 3. (b 7)(b + 3) SOLUTION: Find each product. 1. (x + 5)(x + ). (y )(y + 4) 3. (b 7)(b + 3) 4. (4n + 3)(n + 9) 5. (8h 1)(h 3) 6. (a + 9)(5a 6) 7. FRAME Hugo is designing a frame as shown. The frame has a width of x inches all the

More information

1. Write a polynomial expression that represents the total area of the lawn. Give your answer as a trinomial.

1. Write a polynomial expression that represents the total area of the lawn. Give your answer as a trinomial. Multiplying Polynomials Blue Level Problems Cutting the Lawn Cutting lawns is a popular summer job. It is a great way to earn money and to help others in your community. Your neighbor asks you to cut her

More information

Unit III Factoring Section 3.3 Common Factors of a Polynomial

Unit III Factoring Section 3.3 Common Factors of a Polynomial Sections 3.3-3.8 Math 1201 1 Unit III Factoring Section 3.3 Common Factors of a Polynomial Sections 3.3-3.8 Math 1201 2 Sections 3.3-3.8 Math 1201 3 3.5 Polynomials of the form x 2 + bx + c Goals: Practice

More information

Expansion of a Product of Binomials. Here we look at two ways to approach the expansion or removal of brackets from a product of the form

Expansion of a Product of Binomials. Here we look at two ways to approach the expansion or removal of brackets from a product of the form The Mathematics 11 Competency Test Expansion of a Product of Binomials Here we look at two ways to approach the expansion or removal of brackets from a product of the form (5x + 3)(7x 2) in which two binomials

More information

9.4 Solve Polynomial Equations

9.4 Solve Polynomial Equations 9.4 Solve Polynomial Equations in Factored Form Goal p Solve polynomial equations. Your Notes VOCABULARY Roots Vertical motion model ZERO-PRODUCT PROPERTY Let a and b be real numbers. If ab 5 0, then 5

More information

15.1 Factoring Polynomials

15.1 Factoring Polynomials LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE

More information

Factoring Special Polynomials

Factoring Special Polynomials 6.6 Factoring Special Polynomials 6.6 OBJECTIVES 1. Factor the difference of two squares 2. Factor the sum or difference of two cubes In this section, we will look at several special polynomials. These

More information

8-1 Adding and Subtracting Polynomials

8-1 Adding and Subtracting Polynomials Determine whether each expression is a polynomial. If it is a polynomial, find the degree and determine whether it is a monomial, binomial, or trinomial. 1. 7ab + 6b 2 2a 3 yes; 3; trinomial 2. 2y 5 +

More information

Basic Algebra Practice Test

Basic Algebra Practice Test 1. Exponents and integers: Problem type 2 Evaluate. Basic Algebra Practice Test 2. Exponents and signed fractions Evaluate. Write your answers as fractions. 3. Exponents and order of operations Evaluate.

More information

Chapter 6. Polynomials and Polynomial Functions

Chapter 6. Polynomials and Polynomial Functions Chapter 6 Polynomials and Polynomial Functions Lesson 6-1 Polynomial Functions Polynomials A polynomial is a monomial or the sum of monomials. P( x) a x a x... a x a n n1 n n1 1 0 3 P( x) x 5x x 5 The

More information

Developmental Algebra: Intermediate Preparing for College Mathematics

Developmental Algebra: Intermediate Preparing for College Mathematics Developmental Algebra: Intermediate Preparing for College Mathematics By Paul Pierce Included in this preview: Copyright Page Table of Contents Excerpt of Chapter 1 For additional information on adopting

More information

Factoring Polynomials. Mr. Dave Clausen La Cañada High School

Factoring Polynomials. Mr. Dave Clausen La Cañada High School Factoring Polynomials Mr. Dave Clausen La Cañada High School California State Standard Algebra 2 Standards: 3.0 Students are adept at operations on polynomials, including long division. 4.0 Students factor

More information

( ) FACTORING. x In this polynomial the only variable in common to all is x.

( ) FACTORING. x In this polynomial the only variable in common to all is x. FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated

More information

Factoring Trinomials of the Form x 2 bx c

Factoring Trinomials of the Form x 2 bx c 4.2 Factoring Trinomials of the Form x 2 bx c 4.2 OBJECTIVES 1. Factor a trinomial of the form x 2 bx c 2. Factor a trinomial containing a common factor NOTE The process used to factor here is frequently

More information

Addition and Multiplication of Polynomials

Addition and Multiplication of Polynomials LESSON 0 addition and multiplication of polynomials LESSON 0 Addition and Multiplication of Polynomials Base 0 and Base - Recall the factors of each of the pieces in base 0. The unit block (green) is x.

More information

6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions 6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

More information

Algebra Unit 6 Syllabus revised 2/27/13 Exponents and Polynomials

Algebra Unit 6 Syllabus revised 2/27/13 Exponents and Polynomials Algebra Unit 6 Syllabus revised /7/13 1 Objective: Multiply monomials. Simplify expressions involving powers of monomials. Pre-assessment: Exponents, Fractions, and Polynomial Expressions Lesson: Pages

More information

Pre-Algebra Interactive Chalkboard Copyright by The McGraw-Hill Companies, Inc. Send all inquiries to:

Pre-Algebra Interactive Chalkboard Copyright by The McGraw-Hill Companies, Inc. Send all inquiries to: Pre-Algebra Interactive Chalkboard Copyright by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240 Click the mouse button

More information

Table of Contents. Quadratic Equations Pgs: HW: Pages reverse factoring Pgs: HW: Page 38

Table of Contents. Quadratic Equations Pgs: HW: Pages reverse factoring Pgs: HW: Page 38 Chapter 9B Table of Contents o Day 1: SWBAT: Solve Quadratic Equations by Factoring and Graphically Pgs: 1-6 HW: Pages 7-8 o Day 2: SWBAT: Solve Quadratic Word Problems by Factoring Pgs: 9-15 HW: Pages

More information

Algebra Cheat Sheets

Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

More information

9.2. Multiply Polynomials E XAMPLE 1 E XAMPLE 2. Multiply a monomial and a polynomial. Multiply polynomials using a table

9.2. Multiply Polynomials E XAMPLE 1 E XAMPLE 2. Multiply a monomial and a polynomial. Multiply polynomials using a table 9.2 Multiply Polynomials Before You added and subtracted polynomials. Now You will multiply polynomials. Why? So you can determine areas, as in Eample 7. Key Vocabulary polynomial, p. 554 binomial, p.

More information

Elementary Algebra MATH 97 Practice Test Form B

Elementary Algebra MATH 97 Practice Test Form B Elementary Algebra MATH 97 Practice Test Form B MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the expression for the given values. If necessary,

More information

FACTORING OUT COMMON FACTORS

FACTORING OUT COMMON FACTORS 278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the

More information

7-8 Multiplying Polynomials

7-8 Multiplying Polynomials 7-8 Multiplying Polynomials California Standards 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these

More information

Summer 2016 Math Packet for Rising Geometry Students

Summer 2016 Math Packet for Rising Geometry Students Summer 016 Math Packet for Rising Geometry Students This packet is designed to help you review your Algebra Skills and help you prepare for your Geometry class. Your Geometry teacher will expect you to

More information

x n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent.

x n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent. Rules of Exponents: If n > 0, m > 0 are positive integers and x, y are any real numbers, then: x m x n = x m+n x m x n = xm n, if m n (x m ) n = x mn (xy) n = x n y n ( x y ) n = xn y n 1 Can we make sense

More information

Algebra Placement Test Review

Algebra Placement Test Review Algebra Placement Test Review Recognizing the Relative Position between Real Numbers A. Which number is smaller, or 000? To really appreciate which number is smaller one must view both numbers plotted

More information

Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter)

Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter) Name: Per.: Date: Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter) Your Algebra 1 Final will be on at. You will need to bring your textbook and number 2 pencils with you to the final exam.

More information

CH 8. Polynomials and Factoring

CH 8. Polynomials and Factoring 8.1: Add and Subtract Polynomials 8.2: Multiply Polynomials 8.3: Find Special Products of Polynomials 8.4: Solve Polynomial Equations in Factored Form 8.5: Factor 8.6: Factor 8.7: Factor Special Products

More information

5.6 A GENERAL REVIEW: FACTORING

5.6 A GENERAL REVIEW: FACTORING 5.6 A GENERAL REVIEW: FACTORING Recall: Factor a # that goes into another # without remainder Ex: Factors of 20: 1,2,4,5,10,20 Factoring - undoing multiplication (division) Methods of Factoring The Greatest

More information

What you can do - (Goal Completion) Learning

What you can do - (Goal Completion) Learning What you can do - (Goal Completion) Learning ARITHMETIC READINESS Whole Numbers Order of operations: Problem type 1 Order of operations: Problem type 2 Factors Prime factorization Greatest common factor

More information

MATH LEVEL 1 ARITHMETIC (ACCUPLACER)

MATH LEVEL 1 ARITHMETIC (ACCUPLACER) MATH LEVEL ARITHMETIC (ACCUPLACER) 7 Questions This test measures your ability to perform basic arithmetic operations and to solve problems that involve fundamental arithmetic concepts. There are 7 questions

More information

MATH 90 CHAPTER 6 Name:.

MATH 90 CHAPTER 6 Name:. MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a

More information

Multiplying With Polynomials What do you do? 1. Distribute (or double-distribute/foil, when necessary) 2. Combine like terms

Multiplying With Polynomials What do you do? 1. Distribute (or double-distribute/foil, when necessary) 2. Combine like terms Regents Review Session #1 Polynomials Adding and Subtracting Polynomials What do you do? 1. Add/subtract like terms Example: 1. (8x 3-9x 2 + 6x + 2) - (-7x 3-5x 2 + 1x - 8) Multiplying With Polynomials

More information

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2 Pure Math 0 Notes Unit : Polynomials Unit : Polynomials -: Reviewing Polynomials Epressions: - mathematical sentences with no equal sign. Eample: Equations: - mathematical sentences that are equated with

More information

Mth 95 Module 2 Spring 2014

Mth 95 Module 2 Spring 2014 Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression

More information

Math 9 Unit 5 Polynomials Practice Test

Math 9 Unit 5 Polynomials Practice Test Name: Class: _ Date: _ ID: A Math 9 Unit Polynomials Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A large white square represents an x

More information

Factoring Polynomials

Factoring Polynomials Factoring Polynomials Writing a polynomial as a product of polynomials of lower degree is called factoring. Factoring is an important procedure that is often used to simplify fractional expressions and

More information

Name: Date: Algebra 2/ Trig Apps: Simplifying Square Root Radicals. Arithmetic perfect squares: 1, 4, 9,,,,,,...

Name: Date: Algebra 2/ Trig Apps: Simplifying Square Root Radicals. Arithmetic perfect squares: 1, 4, 9,,,,,,... RADICALS PACKET Algebra 2/ Trig Apps: Simplifying Square Root Radicals Perfect Squares Perfect squares are the result of any integer times itself. Arithmetic perfect squares: 1, 4, 9,,,,,,... Algebraic

More information

Dividing Polynomials VOCABULARY

Dividing Polynomials VOCABULARY - Dividing Polynomials TEKS FOCUS TEKS ()(C) Determine the quotient of a polynomial of degree three and degree four when divided by a polynomial of degree one and of degree two. TEKS ()(A) Apply mathematics

More information

Section 6.1 The Greatest Common Factor and Factoring by Grouping

Section 6.1 The Greatest Common Factor and Factoring by Grouping Greatest Common Factor (GCF): The GCF is an expression of the highest degree that divides each term of the polynomial. The variable part of the greatest common factor always contains the smallest power

More information

Factoring Polynomials

Factoring Polynomials Chapter 13 13.1 Factoring Polynomials The Greatest Common Factor Chapter Sections 13.1 The Greatest Common Factor 13.2 Factoring Trinomials of the Form x 2 + bx + c 13.3 Factoring Trinomials of the Form

More information

Unit 3: Algebra. Date Topic Page (s) Algebra Terminology 2. Variables and Algebra Tiles 3 5. Like Terms 6 8. Adding/Subtracting Polynomials 9 12

Unit 3: Algebra. Date Topic Page (s) Algebra Terminology 2. Variables and Algebra Tiles 3 5. Like Terms 6 8. Adding/Subtracting Polynomials 9 12 Unit 3: Algebra Date Topic Page (s) Algebra Terminology Variables and Algebra Tiles 3 5 Like Terms 6 8 Adding/Subtracting Polynomials 9 1 Expanding Polynomials 13 15 Introduction to Equations 16 17 One

More information

CLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column.

CLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column. SYNTHETIC DIVISION CLASS NOTES When factoring or evaluating polynomials we often find that it is convenient to divide a polynomial by a linear (first degree) binomial of the form x k where k is a real

More information

Planning Guide. Patterns and Relations (Variables and Equations) Specific Outcomes 5, 6 and 7

Planning Guide. Patterns and Relations (Variables and Equations) Specific Outcomes 5, 6 and 7 Mathematics Planning Guide Grade 9 Polynomials Patterns and Relations (Variables and Equations) Specific Outcomes 5, 6 and 7 This Planning Guide can be accessed online at: http://www.learnalberta.ca/content/mepg9/html/pg9_polynomials/index.html

More information

3.3 Power Functions and Polynomial Functions

3.3 Power Functions and Polynomial Functions 362 Chapter 3 Polynomial and Rational Functions 3.3 Power Functions and Polynomial Functions In this section, you will: Learning Objectives 3.3.1 Identify power functions. 3.3.2 Identify end behavior of

More information

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1

FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1 5.7 Factoring ax 2 bx c (5-49) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.

More information

Chapter Exam Review for MAT098 - Prealgebra Chapters 1-2: Whole Numbers and Introduction to Algebra & Integers and Introduction to Solving Equations

Chapter Exam Review for MAT098 - Prealgebra Chapters 1-2: Whole Numbers and Introduction to Algebra & Integers and Introduction to Solving Equations Chapter Exam Review for MAT098 - Prealgebra Chapters 1-2: Whole Numbers and Introduction to Algebra & Integers and Introduction to Solving Equations Chapters 1-2 Learning Objectives: In chapter 1 students

More information

Exponents and Polynomials

Exponents and Polynomials CHAPTER 6 Exponents and Polynomials Solutions Key are you ready?. F 2. B. C 4. D 5. E 6. 4 7 7. 5 2 8. ( -0) 4 9. x 0. k 5. 9 2. 4 = = 8. -2 2 = -(2 2) = -44 5. 2 5 = 2 2 2 2 2 = 2 7. ( -) 6 = (-)(-)(-)(-)(-)(-)

More information

SPECIAL PRODUCTS AND FACTORS

SPECIAL PRODUCTS AND FACTORS CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

More information

Polynomials. 4-4 to 4-8

Polynomials. 4-4 to 4-8 Polynomials 4-4 to 4-8 Learning Objectives 4-4 Polynomials Monomials, binomials, and trinomials Degree of a polynomials Evaluating polynomials functions Polynomials Polynomials are sums of these "variables

More information

A. A Little Review: Multiply the following: 1. 7(2x + 5) 2. 2x(5x 2 + 3) 3. 3x 2 (5x 7) 4. 3x 3 (2x 2 3x + 5)

A. A Little Review: Multiply the following: 1. 7(2x + 5) 2. 2x(5x 2 + 3) 3. 3x 2 (5x 7) 4. 3x 3 (2x 2 3x + 5) Section 5.1: Introduction to Factoring This is the chapter in which we learn the very important skill of factoring polynomials, especially trinomials. Please remember that when we ask you to factor something,

More information

Use the guess and check strategy and the FOIL method to factor a trinomial.

Use the guess and check strategy and the FOIL method to factor a trinomial. A - Factor each polynomial. 5. 7b b 6. 5m n 7mn 0z 0z 6 8s s q 9. 6g gh 0. 6k 5 k 8k. 6y y y. 6 w wz 8w 6z Geometry The area of a rectangle is represented by 0 5 6. Its dimensions are represented by binomials

More information

A Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles

A Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...

More information