Factoring Special Polynomials


 Jacob Marshall
 3 years ago
 Views:
Transcription
1 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 polynomials are special because they fit a recognizable pattern. Pattern recognition is an important element of mathematics. Many mathematical discoveries were made because somebody recognized a pattern. The first pattern, which we saw in Section 6.4, is called the difference of two squares. CAUTION Rules and Properties: The Difference of Two Squares What about the sum of two squares, such as x 2 25 In general, it is not possible to factor (using real numbers) a sum of two squares. So (x 2 25) (x 5)(x 5) a 2 b 2 (a b)(a b) (1) In words: The product of the sum and difference of two terms gives the difference of two squares. Equation (1) is easy to apply in factoring. It is just a matter of recognizing a binomial as the difference of two squares. To confirm this identity, use the FOIL method to multiply (a b)(a b) Example 1 Factoring the Difference of Two Squares NOTE We are looking for perfect squares the exponents must be multiples of 2 and the coefficients perfect squares 1, 4, 9, 16, and so on. (a) Factor x Note that our example has two terms a clue to try factoring as the difference of two squares. x 2 25 (x) 2 (5) 2 (x 5)(x 5) (b) Factor 9a a 2 16 (3a) 2 (4) 2 (3a 4)(3a 4) (c) Factor 25m 4 49n 2. 25m 4 49n 2 (5m 2 ) 2 (7n) 2 (5m 2 7n)(5m 2 7n) CHECK YOURSELF 1 Factor each of the following binomials. (a) y 2 36 (b) 25m 2 n 2 (c) 16a 4 9b 2 423
2 424 CHAPTER 6 POLYNOMIALS AND POLYNOMIAL FUNCTIONS We mentioned earlier that factoring out a common factor should always be considered your first step. Then other steps become obvious. Consider Example 2. Example 2 Factoring the Difference of Two Squares Factor a 3 16ab 2. First note the common factor of a. Removing that factor, we have a 3 16ab 2 a(a 2 16b 2 ) We now see that the binomial factor is a difference of squares, and we can continue to factor as before. So a 3 16ab 2 a(a 4b)(a 4b) CHECK YOURSELF 2 Factor 2x 3 18xy 2. You may also have to apply the difference of two squares method more than once to completely factor a polynomial. Example 3 Factoring the Difference of Two Squares Factor m 4 81n 4. m 4 81n 4 (m 2 9n 2 )(m 2 9n 2 ) Do you see that we are not done in this case? Because m 2 9n 2 is still factorable, we can continue to factor as follows. NOTE The other binomial factor, m 2 9n 2, is a sum of two squares, which cannot be factored further. m 4 81n 4 (m 2 9n 2 )(m 3n)(m 3n) CHECK YOURSELF 3 Factor x 4 16y 4. NOTE Be sure you take the time to expand the product on the righthand side to confirm the identity. Two additional patterns for factoring certain binomials include the sum or difference of two cubes. Rules and Properties: The Sum or Difference of Two Cubes a 3 b 3 (a b)(a 2 ab b 2 ) (2) a 3 b 3 (a b)(a 2 ab b 2 ) (3)
3 FACTORING SPECIAL POLYNOMIALS SECTION Example 4 Factoring the Sum or Difference of Two Cubes NOTE We are now looking for perfect cubes the exponents must be multiples of 3 and the coefficients perfect cubes 1, 8, 27, 64, and so on. (a) Factor x The first term is the cube of x, and the second is the cube of 3, so we can apply equation (2). Letting a x and b 3, we have x 3 27 (x 3)(x 2 3x 9) (b) Factor 8w 3 27z 3. This is a difference of cubes, so use equation (3). 8w 3 27z 3 (2w 3z)[(2w) 2 (2w)(3z) (3z) 2 ] (2w 3z)(4w 2 6wz 9z 2 ) NOTE Again, looking for a common factor should be your first step. NOTE Remember to write the GCF as a part of the final factored form. (c) Factor 5a 3 b 40b 4. First note the common factor of 5b. The binomial is the difference of cubes, so use equation (3). 5a 3 b 40b 4 5b(a 3 8b 3 ) 5b(a 2b)(a 2 2ab 4b 2 ) CHECK YOURSELF 4 Factor completely. (a) 27x 3 8y 3 (b) 3a 4 24ab 3 In each example in this section, we factored a polynomial expression. If we are given a polynomial function to factor, there is no change in the ordered pairs represented by the function after it is factored. Example 5 Factoring a Polynomial Function Given the function f(x) 9x 2 15x, complete the following. (a) Find f(1). f(1) 9(1) 2 15(1) (b) Factor f(x). f(x) 9x 2 15x 3x(3x 5)
4 426 CHAPTER 6 POLYNOMIALS AND POLYNOMIAL FUNCTIONS (c) Find f(1) from the factored form of f(x). f(1) 3(1)(3(1) 5) 3(8) 24 CHECK YOURSELF 5 Given the function f(x) 16x 5 10x 2, complete the following. (a) Find f(1). (c) Find f(1) from the factored form of f(x). (b) Factor f(x). CHECK YOURSELF ANSWERS 1. (a) (y 6)( y 6); (b) (5m n)(5m n); (c) (4a 2 3b)(4a 2 3b) 2. 2x(x 3y)( x 3y) 3. (x 2 4y 2 )(x 2y)( x 2y) 4. (a) (3x 2y)( 9x 2 6xy 4y 2 ); (b) 3a(a 2b)(a 2 2ab 4b 2 ) 5. (a) 26; (b) 2x 2 (8x 3 5); (c) 26
5 Name 6.6 Exercises Section Date For each of the following binomials, state whether the binomial is a difference of squares. 1. 3x 2 2y x 2 7y a 2 25b n 2 16m r p a 2 12b a 2 b 2 16c 2 d 2 9. a 2 b a 3 b 3 Factor the following binomials. 11. x m a b p x a m x 2 y m 2 n 2 9 ANSWERS c 2 25d a 2 49b p 2 64q x 2 36y x 4 16y a 2 25b
6 ANSWERS a 3 4ab p 2 q q a 4 16b x 4 y x y m b a 3 b p 3 q w 3 z c 3 27d r 3 64s x 3 y x 3 27y m 3 27n x 3 y m 6 27n x 3 32y a 3 81b x 3 2xy a 2 b 2b m 3 n 75mn p 4 7p 2 q 2 428
7 ANSWERS For each of the functions in exercises 51 to 56, (a) find f(1), (b) factor f(x), and (c) find f(1) from the factored form of f(x). 51. f(x) 12x 5 21x f(x) 6x 3 10x f(x) 8x 5 20x 54. f(x) 5x 5 35x f(x) x 5 3x f(x) 6x 6 16x 5 Factor each expression. 57. x 2 (x y) y 2 (x y) 58. a 2 (b c) 16b 2 (b c) 59. 2m 2 (m 2n) 18n 2 (m 2n) 60. 3a 3 (2a b) 27ab 2 (2a b) 61. Find a value for k so that kx 2 25 will have the factors 2x 5 and 2x Find a value for k so that 9m 2 kn 2 will have the factors 3m 7n and 3m 7n Find a value for k so that 2x 3 kxy 2 will have the factors 2x, x 3y, and x 3y Find a value for k so that 20a 3 b kab 3 will have the factors 5ab, 2a 3b, and 2a 3b Complete the following statement in complete sentences: To factor a number you Complete this statement: To factor an algebraic expression into prime factors means Verify the formula for factoring the sum of two cubes by finding the product (a b)(a 2 ab b 2 ). 68. Verify the formula for factoring the difference of two cubes by finding the product (a b)(a 2 ab b 2 ). 429
8 ANSWERS What are the characteristics of a monomial that is a perfect cube? 70. Suppose you factored the polynomial 4x 2 16 as follows: 4x 2 16 (2x 4)(2x 4) Would this be in completely factored form? If not, what would be the final form? Answers 1. No 3. Yes 5. No 7. No 9. Yes 11. (x 7)(x 7) 13. (a 9)(a 9) 15. (3p 1)(3p 1) 17. (5a 4)(5a 4) 19. (xy 5)(xy 5) 21. (2c 5d)(2c 5d) 23. (7p 8q)(7p 8q) 25. (x 2 4y)(x 2 4y) 27. a(a 2b)(a 2b) 29. (a 2 4b 2 )(a 2b)(a 2b) 31. (x 4)(x 2 4x 16) 33. (m 5)(m 2 5m 25) 35. (ab 3)(a 2 b 2 3ab 9) 37. (2w z)(4w 2 2wz z 2 ) 39. (r 4s)(r 2 4rs 16s 2 ) 41. (2x 3y)(4x 2 6xy 9y 2 ) 43. (2x y 2 )(4x 2 2xy 2 y 4 ) 45. 4(x 2y)(x 2 2xy 4y 2 ) 47. 2x(3x y)(3x y) 49. 3mn(2m 5n)(2m 5n) 51. (a) 33; (b) 3x 2 (4x 3 7); (c) (a) 12; (b) 4x( 2x 4 5); (c) (a) 4; (b) x 2 (x 3 3); (c) (x y) 2 (x y) 59. 2(m 2n)(m 3n)(m 3n)
Difference of Squares and Perfect Square Trinomials
4.4 Difference of Squares and Perfect Square Trinomials 4.4 OBJECTIVES 1. Factor a binomial that is the difference of two squares 2. Factor a perfect square trinomial In Section 3.5, we introduced some
More informationBy reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms.
SECTION 5.4 Special Factoring Techniques 317 5.4 Special Factoring Techniques OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor
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 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 informationUsing the ac Method to Factor
4.6 Using the ac Method to Factor 4.6 OBJECTIVES 1. Use the ac test to determine factorability 2. Use the results of the ac test 3. Completely factor a trinomial In Sections 4.2 and 4.3 we used the trialanderror
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 informationIn algebra, factor by rewriting a polynomial as a product of lowerdegree polynomials
Algebra 2 Notes SOL AII.1 Factoring Polynomials Mrs. Grieser Name: Date: Block: Factoring Review Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 3 In algebra, factor by rewriting
More informationFactoring (pp. 1 of 4)
Factoring (pp. 1 of 4) Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. C) 6 Simplify 48 using the greatest common
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 information6.4 Special Factoring Rules
6.4 Special Factoring Rules OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor a sum of cubes. By reversing the rules for multiplication
More informationFACTORING POLYNOMIALS
296 (540) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated
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 informationFactoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1)
Factoring Methods When you are trying to factor a polynomial, there are three general steps you want to follow: 1. See if there is a Greatest Common Factor 2. See if you can Factor by Grouping 3. See if
More informationFactoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
More information76. Choosing a Factoring Model. Extension: Factoring Polynomials with More Than One Variable IN T RO DUC E T EACH. Standards for Mathematical Content
76 Choosing a Factoring Model Extension: Factoring Polynomials with More Than One Variable Essential question: How can you factor polynomials with more than one variable? What is the connection between
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 informationThe Greatest Common Factor; Factoring by Grouping
296 CHAPTER 5 Factoring and Applications 5.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.
More informationFactoring. Factoring Polynomial Equations. Special Factoring Patterns. Factoring. Special Factoring Patterns. Special Factoring Patterns
Factoring Factoring Polynomial Equations Ms. Laster Earlier, you learned to factor several types of quadratic expressions: General trinomial  2x 25x12 = (2x + 3)(x  4) Perfect Square Trinomial  x
More information6.1 The Greatest Common Factor; Factoring by Grouping
386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.
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 informationSection 6.1 Factoring Expressions
Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what
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 informationFactoring Algebra Chapter 8B Assignment Sheet
Name: Factoring Algebra Chapter 8B Assignment Sheet Date Section Learning Targets Assignment Tues 2/17 Find the prime factorization of an integer Find the greatest common factor (GCF) for a set of monomials.
More informationMultiplying polynomials  Decimals  Simplify product of monomials and trinomials
1 MATHX.NET Multiplying polynomials  Decimals  Simplify product of monomials and trinomials Simplify decimal product with two variables: 1) 0.4(2.4x 2 6.87xy 4.27y 2 ) 2) 4.8a 4 (6.1a 2 + 2.9ab 5.8b
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 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 informationFactoring. Factoring Monomials Monomials can often be factored in more than one way.
Factoring Factoring is the reverse of multiplying. When we multiplied monomials or polynomials together, we got a new monomial or a string of monomials that were added (or subtracted) together. For example,
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 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 informationSimplifying Algebraic Fractions
5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions
More informationUnit 3 Polynomials Study Guide
Unit Polynomials Study Guide 75 Polynomials Part 1: Classifying Polynomials by Terms Some polynomials have specific names based upon the number of terms they have: # of Terms Name 1 Monomial Binomial
More informationA Systematic Approach to Factoring
A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool
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 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 information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More information6.3 FACTORING ax 2 bx c WITH a 1
290 (6 14) Chapter 6 Factoring e) What is the approximate maximum revenue? f) Use the accompanying graph to estimate the price at which the revenue is zero. y Revenue (thousands of dollars) 300 200 100
More informationFactoring  Trinomials where a = 1
6.3 Factoring  Trinomials where a = 1 Objective: Factor trinomials where the coefficient of x 2 is one. Factoring with three terms, or trinomials, is the most important type of factoring to be able to
More informationPOLYNOMIALS and FACTORING
POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use
More information5.1 FACTORING OUT COMMON FACTORS
C H A P T E R 5 Factoring he sport of skydiving was born in the 1930s soon after the military began using parachutes as a means of deploying troops. T Today, skydiving is a popular sport around the world.
More informationFactoring Flow Chart
Factoring Flow Chart greatest common factor? YES NO factor out GCF leaving GCF(quotient) how many terms? 4+ factor by grouping 2 3 difference of squares? perfect square trinomial? YES YES NO NO a 2 b
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 informationSummer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2
Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level
More informationThis is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0).
This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationFactoring  Factoring Special Products
6.5 Factoring  Factoring Special Products Objective: Identify and factor special products including a difference of squares, perfect squares, and sum and difference of cubes. When factoring there are
More informationA. Factoring out the Greatest Common Factor.
DETAILED SOLUTIONS AND CONCEPTS  FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!
More informationSTUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS
STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationChapter R.4 Factoring Polynomials
Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x
More informationFACTORING TRINOMIALS IN THE FORM OF ax 2 + bx + c
Tallahassee Community College 55 FACTORING TRINOMIALS IN THE FORM OF ax 2 + bx + c This kind of trinomial differs from the previous kind we have factored because the coefficient of x is no longer "1".
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 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 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 informationFactoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai
Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for
More informationAIP Factoring Practice/Help
The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There
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 informationPolynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF
Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials
More informationFactoring  Grouping
6.2 Factoring  Grouping Objective: Factor polynomials with four terms using grouping. The first thing we will always do when factoring is try to factor out a GCF. This GCF is often a monomial like in
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 informationPolynomials  Multiplying Polynomials
5.5 Polynomials  Multiplying Polynomials Objective: Multiply polynomials. Multiplying polynomials can take several different forms based on what we are multiplying. We will first look at multiplying monomials,
More informationPlacement Test Review Materials for
Placement Test Review Materials for 1 To The Student This workbook will provide a review of some of the skills tested on the COMPASS placement test. Skills covered in this workbook will be used on the
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 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 informationTool 1. Greatest Common Factor (GCF)
Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
More informationFactoring a Difference of Two Squares. Factoring a Difference of Two Squares
284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this
More informationMath 25 Activity 6: Factoring Advanced
Instructor! Math 25 Activity 6: Factoring Advanced Last week we looked at greatest common factors and the basics of factoring out the GCF. In this second activity, we will discuss factoring more difficult
More informationUnit: Polynomials and Factoring
Name Unit: Polynomials: Multiplying and Factoring Specific Outcome 10I.A.1 Demonstrate an understanding of factors of whole numbers by determining: Prime factors Greatest common factor Least common multiple
More informationPolynomials  Multiplying
5.5 Polynomials  Multiplying Multiplying polynomials can take several different forms based on what we are multiplying. We will first look at multiplying monomials, then monomials by polynomials and finish
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
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 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 informationName Intro to Algebra 2. Unit 1: Polynomials and Factoring
Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332
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 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 informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationSimplification of Radical Expressions
8. Simplification of Radical Expressions 8. OBJECTIVES 1. Simplify a radical expression by using the product property. Simplify a radical expression by using the quotient property NOTE A precise set of
More informationMath PreCalc 20 Chapter 4 Review of Factoring. Questions to try. 2. x 2 6xy x x x x 2 y + 8xy
Math PreCalc 20 Chapter 4 Review of Factoring Multiplying (Expanding) Type 1: Monomial x Binomial Monomial x Trinomial Ex: 3(x + 4) = 3x + 122(x 2 + 2x 1) = 2x 2 4x + 2 Multiply the following: 1. 5(x
More informationPolynomial Expression
DETAILED SOLUTIONS AND CONCEPTS  POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More informationPERFECT SQUARES AND FACTORING EXAMPLES
PERFECT SQUARES AND FACTORING EXAMPLES 1. Ask the students what is meant by identical. Get their responses and then explain that when we have two factors that are identical, we call them perfect squares.
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More information#6 Opener Solutions. Move one more spot to your right. Introduce yourself if needed.
1. Sit anywhere in the concentric circles. Do not move the desks. 2. Take out chapter 6, HW/notes #1#5, a pencil, a red pen, and your calculator. 3. Work on opener #6 with the person sitting across from
More informationSIMPLIFYING ALGEBRAIC FRACTIONS
Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is
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 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 informationSimplifying Radical Expressions
9.2 Simplifying Radical Expressions 9.2 OBJECTIVES. Simplify expressions involving numeric radicals 2. Simplify expressions involving algebraic radicals In Section 9., we introduced the radical notation.
More informationAlgebra 1 Chapter 08 review
Name: Class: Date: ID: A Algebra 1 Chapter 08 review Multiple Choice Identify the choice that best completes the statement or answers the question. Simplify the difference. 1. (4w 2 4w 8) (2w 2 + 3w 6)
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 information1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style
Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with
More information~ EQUIVALENT FORMS ~
~ EQUIVALENT FORMS ~ Critical to understanding mathematics is the concept of equivalent forms. Equivalent forms are used throughout this course. Throughout mathematics one encounters equivalent forms of
More informationEAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors.
EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationAlgebra 2 PreAP. Name Period
Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
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 informationSect 6.1  Greatest Common Factor and Factoring by Grouping
Sect 6.1  Greatest Common Factor and Factoring by Grouping Our goal in this chapter is to solve nonlinear equations by breaking them down into a series of linear equations that we can solve. To do this,
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 informationThe majority of college students hold credit cards. According to the Nellie May
CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials
More informationPolynomial Equations and Factoring
7 Polynomial Equations and Factoring 7.1 Adding and Subtracting Polynomials 7.2 Multiplying Polynomials 7.3 Special Products of Polynomials 7.4 Dividing Polynomials 7.5 Solving Polynomial Equations in
More informationSIMPLIFYING SQUARE ROOTS
40 (88) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify
More informationPolynomials. Solving Equations by Using the Zero Product Rule
mil23264_ch05_303396 9:21:05 06:16 PM Page 303 Polynomials 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials 5.4 Greatest
More information