Negative Integer Exponents


 Dora Shelton
 1 years ago
 Views:
Transcription
1 7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions that contain negative exponents In Section.4, all the exponents we looked at were positive integers. In this section, we look at the meaning of zero and negative integer exponents. First, let s look at an application of the quotient rule that will yield a zero exponent. Recall that, in the quotient rule, to divide expressions with the same base, keep the base and subtract the exponents. a m a n am n Now, suppose that we allow m to equal n. We then have a m () a m am m a 0 But we know that it is also true that a m (2) a m Comparing equations () and (2), we see that the following definition is reasonable. Rules and Properties: The Zero Exponent NOTE We must have a 0. The form 0 0 is called indeterminate and is considered in later mathematics classes. For any real number a when a 0, a 0 Example The Zero Exponent Use the above definition to simplify each expression. NOTE Notice that in 6x 0 the exponent 0 applies only to x. (a) 7 0 (b) (a 3 b 2 ) 0 (c) 6x (d) 3y McGrawHill Companies CHECK YOURSELF Simplify each expression. (a) 25 0 (b) (m 4 n 2 ) 0 (c) 8s 0 (d) 7t 0 Recall that, in the product rule, to multiply expressions with the same base, keep the base and add the exponents. a m a n a m+n 555
2 556 CHAPTER 7 RATIONAL EXPRESSIONS AND FUNCTIONS Now, what if we allow one of the exponents to be negative and apply the product rule? Suppose, for instance, that m 3 and n 3. Then a m a n a 3 a 3 a 3 ( 3) so a 0 a 3 a 3 Dividing both sides by a 3, we get a 3 a 3 Rules and Properties: Negative Integer Exponents NOTE John Wallis (66 702), an English mathematician, was the first to fully discuss the meaning of 0, negative, and rational exponents. For any nonzero real number a and whole number n, a n a n and a n is the multiplicative inverse of a n. Example 2 illustrates this definition. Example 2 Using Properties of Exponents NOTE From this point on, to simplify will mean to write the expression with positive exponents only. Simplify the following expressions. (a) y 5 y 5 NOTE Also, we will restrict all variables so that they represent nonzero real numbers. (b) (c) ( 3) 3 ( 3) (d) CHECK YOURSELF 2 (a) a 0 (b) 2 4 (c) ( 4) 2 (d) McGrawHill Companies
3 NEGATIVE INTEGER EXPONENTS SECTION Example 3 illustrates the case in which coefficients are involved in an expression with negative exponents. As will be clear, some caution must be used. Example 3 Using Properties of Exponents CAUTION (a) 2x 3 2 x 3 2 x 3 The expressions 4w 2 and (4w) 2 are not the same. Do you see why? The exponent 3 applies only to the variable x, and not to the coefficient 2. (b) 4w 2 4 w 2 4 w 2 (c) (4w) 2 (4w) 2 6w 2 CHECK YOURSELF 3 (a) 3w 4 (b) 0x 5 (c) (2y) 4 (d) 5t 2 Suppose that a variable with a negative exponent appears in the denominator of an expression. Our previous definition can be used to write a complex fraction that can then be simplified. For instance, a 2 a2 a2 a 2 Negative exponent in denominator. Positive exponent in numerator. To divide, we invert and multiply. To avoid the intermediate steps, we can write that, in general, 200 McGrawHill Companies Rules and Properties: For any nonzero real number a and integer n, a n a n Negative Exponents in a Denominator
4 558 CHAPTER 7 RATIONAL EXPRESSIONS AND FUNCTIONS Example 4 Using Properties of Exponents (a) (b) (c) y 3 y x 2 3x2 4 The exponent 2 applies only to x, not to 4. (d) a 3 b 4 b4 a 3 CHECK YOURSELF 4 2 (a) (b) (c) (d) c 5 x a 2 d 7 NOTE To review these properties, return to Section.4. The product and quotient rules for exponents apply to expressions that involve any integer exponent positive, negative, or 0. Example 5 illustrates this concept. Example 5 Using Properties of Exponents Simplify each of the following expressions, and write the result, using positive exponents only. (a) x 3 x 7 x 3 ( 7) (b) x 4 x 4 m 5 m 3 m 5 ( 3) m 5 3 Add the exponents by the product rule. Subtract the exponents by the quotient rule. m 2 m 2 NOTE Notice that m 5 in the numerator becomes m 5 in the denominator, and m 3 in the denominator becomes m 3 in the numerator. We then simplify as before. (c) x 5 x 3 x 7 x5 ( 3) x 7 x2 x 7 x2 ( 7) x 9 In simplifying expressions involving negative exponents, there are often alternate approaches. For instance, in Example 5(b), we could have made use of our earlier work to write m 5 m 3 m3 m 5 m 3 5 m 2 m 2 We apply first the product rule and then the quotient rule. 200 McGrawHill Companies
5 NEGATIVE INTEGER EXPONENTS SECTION CHECK YOURSELF 5 y 7 (a) (b) (c) a 3 a 2 x 9 x 5 y 3 a 5 The properties of exponents can be extended to include negative exponents. One of these properties, the quotientpower rule, is particularly useful when rational expressions are raised to a negative power. Let s look at the rule and apply it to negative exponents. Rules and Properties: a n a, b 0 b n b n QuotientPower Rule Rules and Properties: a b n a n bn n b a n b a n Raising Quotients to a Negative Power a 0, b 0 Example 6 Extending the Properties of Exponents Simplify each expression. (a) s3 t 2 2 t 2 s 3 2 t4 s 6 (b) m 2 n 2 3 n 2 CHECK YOURSELF 6 Simplify each expression. s3 m 3 2 n 6 m 6 n 6 m 6 x (a) (b) 5 y 3 3t As you might expect, more complicated expressions require the use of more than one of the properties for simplification. Example 7 illustrates such cases. 200 McGrawHill Companies Example 7 Using Properties of Exponents (a) (a 2 ) 3 (a 3 ) 4 (a 3 ) 3 a 6 a 2 a 9 a 6 2 a 9 a6 a 9 a 6 ( 9) a 6 9 a 5 Apply the power rule to each factor. Apply the product rule. Apply the quotient rule.
6 560 CHAPTER 7 RATIONAL EXPRESSIONS AND FUNCTIONS NOTE It may help to separate the problem into three fractions, one for the coefficients and one for each of the variables. CAUTION (b) (c) 8x 2 y 5 2x 4 y x 2 x y 5 4 y x 2 ( 4) y x2 y 8 2x2 3y 8 pr3 s 5 p 3 r 3 s 2 2 p 3 r 3 s 2 pr 3 s 5 2 p6 r 6 s 4 p 2 r 6 s 0 p 4 r 2 s 6 p4 s 6 r 2 Be Careful! Another possible first step (and generally an efficient one) is to rewrite an expression by using our earlier definitions. a n a n and n an a For instance, in Example 8(b), we would correctly write 8x 2 y 5 2x 4 y 3 8x4 2x 2 y 3 y 5 A common error is to write 8x 2 y 5 2x 4 y 3 2x4 8x 2 y 3 y 5 This is not correct. The coefficients should not be moved along with the factors in x. Keep in mind that the negative exponents apply only to the variables. The coefficients remain where they were in the original expression when the expression is rewritten by using this approach. CHECK YOURSELF 7 (x 5 ) 2 (x 2 ) 3 2a 3 b 2 xy (a) (b) (c) 3 z 5 x 4 y 2 z 3 (x 4 ) 3 6a 2 b 3 3 CHECK YOURSELF ANSWERS 4. (a) ; (b) ; (c) 8; (d) 7 2. (a) ; (b) ; (c) ; (d) a a 2 d 7 3. (a) ; (b) ; (c) ; (d) 5 4. (a) x 4 ; (b) 27; (c) ; (d) w 4 x 5 6y 4 t 2 3 c 5 27t (a) x 4 ; (b) ; (c) a 4 6. (a) ; (b) 7. (a) x 8 ; (b) ; (c) y3 z 24 y 4 s 9 x 5 y 6 4ab 5 x McGrawHill Companies
7 Name 7.7 Exercises Section Date In exercises to 22, simplify each expression.. x ANSWERS x 8 5. ( 5) 2 6. ( 3) ( 2) 3 8. ( 2) x x x 4 4. ( 2x) 4 5. ( 3x) x x 3 x x 3 x 5 y 3 4x 4 x 3 y In exercises 23 to 32, use the properties of exponents to simplify the expressions. 23. x 5 x y 4 y a 9 a w 5 w z 2 z b 7 b McGrawHill Companies 29. a 5 a x 4 x x 3 x 6 x 5 x 2 56
8 ANSWERS In exercises 33 to 58, use the properties of exponents to simplify the following. 33. (x 5 ) (w 4 ) (2x 3 )(x 2 ) (p 4 )(3p 3 ) (3a 4 )(a 3 )(a 2 ) 38. (5y 2 )(2y)(y 5 ) (x 4 y)(x 2 ) 3 (y 3 ) (r 4 ) 2 (r 2 s)(s 3 ) 2 4. (ab 2 c)(a 4 ) 4 (b 2 ) 3 (c 3 ) (p 2 qr 2 )(p 2 )(q 3 ) 2 (r 2 ) (x 5 ) (x 2 ) (b 4 ) (a 0 b 4 ) (x 5 y 3 ) (p 3 q 2 ) (x 4 y 2 ) (3x 2 y 2 ) a 6 b 4 5. (2x 3 y 0 ) x 2 y x 3 x y 3 2 y (4x 2 ) 2 (3x 4 ) 58. (5x 4 ) 4 (2x 3 ) 5 (3x 4 ) 2 (2x 2 ) x 6 In exercises 59 to 90, simplify each expression. 59. (2x 5 ) 4 (x 3 ) (3x 2 ) 3 (x 2 ) 4 (x 2 ) 6. (2x 3 ) 3 (3x 3 ) (x 2 y 3 ) 4 (xy 3 ) (xy 5 z) 4 (xyz 2 ) 8 (x 6 yz) (x 2 y 2 z 2 ) 0 (xy 2 z) 2 (x 3 yz 2 ) 65. (3x 2 )(5x 2 ) (2a 3 ) 2 (a 0 ) (2w 3 ) 4 (3w 5 ) McGrawHill Companies 562
9 ANSWERS 3x 6 x (3x 3 ) 2 (2x 4 ) y 6 2y9 2y 9 y5 x 3 x 3 7. ( 7x 2 y)( 3x 5 y 6 ) w5 z (2x 2 y 3 )(3x 4 y 2 ) 3x 3 y x5 y 4 9 w 4 z (x 3 )(y 2 ) 74. ( 5a 2 b 4 )(2a 5 b 0 ) y 3 6x 3 y 4 24x 2 y x 3 y 2 z 4 24x 5 y 3 z x 4 y 3 z 2 36x 2 y 3 z x 2 y 2 8. x 3 y 2 x 4 y xy3 z 4 2 x 3 y 2 z 2 2 x 2 y x 5 y 7 x 0 y 4 x 3 y 3 x 4 y 3 x 2 2 y 2 xy x 2n x 3n 84. x n x 3n 85. x n 4 x n x n 3 x n (y n ) 3n 88. (x n ) n x 2n x n x 3n x n x 3n 5 x 4n Can (a b) be written as by using the properties of exponents? If not, why a b not? Explain Write a short description of the difference between ( 4) 3, 4 3, ( 4) 3, and 4 3. Are any of these equal? McGrawHill Companies 93. If n 0, which of the following expressions are negative? ( n) 3, n 3, n 3, ( n) 3, ( n) 3, n 3 If n 0, which of these expressions are negative? Explain what effect a negative in the exponent has on the sign of the result when an exponential expression is simplified
10 Answers x x 2 x x 3 2x 3 y x x 2 5 x 3 a 3 z x x x a 39. x 0 y 4. a 7 b 8 c b 8 x x 2 y 6 x 5 5. x 5 y 6 32 y 4 y x x 42 y 33 z 25 x x w 2 3x x 22 y y 4 x 2 y 5 y 5 3xy 5 y 8 5n x 85. x 2 x 3 4z 6 x 5 y 3 x x y 3n2 x 4 x 8 x McGrawHill Companies 564
Chapter 5. Rational Expressions
5.. Simplify Rational Expressions KYOTE Standards: CR ; CA 7 Chapter 5. Rational Expressions Definition. A rational expression is the quotient P Q of two polynomials P and Q in one or more variables, where
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More informationSection A3 Polynomials: Factoring APPLICATIONS. A22 Appendix A A BASIC ALGEBRA REVIEW
A Appendi A A BASIC ALGEBRA REVIEW C In Problems 53 56, perform the indicated operations and simplify. 53. ( ) 3 ( ) 3( ) 4 54. ( ) 3 ( ) 3( ) 7 55. 3{[ ( )] ( )( 3)} 56. {( 3)( ) [3 ( )]} 57. Show by
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
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 informationBEGINNING ALGEBRA ACKNOWLEDMENTS
BEGINNING ALGEBRA The Nursing Department of Labouré College requested the Department of Academic Planning and Support Services to help with mathematics preparatory materials for its Bachelor of Science
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 informationFree PreAlgebra Lesson 55! page 1
Free PreAlgebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
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 informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationMATH 90 CHAPTER 1 Name:.
MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.
More informationSection 1. Finding Common Terms
Worksheet 2.1 Factors of Algebraic Expressions Section 1 Finding Common Terms In worksheet 1.2 we talked about factors of whole numbers. Remember, if a b = ab then a is a factor of ab and b is a factor
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 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 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 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 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 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 informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationTransition To College Mathematics
Transition To College Mathematics In Support of Kentucky s College and Career Readiness Program Northern Kentucky University Kentucky Online Testing (KYOTE) Group Steve Newman Mike Waters Janis Broering
More information4.1. COMPLEX NUMBERS
4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
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 informationSimplification Problems to Prepare for Calculus
Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.
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 informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
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 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 informationCollege Algebra and Trigonometry
7th edition College Algebra and Trigonometry Richard N. Aufmann Vernon C. Barker Richard D. Nation College Algebra and Trigonometry, Seventh Edition Richard N. Aufmann, Vernon C. Barker, Richard D. Nation
More informationSECTION A3 Polynomials: Factoring
A3 Polynomials: Factoring A23 thick, write an algebraic epression in terms of that represents the volume of the plastic used to construct the container. Simplify the epression. [Recall: The volume 4
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More informationFACTORISATION YEARS. A guide for teachers  Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project
9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers  Years 9 10 June 2011 Factorisation (Number and Algebra : Module
More informationBoolean Algebra Part 1
Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems
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  Greatest Common Factor
6.1 Factoring  Greatest Common Factor Objective: Find the greatest common factor of a polynomial and factor it out of the expression. The opposite of multiplying polynomials together is factoring polynomials.
More informationReview of Basic Algebraic Concepts
Section. Sets of Numbers and Interval Notation Review of Basic Algebraic Concepts. Sets of Numbers and Interval Notation. Operations on Real Numbers. Simplifying Expressions. Linear Equations in One Variable.
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationPolynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005
Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division
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 informationLicensed to: Printed in the United States of America 1 2 3 4 5 6 7 16 15 14 13 12
Licensed to: CengageBrain User This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
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 informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More information86 Radical Expressions and Rational Exponents. Warm Up Lesson Presentation Lesson Quiz
86 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt Algebra ALgebra2 2 Warm Up Simplify each expression. 1. 7 3 7 2 16,807 2. 11 8 11 6 121 3. (3 2 ) 3 729 4. 5.
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 informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationPERT Mathematics Test Review
PERT Mathematics Test Review Prof. Miguel A. Montañez ESL/Math Seminar Math Test? NO!!!!!!! I am not good at Math! I cannot graduate because of Math! I hate Math! Helpful Sites Math Dept Web Site Wolfson
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More informationMATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab
MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring noncourse based remediation in developmental mathematics. This structure will
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationFactoring Polynomials
Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent
More informationMTH41061. actoring. and. Algebraic Fractions
MTH41061 C1C4 Factorization 1/31/12 11:38 AM Page 1 F MTH41061 actoring and Algebraic Fractions FACTORING AND ALGEBRAIC FUNCTIONS Project Coordinator: JeanPaul Groleau Authors: Nicole Perreault
More informationSOLVING POLYNOMIAL EQUATIONS
C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra
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 informationMaple for Math Majors. 8. Manipulating and Simplifying Expressions 8.1. Introduction
Maple for Math Majors Roger Kraft Department of Mathematics, Computer Science, and Statistics Purdue University Calumet roger@calumet.purdue.edu 8. Manipulating and Simplifying Expressions 8.1. Introduction
More informationHIBBING COMMUNITY COLLEGE COURSE OUTLINE
HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE:  Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
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 informationSample Test Questions
mathematics Numerical Skills/PreAlgebra Algebra Sample Test Questions A Guide for Students and Parents act.org/compass Note to Students Welcome to the ACT Compass Sample Mathematics Test! You are about
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 informationSome facts about polynomials modulo m (Full proof of the Fingerprinting Theorem)
Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem) In order to understand the details of the Fingerprinting Theorem on fingerprints of different texts from Chapter 19 of the
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 information2 Integrating Both Sides
2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation
More informationPolynomials and Factoring; More on Probability
Polynomials and Factoring; More on Probability Melissa Kramer, (MelissaK) Anne Gloag, (AnneG) Andrew Gloag, (AndrewG) Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required)
More informationFactor Diamond Practice Problems
Factor Diamond Practice Problems 1. x 2 + 5x + 6 2. x 2 +7x + 12 3. x 2 + 9x + 8 4. x 2 + 9x +14 5. 2x 2 7x 4 6. 3x 2 x 4 7. 5x 2 + x 18 8. 2y 2 x 1 9. 613x + 6x 2 10. 15 + x 2x 2 Factor Diamond Practice
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More information9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.
9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n1 x n1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role
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 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 informationPROOFS BY DESCENT KEITH CONRAD
PROOFS BY DESCENT KEITH CONRAD As ordinary methods, such as are found in the books, are inadequate to proving such difficult propositions, I discovered at last a most singular method... that I called the
More informationCollege Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
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 informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationparent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL
parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL HS America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does
More informationLimits. Graphical Limits Let be a function defined on the interval [6,11] whose graph is given as:
Limits Limits: Graphical Solutions Graphical Limits Let be a function defined on the interval [6,11] whose graph is given as: The limits are defined as the value that the function approaches as it goes
More informationGouvernement du Québec Ministère de l Éducation, 2004 0400813 ISBN 2550435451
Gouvernement du Québec Ministère de l Éducation, 004 0400813 ISBN 550435451 Legal deposit Bibliothèque nationale du Québec, 004 1. INTRODUCTION This Definition of the Domain for Summative Evaluation
More informationSECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31
More informationSOLUTIONS FOR PROBLEM SET 2
SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
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 informationSolving Quadratic Equations by Factoring
4.7 Solving Quadratic Equations by Factoring 4.7 OBJECTIVE 1. Solve quadratic equations by factoring The factoring techniques you have learned provide us with tools for solving equations that can be written
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationDeterminants can be used to solve a linear system of equations using Cramer s Rule.
2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution
More informationPolynomial Operations and Factoring
Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of 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 information3.2 The Factor Theorem and The Remainder Theorem
3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationPURSUITS IN MATHEMATICS often produce elementary functions as solutions that need to be
Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions 2007 Ron Doerfler http://www.myreckonings.com June 27, 2007 Abstract There are some of us who enjoy using our
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 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 information