Polynomials INTRODUCTION CHAPTER 3 OUTLINE. Chapter 3 :: Prerequisite Test 182. Exponents and Polynomials 183

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1 bar92103_h03_a_ qxd 9/19/09 12:03 PM Page 181 C H A P T E R hapter The MGraw-Hill Companies. All Rights Reserved. The Streeter/Huthison Series in Mathematis Beginning Algebra 3 > Make the Connetion 3 INTRODUCTION Polynomials are used in many disiplines and industries to model appliations and solve problems. For example, aerospae engineers use omplex formulas to plan and guide spae shuttle flights, and teleommuniations engineers use them to improve digital signal proessing. Equations expressing relationships among variables play a signifiant role in building onstrution, estimating eletrial power generation needs and onsumption, astronomy, mediine and pharmaologial measurements, determining manufaturing osts, and projeting retail revenue. The field of personal investments and savings presents an opportunity to estimate the future value of savings aounts, Individual Retirement Aounts, and other investment produts. In the hapter ativity we explore the power of ompound interest. Polynomials CHAPTER 3 OUTLINE Chapter 3 :: Prerequisite Test Exponents and Polynomials Adding and Subtrating Polynomials 210 Negative Exponents and Sientifi Notation 198 Multiplying Polynomials 220 Dividing Polynomials 236 Chapter 3 :: Summary / Summary Exerises / Self-Test / Cumulative Review :: Chapters

2 3 prerequisite test CHAPTER 3 Name Setion Date This prerequisite test provides some exerises requiring skills that you will need to be suessful in the oming hapter. The answers for these exerises an be found in the bak of this text. This prerequisite test an help you identify topis that you will need to review before beginning the hapter. Answers Evaluate eah expression ( 3) Simplify eah expression x 2(3x 4) 8. 2x 5y y Evaluate eah expression. 9. 7x 2 4x 3 for x x 2 3xy y 2 for x 3 and y 2 Solve eah appliation. 11. NUMBER PROBLEM Find two onseutive odd integers suh that 3 times the first integer is 5 more than twie the seond integer. 12. ELECTRICAL ENGINEERING Resistane (in ohms, Ω) is given by the formula R V 2 D in whih D is the power dissipation (in watts) and V is the voltage. Determine the power dissipation when 13.2 volts pass through a 220-Ω resistor. 182

3 bar92103_h03_a_ qxd 9/19/ < 3.1 Objetives > 12:03 PM Page 183 Exponents and Polynomials 1> 2> 3> 4> 5> Use the properties of exponents to simplify expressions Identify types of polynomials Find the degree of a polynomial Write a polynomial in desending order Evaluate a polynomial Preparing for a Test Preparing for a test begins on the first day of lass. Everything you do in lass and at home is part of that preparation. In fat, if you attend lass every day, take good notes, and keep up with the homework, then you will already be prepared and not need to ram for your exam. Instead of ramming, here are a few things to fous on in the days before a sheduled test. 1. Study for your exam, but finish studying 24 hours before the test. Make ertain to get some rest before taking a test. 2. Study for an exam by going over homework and lass notes. Write down all of the problem types, formulas, and definitions that you think might give you trouble on the test. 3. The last item before you finish studying is to take the notes you made in step 2 and transfer the most important ideas to a 3 5 (index) ard. You should omplete this step a full 24 hours before your exam. 4. One hour before your exam, review the information on the 3 5 ard you made in step 3. You will be surprised at how muh you remember about eah onept. 5. The biggest obstale for many students is believing that they an be suessful on the test. You an overome this obstale easily enough. If you have been ompleting the homework and keeping up with the lasswork, then you should perform quite well on the test. Truly anxious students are often surprised to sore well on an exam. These students attribute a good test sore to blind luk when it is not luk at all. This is the first sign that they get it. Enjoy the suess! Reall that exponential notation indiates repeated multipliation; the exponent or power tells us how many times the base is to be used as a fator. Exponent or Power The MGraw-Hill Companies. All Rights Reserved. The Streeter/Huthison Series in Mathematis Beginning Algebra Tips for Student Suess 5 fators Base 183

4 184 CHAPTER 3 Polynomials NOTE In order to effetively use exponential notation, we need to understand how to evaluate and simplify expressions that ontain exponents. To do this, we need to understand some properties assoiated with exponents. 2 3 # # ; Another way to look at this same produt is to expand eah exponential expression. 2 3 # 2 2 (2 # 2 # 2) # (2 # 2) 2 # 2 # 2 # 2 # 2 We an remove the parentheses. 2 5 There are 5 fators (of 2). Now onsider what happens when we replae 2 by a variable. a 3 # a 2 (a # a # a) (a # a) The base must be the same in both fators. We annot ombine a 2 b 3 any further. Property > CAUTION Produt Property of Exponents NOTE Example 1 In every ase, the base stays the same. RECALL If a fator has no exponent, it is understood to be to the first power (the exponent is one). a 3 a 2 a # a # a # a # a Five fators. a 5 You should see that the result, a 5, an be found by simply adding the exponents beause this gives the number of times the base appears as a fator in the final produt. a 3 # a 2 a 3 2 a 5 Add the exponents. We an now state our first property, the produt property of exponents, for the general ase. For any real number a and positive integers m and n, a m a n a m n In words, the produt of two terms with the same base is the base taken to the power that is the sum of the exponents. For example, Here is an example illustrating the produt property of exponents. Using the Produt Property of Exponents Write eah expression as a single base to a power. (a) (b) () (d) b 4 # b 6 b 4 6 b 10 ( 2) 5 ( 2) 4 ( 2) # x 5 # x x 5 1 x Add the exponents. ( 2) 9 x x1 The base does not hange; we are already multiplying the base by adding the exponents.

5 Exponents and Polynomials SECTION Chek Yourself 1 Write eah expression as a single base to a power. (a) (b) () (d) y # y 6 x 7 # x 3 ( 3) 4 ( 3) 3 (x 2 y) 3 (x 2 y) 5 RECALL Example 2 Multiply the oeffiients but add the exponents. With pratie, you will not need to write the regrouping step. By applying the ommutative and assoiative properties of multipliation, we an simplify produts that have oeffiients. Consider the following ase. 2x 3 # 3x 4 (2 # 3)(x 3 # x 4 ) We an group the fators any way we want. 6x 7 The next example expands on this idea. Using the Properties of Exponents Simplify eah expression. (a) (b) (3x 4 )(5x 2 ) (3 # 5)(x 4 # x 2 ) 15x 6 (2x 5 y)(9x 3 y 4 ) (2 # 9)(x 5 # x 3 )(y # y 4 ) 18x 8 y 5 Regroup the fators. Add the exponents. Chek Yourself 2 Property Quotient Property of Exponents Simplify eah expression. (a) (7x 5 )(2x 2 ) (b) ( 2x 3 y)(x 2 y 2 ) () (d) x # x 5 # x 3 ( 5x 3 y 2 )( 3x 2 y 3 ) What happens when we divide two exponential expressions with the same base? Consider the following ases # 2 # 2 # 2 # 2 2 # 2 2 # 2 # Expand and simplify. You should immediately see that the final exponent is the differene between the two exponents: This is true in the more general ase: a 6 a 4 a # a # a # a # a # a a # a # a # a a 2 We an now state our seond rule, the quotient property of exponents. For any nonzero real number a and positive integers m and n, with m n, a m n am n a For example,

6 186 CHAPTER 3 Polynomials Example 3 Using the Quotient Properties of Exponents Simplify eah expression. (a) (b) x 10 x 4 x10 4 x 6 a 8 7 a8 7 a a Subtrat the exponents. a 1 a; we do not need to write the exponent. () 32a 4 b 5 32 # a 4 # b 5 8a 2 b 8 a 2 b 4a 4 2 b 5 1 4a 2 b 4 Chek Yourself 3 Use the properties of frations to regroup the fators. Apply the quotient property to eah grouping. Simplify eah expression. y 12 x 9 45r 8 (a) (b) () (d) 56m6 n 7 y 5 x 9r 7 7mn 3 NOTE This means that the base, x 2, is used as a fator 4 times. Property Power to a Power Property of Exponents < Objetive 1 > Example 4 Be sure to distinguish between the orret use of the produt property and the power to a power property. (x 4 ) 5 x 4 5 x 20 but x 4 x 5 x 4 5 x 9 > CAUTION Consider the following: (x 2 ) 4 x 2 x 2 x 2 x 2 x 8 This leads us to our third property for exponents. For any real number a and positive integers m and n, (a m ) n a m n For example, (2 3 ) We illustrate this property in the next example. Using the Power to a Power Property of Exponents Simplify eah expression. (a) (x 4 ) 5 x 4 5 x 20 (b) (2 3 ) Chek Yourself 4 Multiply the exponents. Simplify eah expression. (a) (m 5 ) 6 (b) (m 5 )(m 6 ) () (3 2 ) 4 (d) (3 2 )(3 4 )

7 Exponents and Polynomials SECTION NOTES Here the base is 3x. We apply the ommutative and assoiative properties. Suppose we have a produt raised to a power, suh as (3x) 4. We know that (3x) 4 (3x)(3x)(3x)(3x) ( )(x x x x) 3 4 x 4 81x 4 Note that the power, here 4, has been applied to eah fator, 3 and x. In general, we have: Property Produt to a Power Property of Exponents For any real numbers a and b and positive integer m, (ab) m a m b m For example, (3x) x 3 27x 3 The use of this property is shown in Example 5. NOTE Example 5 (2x) 5 and 2x 5 are different expressions. For (2x) 5, the base is 2x, so we raise eah fator to the fifth power. For 2x 5, the base is x, and so the exponent applies only to x. NOTE Example 6 To help you understand eah step of the simplifiation, we refer to the property being applied. Make a list of the properties now to help you as you work through the remainder of this setion and Setion 3.2. Using the Produt to a Power Property of Exponents Simplify eah expression. (a) (2x) x 5 32x 5 (b) (3ab) a 4 b 4 81a 4 b 4 () 5( 2r) 3 5 ( 2) 3 (r) 3 5 ( 8) r 3 40r 3 Chek Yourself 5 Simplify eah expression. (a) (3y) 4 (b) (2mn) 6 () 3(4x) 2 (d) 6( 2x) 3 We may have to use more than one property when simplifying an expression involving exponents, as shown in Example 6. Using the Properties of Exponents Simplify eah expression. (a) (r 4 s 3 ) 3 (r 4 ) 3 (s 3 ) 3 r 12 s 9 (b) (3x 2 ) 2 (2x 3 ) 3 () 3 2 (x 2 ) (x 3 ) 3 9x 4 8x 9 72x 13 (a 3 ) 5 a 4 a15 a 4 a 11 Produt to a power property Power to a power property Produt to a power property Power to a power property Multiply the oeffiients and apply the produt property. Power to a power property Quotient property

8 188 CHAPTER 3 Polynomials Chek Yourself 6 Simplify eah expression. (a) (m 5 n 2 ) 3 (b) (2p) 4 ( 4p 2 ) 2 () (s4 ) 3 s 5 We have one final exponent property to develop. Suppose we have a quotient raised to a power. Consider the following: 3 x 3 x # x # x x # x # x x3 3 # 3 # Note that the power, here 3, has been applied to the numerator x and to the denominator 3. This gives us our fifth property of exponents. Property Quotient to a Power Property of Exponents For any real numbers a and b, when b is not equal to 0, and positive integer m, a b m am b m For example, Example 7 Example 7 illustrates the use of this property. Again note that the other properties may also be applied when simplifying an expression. Using the Quotient to a Power Property of Exponents Simplify eah expression. (a) (b) () x3 y 2 4 r2 s 3 t (x3 ) 4 (y 2 ) 4 x12 y 8 (r2 s 3 ) 2 (t 4 ) 2 (r2 ) 2 (s 3 ) 2 (t 4 ) 2 r4 s 6 t 8 Chek Yourself 7 Simplify eah expression Quotient to a power property Quotient to a power property Power to a power property Quotient to a power property Produt to a power property Power to a power property m3 n 4 5 (a) (b) () a 2 b 3 5 2

9 Exponents and Polynomials SECTION The following table summarizes the five properties of exponents that were disussed in this setion: Property General Form Example Produt a m a n a m n x 2 x 3 x 5 Quotient a m a n am n (m n) Power to a power (a m ) n a mn (z 5 ) 4 z 20 Produt to a power (ab) m a m b m (4x) x 3 64x 3 Quotient to a power a b m am 23 b m Our work in this hapter deals with the most ommon kind of algebrai expression, a polynomial. To define a polynomial, we reall our earlier definition of the word term. Definition Term A term an be written as a number or the produt of a number and one or more variables. Definition Polynomial < Objetive 2 > NOTE Example 8 In a polynomial, terms are separated by and signs. This definition indiates that onstants, suh as the number 3, and single variables, suh as x, are terms. For instane, x 5, 3x, 4xy 2, and 8 are all examples of terms. You should reall that the number fator of a term is alled the numerial oeffiient or simply the oeffiient. In the terms above, 1 is the oeffiient of x 5, 3 is the oeffiient of 3x, 4 is the oeffiient of 4xy 2 beause the negative sign is part of the oeffiient, and 8 is the oeffiient of the term 8. We ombine terms to form expressions alled polynomials. Polynomials are one of the most ommon expressions in algebra. A polynomial is an algebrai expression that an be written as a term or as the sum or differene of terms. Any variable fators with exponents must be to whole number powers. Identifying Polynomials State whether eah expression is a polynomial. List the terms of eah polynomial and the oeffiient of eah term. (a) x 3 is a polynomial. The terms are x and 3. The oeffiients are 1 and 3. (b) 3x 2 2x 5, or 3x 2 ( 2x) 5, is also a polynomial. Its terms are 3x 2, 2x, and 5. The oeffiients are 3, 2, and 5. () 5x x is not a polynomial beause of the division by x in the third term.

10 190 CHAPTER 3 Polynomials Chek Yourself 8 Whih expressions are polynomials? 2 (a) 5x 2 (b) 3y 3 2y 5 () 4x 2 x 3 y 3 Definition Monomial, Binomial, and Trinomial Certain polynomials are given speial names beause of the number of terms that they have. A polynomial with one term is alled a monomial. The prefix mono- means 1. A polynomial with two terms is alled a binomial. The prefix bi- means 2. A polynomial with three terms is alled a trinomial. The prefix tri- means 3. Example 9 We do not use speial names for polynomials with more than three terms. Identifying Types of Polynomials (a) 3x 2 y is a monomial. It has one term. (b) 2x 3 5x is a binomial. It has two terms, 2x 3 and 5x. () 5x 2 4x 3 is a trinomial. Its three terms are 5x 2, 4x, and 3. Chek Yourself 9 < Objetive 3 > NOTE Example 10 We will see in the next setion that x 0 1. Classify eah polynomial as a monomial, binomial, or trinomial. (a) 5x 4 2x 3 (b) 4x 7 () 2x 2 5x 3 We also lassify polynomials by their degree. The degree of a polynomial that has only one variable is the highest power appearing in any one term. Classifying Polynomials by Their Degree The highest power (a) 5x 3 3x 2 4x has degree 3. (b) 4x 5x 4 3x 3 2 has degree 4. () 8x has degree 1. Beause 8x 8x 1 (d) 7 has degree 0. The highest power The degree of any nonzero onstant expression is zero. Note: Polynomials an have more than one variable, suh as 4x 2 y 3 5xy 2. The degree is then the highest sum of the powers in any single term (here 2 3, or 5). In general, we will be working with polynomials in a single variable, suh as x. Chek Yourself 10 Find the degree of eah polynomial. (a) 6x 5 3x 3 2 (b) 5x () 3x 3 2x 6 1 (d) 9 Working with polynomials is muh easier if you get used to writing them in desending order (sometimes alled desending-exponent form). This simply means that the term with the highest exponent is written first, then the term with the next highest exponent, and so on.

11 Exponents and Polynomials SECTION Example 11 Writing Polynomials in Desending Order < Objetive 4 > The exponents get smaller from left to right. (a) 5x 7 3x 4 2x 2 is in desending order. (b) 4x 4 5x 6 3x 5 is not in desending order. The polynomial should be written as 5x 6 3x 5 4x 4 The degree of the polynomial is the power of the first, or leading, term one the polynomial is arranged in desending order. Chek Yourself 11 Write eah polynomial in desending order. (a) 5x 4 4x 5 7 (b) 4x 3 9x 4 6x 8 A polynomial an represent any number. Its value depends on the value given to the variable. < Objetive 5 > RECALL Example 12 We use the rules for order of operations to evaluate eah polynomial. > CAUTION Be partiularly areful when dealing with powers of negative numbers! Evaluating Polynomials Given the polynomial 3x 3 2x 2 4x 1 (a) Find the value of the polynomial when x 2. To evaluate the polynomial, substitute 2 for x. 3(2) 3 2(2) 2 4(2) 1 3(8) 2(4) 4(2) (b) Find the value of the polynomial when x 2. Now we substitute 2 for x. 3( 2) 3 2( 2) 2 4( 2) 1 3( 8) 2(4) 4( 2) Chek Yourself 12 Find the value of the polynomial 4x 3 3x 2 2x 1 when (a) x 3 (b) x 3 Polynomials are used in almost every professional field. Many appliations are related to preditions and foreasts. In allied health, polynomials an be used to alulate the onentration of a mediation in the bloodstream after a given amount of time, as the next example demonstrates.

12 192 CHAPTER 3 Polynomials Example 13 An Allied Health Appliation The onentration of digoxin, a mediation presribed for ongestive heart failure, in a patient s bloodstream t hours after injetion is given by the polynomial t t where onentration is measured in nanograms per milliliter (ng/ml). Determine the onentration of digoxin in a patient s bloodstream 19 hours after injetion. We are asked to evaluate the polynomial t t for the variable value t = 19. We substitute 19 for t in the polynomial (19) (19) (361) The onentration is nanograms per milliliter. Chek Yourself 13 The onentration of a sedative, in mirograms per milliliter (mg/ml), in a patient s bloodstream t hours after injetion is given by the polynomial 1.35t t Determine the onentration of the sedative in a patient s bloodstream 3.5 hours after injetion. Round to the nearest tenth. Chek Yourself ANSWERS 1. (a) x 10 ; (b) ( 3) 7 ; () (x 2 y) 8 ; (d) y 7 2. (a) 14x 7 ; (b) 2x 5 y 3 ; () 15x 5 y 5 ; (d) 3. (a) y 7 ; (b) x 8 ; () 5r; (d) 8m 5 n 4 4. (a) m 30 ; (b) m 11 ; () 3 8 ; (d) (a) 81y 4 ; (b) 64m 6 n 6 ; () 48x 2 ; (d) 48x 3 6. (a) m 15 n 6 ; (b) 256p 8 ; () s 7 16 m 15 a 4 b 6 7. (a) ; (b) ; () 8. (a) polynomial; (b) not a polynomial; 81 n () polynomial 9. (a) binomial; (b) monomial; () trinomial 10. (a) 5; (b) 1; () 6; (d) (a) 4x 5 5x 4 7; (b) 6x 8 9x 4 4x (a) 86; (b) mg/ml Reading Your Text The following fill-in-the-blank exerises are designed to ensure that you understand some of the key voabulary used in this setion. SECTION 3.1 (a) Exponential notation indiates repeated. (b) A an be written as a number or produt of a number and one or more variables. () In eah term of a polynomial, the number fator is alled the numerial. (d) The of a polynomial in one variable is the highest power of the variable that appears in a term. 9 x b

13 Basi Skills Challenge Yourself Calulator/Computer Career Appliations Above and Beyond < Objetive 1 > Simplify eah expression. 3.1 exerises Boost your GRADE at ALEKS.om! 1. (x 2 ) 3 2. (a 5 ) 3 3. (m 4 ) 4 4. ( p 7 ) 2 Pratie Problems Self-Tests NetTutor e-professors Videos 5. (2 4 ) 2 6. (3 3 ) 2 7. (5 3 ) 5 8. (7 2 ) 4 9. (3x) (4m) (2xy) (5pq) 3 Name Setion Date Answers x (2x 2 ) (3y 2 ) (a 8 b 6 ) (p 3 q 4 ) (4x 2 y) (4m 4 n 4 ) (3m 2 ) 4 ( 2m 3 ) ( 2y 4 ) 3 (4y 3 ) 2 (x 4 ) x 2 (s 3 ) 2 (s 2 ) (s 5 ) 2 m3 n x5 y a3 b a 2 5 (m 5 ) 3 m 6 (y 5 ) 3 (y 3 ) 2 (y 4 ) 4 a4 b 3 4 z 4 3 > Videos > Videos SECTION

14 3.1 exerises Answers < Objetive 2 > Whih expressions are polynomials? x x 3 3 x x 3 x 3 x a 2 2a 7 x 2 For eah polynomial, list the terms and their oeffiients x 2 3x 40. 5x 3 x 41. 4x 3 3x 2 > Videos 42. 7x Classify eah expression as a monomial, binomial, or trinomial, where possible x 3 3x x y 2 4y x 2 1 xy y x 4 3x 2 5x x 4 5 x y x 4 2x 2 x x 5 3 x x 2 9 < Objetives 3 4 > Arrange in desending order if neessary, and give the degree of eah polynomial x 5 3x x 2 3x x 7 5x 9 4x x 57. 4x 58. x 17 3x x 2 3x 5 x 6 7 > Videos SECTION 3.1

15 3.1 exerises < Objetive 5 > Evaluate eah polynomial for the given values of the variable x 1, x 1 and x x 5, x 2 and x x 3 2x, x 2 and x x 2 7, x 3 and x 3 > Videos Answers x 2 4x 2, x 4 and x x 2 5x 1, x 2 and x x 2 2x 3, x 1 and x x 2 5x 6, x 3 and x Basi Skills Challenge Yourself Calulator/Computer Career Appliations Above and Beyond 66. Complete eah statement with never, sometimes, or always. 69. A polynomial is a trinomial A trinomial is a polynomial The produt of two monomials is a monomial. 72. A term is a binomial. Determine whether eah statement is always true, sometimes true, or never true. 73. A monomial is a polynomial. 74. A binomial is a trinomial. 75. The degree of a trinomial is A trinomial has three terms. 77. A polynomial has four or more terms. 78. A binomial must have two oeffiients. Basi Skills Challenge Yourself Calulator/Computer Career Appliations Above and Beyond Solve eah problem. 79. Write x 12 as a power of x Write y 15 as a power of y Write a 16 as a power of a Write m 20 as a power of m SECTION

16 3.1 exerises 83. Write eah expression as a power of 8. (Remember that ) Answers , 2 18, (2 5 ) 3, (2 7 ) Write eah expression as a power of , 3 14, (3 5 ) 8, (3 4 ) What expression raised to the third power is 8x 6 y 9 z 15? 86. What expression raised to the fourth power is 81x 12 y 8 z 16? The formula (1 R) y G gives us useful information about the growth of a population. Here R is the rate of growth expressed as a deimal, y is the time in years, and G is the growth fator. If a ountry has a 2% growth rate for 35 years, then its population will double: (1.02) SOCIAL SCIENCE (a) With a 2% growth rate, how many doublings will our in 105 years? How muh larger will the ountry s population be to the nearest whole number? (b) The less-developed ountries of the world had an average growth rate of 2% in If their total population was 3.8 billion, what will their population be in 105 years if this rate remains unhanged? 88. SOCIAL SCIENCE The United States has a growth rate of 0.7%. What will be its growth fator after 35 years (to the nearest perent)? 89. Write an explanation of why (x 3 )(x 4 ) is not x Your algebra study partners are onfused. Why isn t x 2 x 3 2x 5? they ask you. Write an explanation that will onvine them. Capital itali letters suh as P and Q are often used to name polynomials. For example, we might write P(x) 3x 3 5x 2 2 in whih P(x) is read P of x. The notation permits a onvenient shorthand. We write P(2), read P of 2, to indiate the value of the polynomial when x 2. Here P(2) 3(2) 3 5(2) Use the preeding information to omplete exerises If P(x) x 3 2x 2 5 and Q(x) 2x 2 3, find: 91. P(1) 92. P( 1) 93. Q(2) 94. Q( 2) 95. P(3) 96. Q( 3) 196 SECTION 3.1

17 3.1 exerises 97. P(0) 98. Q(0) 99. P(2) Q( 1) 100. P( 2) Q(3) 101. P(3) Q( 3) Q(0) 102. Q( 2) Q(2) P(0) Answers Q(4) P(4) 104. P( 1) Q(0) P(0) BUSINESS AND FINANCE The ost, in dollars, of typing a term paper is given as 3 times the number of pages plus 20. Use y as the number of pages to be typed and write a polynomial to desribe this ost. Find the ost of typing a 50-page paper BUSINESS AND FINANCE The ost, in dollars, of making suits is desribed as 20 times the number of suits plus 150. Use s as the number of suits and write a polynomial to desribe this ost. Find the ost of making seven suits Answers 1. x 6 3. m x x 4 y 4 9 x x a 16 b x 6 y m x s 2 m 9 a 6 b Polynomial 35. Polynomial 37. Not a polynomial 39. 2x 2, 3x; 2, x 3, 3x, 2; 4, 3, Binomial 45. Trinomial 47. Not lassified 49. Monomial 51. Not a polynomial 53. 4x 5 3x 2 ; x 9 7x 7 4x 3 ; x; x 6 3x 5 5x 2 7; , , , , sometimes 71. always 73. Always 75. Sometimes 77. Sometimes 79. (x 2 ) (a 2 ) , 8 6, 8 5, x 2 y 3 z (a) Three doublings, 8 times as large; (b) 30.4 billion 89. Above and Beyond y 20, $170 n SECTION

18 3.2 Negative Exponents and Sientifi Notation < 3.2 Objetives > 1 > Evaluate expressions involving a zero or negative exponent 2 > Simplify expressions involving a zero or negative exponent 3 > Write a number in sientifi notation 4 > Solve appliations involving sientifi notation RECALL By the quotient property, a m n am n a when m n. Here m and n are both 5, so m n. In Setion 3.1, we disussed exponents. We now want to extend our exponent notation to inlude 0 and negative integers as exponents. First, what do we do with x 0? It will help to look at a problem that gives us x 0 as a result. What if the numerator and denominator of a fration have the same base raised to the same power and we extend our division rule? For example, a 5 a 5 a5 5 a 0 But from our experiene with frations we know that Definition Zero Power < Objetive 1 > Example 1 > CAUTION In part (d) the 0 exponent applies only to the x and not to the fator 6, beause the base is x. a 5 a 5 1 By omparing these equations, it seems reasonable to make the following definition: For any nonzero number a, a 0 1 In words, any expression, exept 0, raised to the 0 power is 1. Example 1 illustrates the use of this definition. Raising Expressions to the Zero Power Evaluate eah expression. Assume all variables are nonzero. (a) (b) ( 27) 0 1 The exponent is applied to 27. () (x 2 y) 0 1 (d) 6x (e) The exponent is applied to 27, but not to the silent 1. Chek Yourself 1 Evaluate eah expression. Assume all variables are nonzero. (a) 7 0 (b) ( 8) 0 () (xy 3 ) 0 (d) 3x 0 (e)

19 Negative Exponents and Sientifi Notation SECTION Example 2 Before we introdue the next property, we look at some examples that use the properties of Setion 3.1. Evaluating Expressions Evaluate eah expression. (a) From our earlier work, we get (b) # # 5 # 5 # 5 # 5 # () # 10 # # 10 # 10 # 10 # 10 # 10 # 10 # 10 # or 1 1,000,000 NOTES John Wallis ( ), an English mathematiian, was the first to fully disuss the meaning of 0 and negative exponents. Divide the numerator and denominator by the two ommon x fators. Definition Negative Powers Chek Yourself 2 Evaluate eah expression (a) (b) () (d) x The quotient property of exponents allows us to define a negative exponent. Suppose that the exponent in the denominator is greater than the exponent in the numerator. Consider the expression. x 2 x 5 Our previous work with frations tells us that x # x x # x # x # x # x 1 x However, if we extend the quotient property to let n be greater than m, we have x 2 x 5 x2 5 x 3 x 2 x 5 1 Now, by omparing these equations, it seems reasonable to define x 3 as. In general, we have the following results. For any nonzero number a, a 1 1 a For any nonzero number a, and any integer n, a n 1 a n This definition tells us that if we have a base a raised to a negative integer power, suh as a 5 1, we may rewrite this as 1 over the base a raised to a positive integer power:. We work with this in Example 3. x 5 x 3 a 5

20 200 CHAPTER 3 Polynomials < Objetive 2 > Example 3 Rewriting Expressions That Contain Negative Exponents Rewrite eah expression using only positive exponents. Simplify when possible. Negative exponent in numerator (a) x 4 1 x 4 (b) m 7 1 m 7 Positive exponent in denominator () or 1 9 > CAUTION 2x 3 is not the same as (2x) 3. (d) (e) x # 1 x 3 2 x 3 1 # # ,000 A negative power in the denominator is equivalent to a positive power in the numerator. So, 1 3 x3 x RECALL Example 4 a m a n a m n for any integers m and n. So add the exponents. (f) The 3 exponent applies only to x, beause x is the base (g) 4 # 1 4x 5 x 5 4 x 5 Chek Yourself 3 Write eah expression using only positive exponents. (a) a 10 (b) 4 3 () 3x 2 (d) We an now use negative integers as exponents in our produt property for exponents. Consider Example 4. Simplifying Expressions Containing Exponents Rewrite eah expression using only positive exponents. (a) x 5 x 2 x 5 ( 2) x 3 Note: An alternative approah would be x 5 x 2 x 5 # 1 x5 2 2 x3 x x

21 Negative Exponents and Sientifi Notation SECTION (b) a 7 a 5 a 7 ( 5) a 2 () y 5 y 9 y 5 ( 9) y 4 1 y 4 Chek Yourself 4 Rewrite eah expression using only positive exponents. (a) x 7 x 2 (b) b 3 b 8 Example 5 Example 5 shows that all the properties of exponents introdued in the last setion an be extended to expressions with negative exponents. Simplifying Expressions Containing Exponents Simplify eah expression. (a) m 3 m 4 m 3 4 Quotient property m 7 1 m 7 a 2 b 6 (b) a 2 5 b 6 ( 4) Apply the quotient property to eah variable. a 5 b 4 NOTE We an also omplete () by using the power to a power property first, so (2x 4 ) (x 4 ) x x x 12 > Calulator () (d) (2x 4 ) 3 (y 2 ) 4 (y 3 ) 2 a 7 b 10 b10 a 7 1 (2x 4 ) (x 4 ) 3 1 8x 12 y 8 y 6 y 8 ( 6) y 2 1 y 2 Chek Yourself 5 Definition of a negative exponent Produt to a power property Power to a power property Power to a power property Quotient property Simplify eah expression. x 5 m 3 n 5 (a) (b) () (3a 3 ) 4 (d) (r 3 ) 2 x 3 m 2 n 3 (r 4 ) 2 Sientifi notation is one important use of exponents. We begin the disussion with a alulator exerise. On most alulators, if you multiply 2.3 times 1,000, the display reads 2300 Multiply by 1,000 a seond time and you see

22 202 CHAPTER 3 Polynomials NOTE 2.3 E09 must equal 2,300,000,000. NOTE Consider the following table: , , Definition Sientifi Notation < Objetive 3 > NOTE Example 6 The exponent on 10 shows the number of plaes we must move the deimal point. A positive exponent tells us to move right, and a negative exponent indiates a move to the left. On most alulators, multiplying by 1,000 a third time results in the display or 2.3 E09 Multiplying by 1,000 again yields or 2.3 E12 Can you see what is happening? This is the way alulators display very large numbers. The number on the left is always between 1 and 10, and the number on the right indiates the number of plaes the deimal point must be moved to the right to put the answer in standard (or deimal) form. This notation is used frequently in siene. It is not unommon in sientifi appliations of algebra to find yourself working with very large or very small numbers. Even in the time of Arhimedes ( B.C.E.), the study of suh numbers was not unusual. Arhimedes estimated that the universe was 23,000,000,000,000,000 m in diameter, whih is the approximate distane light travels in 2 1 years. By omparison, 2 Polaris (the North Star) is atually 680 light-years from Earth. Example 7 looks at the idea of light-years. In sientifi notation, Arhimedes estimate for the diameter of the universe would be m If a number is divided by 1,000 again and again, we get a negative exponent on the alulator. In sientifi notation, we use positive exponents to write very large numbers, suh as the distane of stars. We use negative exponents to write very small numbers, suh as the width of an atom. Any number written in the form a 10 n in whih 1 a 10 and n is an integer, is written in sientifi notation. Sientifi notation is one of the few plaes that we still use the multipliation symbol. Using Sientifi Notation Write eah number in sientifi notation. (a) 120, plaes The power is 5. (b) 88,000, plaes () 520,000, plaes (d) 4000,000, plaes The power is 7.

23 Negative Exponents and Sientifi Notation SECTION NOTE To onvert bak to standard or deimal form, the proess is simply reversed. (e) plaes (f) plaes If the deimal point is to be moved to the left, the exponent is negative. Chek Yourself 6 Write in sientifi notation. (a) 212,000,000,000,000,000 (b) () 5,600,000 (d) < Objetive 4 > NOTE Example NOTE We divide the distane (in meters) by the number of meters in 1 light-year. An Appliation of Sientifi Notation (a) Light travels at a speed of meters per seond (m/s). There are approximately s in a year. How far does light travel in a year? We multiply the distane traveled in 1 s by the number of seonds in a year. This yields ( )( ) ( )( ) For our purposes we round the distane light travels in 1 year to m. This unit is alled a light-year, and it is used to measure astronomial distanes. (b) The distane from Earth to the star Spia (in Virgo) is m. How many light-years is Spia from Earth? light-years Chek Yourself 7 Earth Multiply the oeffiients, and add the exponents m Spia The farthest objet that an be seen with the unaided eye is the Andromeda galaxy. This galaxy is m from Earth. What is this distane in light-years?

24 204 CHAPTER 3 Polynomials Chek Yourself ANSWERS (a) 1; (b) 1; () 1; (d) 3; (e) 1 2. (a) 125; (b) () ; (d) 125 ; 10,000 x (a) ; (b) ; () ; (d) 4. (a) x 5 ; (b) 4 3 or 1 a x 2 9 b 5 m (a) x 8 ; (b) ; () ; (d) r 2 6. (a) ; (b) ; n 8 81a 12 () ; (d) ,300,000 light-years Reading Your Text The following fill-in-the-blank exerises are designed to ensure that you understand some of the key voabulary used in this setion. SECTION 3.2 (a) A nonzero number raised to the zero power is always equal to. b (b) A negative exponent in the denominator is equivalent to a exponent in the numerator. () All of the properties of negative exponents. an be extended to terms with (d) The base a in a number written in sientifi notation annot be greater than or equal to.

25 Basi Skills Challenge Yourself Calulator/Computer Career Appliations Above and Beyond < Objetive 1 > Evaluate (assume any variables are nonzero). 3.2 exerises Boost your GRADE at ALEKS.om! ( 7) 0 3. ( 29) Pratie Problems Self-Tests NetTutor Name e-professors Videos 5. (x 3 y 2 ) m 0 Setion Date 7. 11x 0 > 8. (2a 3 b 7 ) 0 Videos 9. ( 3p 6 q 8 ) x 0 Answers < Objetive 2 > Write eah expression using positive exponents; simplify when possible. 11. b p x a (5x) (3a) x x ( 2x) (3x) 4 > Videos SECTION

26 3.2 exerises Simplify eah expression and write your answers with only positive exponents. Answers a 5 a m 5 m x 8 x a 12 a x 0 x r 3 r a 8 a x 7 x > Videos 36. m 9 m 4 a 3 a Basi Skills Challenge Yourself Calulator/Computer Career Appliations Above and Beyond 38. Determine whether eah statement is true or false Zero raised to any power is one. 38. One raised to any power is one. 39. When multiplying two terms with the same base, add the exponents to find the power of that base in the produt. 40. When multiplying two terms with the same base, multiply the exponents to find the power of that base in the produt. Simplify eah expression. Write your answers with positive exponents only. x 4 yz x 5 yz m 5 n m 4 n ( 2a 3 ) (3x 2 ) (x 2 y 3 ) ( a 5 b 3 ) 3 (r 2 ) r 4 p 6 q 3 p 3 q 6 p 3 q 2 p 4 q 3 (y 3 ) 4 y 6 > Videos 206 SECTION 3.2

27 3.2 exerises 51. m 2 n 3 m 2 n r 3 s 3 s 4 t a 5 (b 2 ) 3 1 a(b 4 ) (p 0 q 2 ) 3 p(q 0 ) 2 (p 1 q) d 3 4 d 5 x 3 yz 2 x 2 y 3 z 4 x 4 y 3 z (xy 2 ) 2 z 1 x 1 (x 2 y 2 ) 3 z 2 xy 3 z 0 Answers (2x 2 ) ab 2 (a 3 b 0 ) a 6 (3a 4 ) b 1 (2b 3 ) 2 m 1 (m 2 n 3 ) 2 4x 2 y 1 (2x 2 y 3 ) [( 2 d 0 ) 2 ] [x 2 y(x 4 y 3 ) 1 ] w(w 2 ) (w 2 ) 2 a 5 (a 2 ) a(a 4 ) 3 < Objetive 3 > (2n 2 ) 3 (2n 2 ) 4 y 2 (y 2 ) 2 (y 3 ) 2 (y 0 ) 2 In exerises 71 74, express eah number in sientifi notation. 71. SCIENCE AND MEDICINE The distane from Earth to the Sun: 93,000,000 mi. > Videos SECTION

28 3.2 exerises 72. SCIENCE AND MEDICINE The diameter of a grain of sand: m. Answers SCIENCE AND MEDICINE The diameter of the Sun: 130,000,000,000 m. 74. SCIENCE AND MEDICINE The number of moleules in 22.4 L of a gas: 602,000,000,000,000,000,000,000 (Avogadro s number). 75. SCIENCE AND MEDICINE The mass of the Sun is approximately kg. If this were written in standard or deimal form, how many 0 s would follow the seond 9 s digit? 76. SCIENCE AND MEDICINE Arhimedes estimated the universe to be millimeters (mm) in diameter. If this number were written in standard or deimal form, how many 0 s would follow the digit 3? Write eah expression in standard notation Write eah number in sientifi notation Evaluate the expressions using sientifi notation, and write your answers in that form. 85. ( )( ) 86. ( )( ) Evaluate eah expression. Write your results in sientifi notation. 89. ( )( ) 90. ( )( ) ( )( ) ( )( ) In 1975 the population of Earth was approximately 4 billion and doubling every 35 years. The formula for the population P in year y for this doubling rate is P (in billions) 4 2 ( y 1975) 35 > Videos ( )( ) ( )( ) 95. SOCIAL SCIENCE What was the approximate population of Earth in 1960? 96. SOCIAL SCIENCE What will Earth s population be in 2025? 208 SECTION 3.2

29 3.2 exerises The U.S. population in 1990 was approximately 250 million, and the average growth rate for the past 30 years gives a doubling time of 66 years. The formula just given for the United States then beomes P (in millions) ( y 1990) SOCIAL SCIENCE What was the approximate population of the United States in 1960? 98. SOCIAL SCIENCE What will the population of the United States be in 2025 if this growth rate ontinues? Answers < Objetive 4 > 99. SCIENCE AND MEDICINE Megrez, the nearest of the Big Dipper stars, is m from Earth. Approximately how long does it take light, m traveling at 10 16, to travel from Megrez to Earth? year 100. SCIENCE AND MEDICINE Alkaid, the most distant star in the Big Dipper, is m from Earth. Approximately how long does it take light to travel from Alkaid to Earth? 101. SOCIAL SCIENCE The number of liters of water on Earth is 15,500 followed by 19 zeros. Write this number in sientifi notation. Then use the number of liters of water on Earth to find out how muh water is available for eah person on Earth. The population of Earth is 6 billion SOCIAL SCIENCE If there are people on Earth and there is enough freshwater to provide eah person with L, how muh freshwater is on Earth? 103. SOCIAL SCIENCE The United States uses an average of L of water per person eah year. The United States has people. How many liters of water does the United States use eah year? Answers b , x 5x x a x x a x 5 x 2 m 9 16 x False 39. True 41. x n 8 a 12 y r 3 t 2 b 6 q 6 3x r 2 m 4 n s 7 a 6 p 8 a b 2 a 2 w mi m billion million years L; L L SECTION

30 3.3 Adding and Subtrating Polynomials < 3.3 Objetives > 1 > Add polynomials 2 > Distribute a negative sign over a polynomial 3 > Subtrat polynomials Addition is always a matter of ombining like quantities (two apples plus three apples, four books plus five books, and so on). If you keep that basi idea in mind, adding polynomials is easy. It is just a matter of ombining like terms. Suppose that you want to add 5x 2 3x 4 and 4x 2 5x 6 RECALL The plus sign between the parentheses indiates addition. Parentheses are sometimes used when adding, so for the sum of these polynomials, we an write (5x 2 3x 4) (4x 2 5x 6) Now what about the parentheses? You an use the following rule. Property Removing Signs of Grouping Case 1 NOTES Remove the parentheses. No other hanges are neessary. We use the assoiative and ommutative properties in reordering and regrouping. We use the distributive property. For example, 5x 2 4x 2 (5 4)x 2 9x 2 When finding the sum of two polynomials, if a plus sign ( ) or nothing at all appears in front of parentheses, simply remove the parentheses. No other hanges are neessary. Now let s return to the addition. (5x 2 3x 4) (4x 2 5x 6) 5x 2 3x 4 4x 2 5x 6 Like terms Like terms Like terms Collet like terms. (Remember: Like terms have the same variables raised to the same power). (5x 2 4x 2 ) (3x 5x) (4 6) Combine like terms for the result: 9x 2 8x 2 As should be lear, muh of this work an be done mentally. You an then write the sum diretly by loating like terms and ombining. Example 1 illustrates this property. 210

31 Adding and Subtrating Polynomials SECTION < Objetive 1 > NOTE Example 1 We all this the horizontal method beause the entire problem is written on one line is the horizontal method is the vertial method. Combining Like Terms Add 3x 5 and 2x 3. Write the sum. (3x 5) (2x 3) 3x 5 2x 3 5x 2 Like terms Like terms Chek Yourself 1 Add 6x 2 2x and 4x 2 7x. The same tehnique is used to find the sum of two trinomials. Example 2 Adding Polynomials Using the Horizontal Method Add 4a 2 7a 5 and 3a 2 3a 4. RECALL Only the like terms are ombined in the sum. Example 3 Write the sum. (4a 2 7a 5) (3a 2 3a 4) 4a 2 7a 5 3a 2 3a 4 7a 2 4a 1 Like terms Like terms Like terms Chek Yourself 2 Add 5y 2 3y 7 and 3y 2 5y 7. Adding Polynomials Using the Horizontal Method Add 2x 2 7x and 4x 6. Write the sum. (2x 2 7x) (4x 6) 2x 2 7x 4x 6 These are the only like terms; 2x 2 and 6 annot be ombined. 2x 2 11x 6

32 212 CHAPTER 3 Polynomials Chek Yourself 3 Add 5m 2 8 and 8m 2 3m. Writing polynomials in desending order usually makes the work easier. Example 4 Adding Polynomials Using the Horizontal Method Add 3x 2x 2 7 and 5 4x 2 3x. Write the polynomials in desending order and then add. ( 2x 2 3x 7) (4x 2 3x 5) 2x 2 12 Chek Yourself 4 Add 8 5x 2 4x and 7x 8 8x 2. Subtrating polynomials requires another rule for removing signs of grouping. Property Removing Signs of Grouping Case 2 < Objetive 2 > NOTE Example 5 We are using the distributive property in part (a), beause (2x 3y) ( 1)(2x 3y) ( 1)(2x) ( 1)(3y) 2x 3y When finding the differene of two polynomials, if a minus sign ( ) appears in front of a set of parentheses, the parentheses an be removed by hanging the sign of eah term inside the parentheses. We illustrate this rule in Example 5. Removing Parentheses Remove the parentheses in eah expression. (a) (2x 3y) 2x 3y (b) m (5n 3p) m 5n 3p Sign hanges () 2x ( 3y z) 2x 3y z Sign hanges Chek Yourself 5 Change eah sign to remove the parentheses. In eah expression, remove the parentheses. (a) (3m 5n) (b) (5w 7z) () 3r (2s 5t) (d) 5a ( 3b 2) Subtrating polynomials is now a matter of using the previous rule to remove the parentheses and then ombining the like terms. Consider Example 6.

33 Adding and Subtrating Polynomials SECTION < Objetive 3 > Example 6 Subtrating Polynomials Using the Horizontal Method (a) Subtrat 5x 3 from 8x 2. RECALL The expression following from is written first in the problem. Write (8x 2) (5x 3) 8x 2 5x 3 Sign hanges 3x 5 Reall that subtrating 5x is the same as adding 5x. (b) Subtrat 4x 2 8x 3 from 8x 2 5x 3. Write (8x 2 5x 3) (4x 2 8x 3) 8x 2 5x 3 4x 2 8x 3 Sign hanges 4x 2 13x 6 Chek Yourself 6 (a) Subtrat 7x 3 from 10x 7. (b) Subtrat 5x 2 3x 2 from 8x 2 3x 6. Example 7 Again, writing all polynomials in desending order makes loating and ombining like terms muh easier. Look at Example 7. Subtrating Polynomials Using the Horizontal Method (a) Subtrat 4x 2 3x 3 5x from 8x 3 7x 2x 2. Write (8x 3 2x 2 7x) ( 3x 3 4x 2 5x) =8x 3 2x 2 7x 3x 3 4x 2 5x 11x 3 2x 2 12x Sign hanges (b) Subtrat 8x 5 from 5x 3x 2. Write (3x 2 5x) (8x 5) 3x 2 5x 8x 5 Only the like terms an be ombined. 3x 2 13x 5 Chek Yourself 7 (a) Subtrat 7x 3x 2 5 from 5 3x 4x 2. (b) Subtrat 3a 2 from 5a 4a 2.

34 214 CHAPTER 3 Polynomials If you think bak to addition and subtration in arithmeti, you should remember that the work was arranged vertially. That is, the numbers being added or subtrated were plaed under one another so that eah olumn represented the same plae value. This meant that in adding or subtrating olumns you were always dealing with like quantities. It is also possible to use a vertial method for adding or subtrating polynomials. First rewrite the polynomials in desending order, and then arrange them one under another, so that eah olumn ontains like terms. Then add or subtrat in eah olumn. Example 8 Adding Using the Vertial Method Add 2x 2 5x, 3x 2 2, and 6x 3. Like terms are plaed in olumns. 2x 2 5x 3x 2 2 6x 3 5x 2 x 1 Chek Yourself 8 Add 3x 2 5, x 2 4x, and 6x 7. Example 9 Example 9 illustrates subtration by the vertial method. Subtrating Using the Vertial Method (a) Subtrat 5x 3 from 8x 7. Write 8x 7 ( ) (5x 3) 3x 4 8x 7 5x 3 3x 4 (b) Subtrat 5x 2 3x 4 from 8x 2 5x 3. Write 8x 2 5x 3 ( ) (5x 2 3x 4) 3x 2 8x 7 8x 2 5x 3 5x 2 3x 4 To subtrat, hange eah sign of 5x 3 to get 5x 3 and then add. To subtrat, hange eah sign of 5x 2 3x 4 to get 5x 2 3x 4 and then add. 3x 2 8x 7 Subtrating using the vertial method takes some pratie. Take time to study the method arefully. You will use it in long division in Setion 3.5.

35 Adding and Subtrating Polynomials SECTION Chek Yourself 9 Subtrat, using the vertial method. (a) 4x 2 3x from 8x 2 2x (b) 8x 2 4x 3 from 9x 2 5x 7 Chek Yourself ANSWERS 1. 10x 2 5x 2. 8y 2 8y 3. 13m 2 3m x 2 11x 5. (a) 3m 5n; (b) 5w 7z; () 3r 2s 5t; (d) 5a 3b 2 6. (a) 3x 10; (b) 3x (a) 7x 2 10x; (b) 4a 2 2a x 2 2x (a) 4x 2 5x; (b) x 2 9x 10 Reading Your Text The following fill-in-the-blank exerises are designed to ensure that you understand some of the key voabulary used in this setion. SECTION 3.3 (a) If a sign appears in front of parentheses, simply remove the parentheses. b (b) If a minus sign appears in front of parentheses, the subtration an be hanged to addition by hanging the in front of eah term inside the parentheses. () When subtrating polynomials, the expression following the word from is written when writing the problem. (d) When adding or subtrating polynomials, we an only ombine terms.

36 3.3 exerises Boost your GRADE at ALEKS.om! Basi Skills Challenge Yourself Calulator/Computer Career Appliations Above and Beyond < Objetive 1 > Add. Pratie Problems Self-Tests NetTutor Name e-professors Videos 1. 6a 5 and 3a x 3 and 3x b 2 11b and 5b 2 7b 4. 2m 2 3m and 6m 2 8m Setion Date 5. 3x 2 2x and 5x 2 2x 6. 3p 2 5p and 7p 2 5p Answers 7. 2x 2 5x 3 and > Videos 8. 4d 2 8d 7 and 3x 2 7x 4 5d 2 6d b 2 8 and 5b x 3 and 3x 2 9x y 3 5y 2 and 5y 2 2y 12. 9x 4 2x 2 and 2x a 2 4a 3 and 3a 3 2a m 3 2m and 6m 4m x 2 2 7x and 16. 5b 3 8b 2b 2 and 5 8x 6x 2 3b 2 7b 3 5b < Objetive 2 > Remove the parentheses in eah expression and simplify when possible. 17. (2a 3b) 18. (7x 4y) 19. 5a (2b 3) 20. 7x (4y 3z) 21. 9r (3r 5s) m (3m 2n) > Videos 23. 5p ( 3p 2q) 24. 8d ( 7 2d) 216 SECTION 3.3

37 3.3 exerises < Objetive 3 > Subtrat. 25. x 4 from 2x x 2 from 3x m 2 2m from 4m 2 5m 28. 9a 2 5a from 11a 2 10a 29. 6y 2 5y from 4y 2 5y 30. 9n 2 4n from 7n 2 4n 31. x 2 4x 3 from 3x 2 5x x 2 2x 4 from 5x 2 8x a 7 from 8a 2 9a 34. 3x 3 x 2 from 4x 3 5x 35. 4b 2 3b from 5b 2b y 3y 2 from 3y 2 2y Answers x 2 5 8x from 38. 4x 2x 2 4x 3 from 3x 2 8x 7 4x 3 x 3x 2 > Videos 34. Perform the indiated operations. 39. Subtrat 3b 2 from the sum of 4b 2 and 5b Subtrat 5m 7 from the sum of 2m 8 and 9m Subtrat 3x 2 2x 1 from the sum of x 2 5x 2 and 2x 2 7x Subtrat 4x 2 5x 3 from the sum of x 2 3x 7 and 2x 2 2x Subtrat 2x 2 3x from the sum of 4x 2 5 and 2x Subtrat 5a 2 3a from the sum of 3a 3 and 5a Subtrat the sum of 3y 2 3y and 5y 2 3y from 2y 2 8y. 46. Subtrat the sum of 7r 3 4r 2 and 3r 3 + 4r 2 from 2r 3 +3r 2. Add using the vertial method w 2 + 7, 3w 5, and 4w 2 5w 48. 3x 2 4x 2, 6x 3, and 2x x 2 3x 4, 4x 2 3x 3, and 2x 2 x x 2 2x 4, x 2 2x 3, and 2x 2 4x 3 > Videos SECTION