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


 Silas McGee
 4 months ago
 Views:
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 highestpower 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. 1 What You Will Learn Find products with monomial multipliers Multiplying binomials using the Distributive
More informationWhen 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 information6.2 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 informationDefinitions 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 informationFactoring 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 informationGreatest 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 informationInvestigation: 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 information72 Factoring by GCF. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 1
72 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 informationAlgebra 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 informationTo 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 informationMTH 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 informationThe 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 informationAlum 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 information1.3 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 informationFactor 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 informationName 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 informationMTH 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 informationFactoring Quadratic Expressions VOCABULARY
55 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 informationIntegration 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 informationFactoring 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 informationOperations 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 informationFactoring 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 informationChapter 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 informationAdding, 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 informationFactoring  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 informationNSM100 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 informationNote 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 informationFactors and Greatest Common Factors (Pages )
A 101 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 informationMATH Fundamental Mathematics II.
MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/funmath2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics
More informationFactoring 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 informationExponents, 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 information2x 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 informationState 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 information1.4 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 informationMath 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 informationPolynomials 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 informationFactoring, Solving. Equations, and Problem Solving REVISED PAGES
05W4801AM1.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 informationexpression 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 informationLESSON 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 information6 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 informationExample: 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 informationFactors 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 informationFactoring 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 information43 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 informationLesson 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 informationMath 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 information83 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 information1. 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 informationUnit III Factoring Section 3.3 Common Factors of a Polynomial
Sections 3.33.8 Math 1201 1 Unit III Factoring Section 3.3 Common Factors of a Polynomial Sections 3.33.8 Math 1201 2 Sections 3.33.8 Math 1201 3 3.5 Polynomials of the form x 2 + bx + c Goals: Practice
More informationExpansion 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 information9.4 Solve Polynomial Equations
9.4 Solve Polynomial Equations in Factored Form Goal p Solve polynomial equations. Your Notes VOCABULARY Roots Vertical motion model ZEROPRODUCT PROPERTY Let a and b be real numbers. If ab 5 0, then 5
More information15.1 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 informationFactoring 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 information81 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 informationBasic 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 informationChapter 6. Polynomials and Polynomial Functions
Chapter 6 Polynomials and Polynomial Functions Lesson 61 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 informationDevelopmental 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 informationFactoring 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 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 informationFactoring 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 informationAddition 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 information6.1 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 informationAlgebra 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. Preassessment: Exponents, Fractions, and Polynomial Expressions Lesson: Pages
More informationPreAlgebra Interactive Chalkboard Copyright by The McGrawHill Companies, Inc. Send all inquiries to:
PreAlgebra Interactive Chalkboard Copyright by The McGrawHill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGrawHill 8787 Orion Place Columbus, Ohio 43240 Click the mouse button
More informationTable 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: 16 HW: Pages 78 o Day 2: SWBAT: Solve Quadratic Word Problems by Factoring Pgs: 915 HW: Pages
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More information9.2. 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 informationElementary 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 informationFACTORING 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 information78 Multiplying Polynomials
78 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 informationSummer 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 informationx 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 informationAlgebra 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 informationAlgebra 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 informationCH 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 information5.6 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 informationWhat 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 informationMATH 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 informationMATH 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 informationMultiplying With Polynomials What do you do? 1. Distribute (or doubledistribute/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 39x 2 + 6x + 2)  (7x 35x 2 + 1x  8) Multiplying With Polynomials
More informationUnit 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 informationMth 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 informationMath 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 informationFactoring 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 informationName: 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 informationDividing 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 informationSection 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 informationFactoring 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 informationUnit 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 informationCLASS 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 informationPlanning 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 information3.3 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 informationFACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1
5.7 Factoring ax 2 bx c (549) 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 informationChapter Exam Review for MAT098  Prealgebra Chapters 12: Whole Numbers and Introduction to Algebra & Integers and Introduction to Solving Equations
Chapter Exam Review for MAT098  Prealgebra Chapters 12: Whole Numbers and Introduction to Algebra & Integers and Introduction to Solving Equations Chapters 12 Learning Objectives: In chapter 1 students
More informationExponents 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 informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
More informationPolynomials. 44 to 48
Polynomials 44 to 48 Learning Objectives 44 Polynomials Monomials, binomials, and trinomials Degree of a polynomials Evaluating polynomials functions Polynomials Polynomials are sums of these "variables
More informationA. 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 informationUse 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 informationA 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