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Exponents and Polynomials For use by Palm Beach State College only. CHAPTER 5 Exponents and Polynomials 5.1 Exponents 5.2 Adding and Subtracting Polynomials 5.3 Multiplying Polynomials 5.4 Special Products Integrated Review—Exponents and Operations on Polynomials 5.5 Negative Exponents and Scientific Notation 5.6 Dividing Polynomials CHECK YOUR PROGRESS Vocabulary Check Chapter Highlights Chapter Review Getting Ready for the Test Can You Imagine a World Without the Internet? Chapter Test In 1995, less than 1% of the world population was connected to the Internet. By 2015, Cumulative Review that number had increased to 40%. Technology changes so fast that, if this trend continues, by the time you read this, far more than 40% of the world population will be connected to the Internet. The circle graph below shows Internet users by region of Recall from Chapter 1 that an exponent is a the world in 2015. In Section 5.2, Exercises 103 and 104, we explore more about the shorthand notation for repeated factors. This growth of Internet users. chapter explores additional concepts about exponents and exponential expressions. An especially useful type of exponential expression is a polynomial. Polynomials model many real- world phenomena. In this chapter, we focus on polynomials and operations on polynomials. Worldwide Internet Users 3500 Worldwide Internet Users 3000 (by region of the world) (in milions) 2500 Oceania Europe 2000 Asia 1500 Africa 1000 500 Number of Internet Users Americas 0 1995 2000 2005 2010 2015 Year Data from International Telecommunication Union and United Nations Population Division 310 Copyright Pearson. All rights reserved. M05_MART8978_07_AIE_C05.indd 310 11/9/15 5:45 PM For use by Palm Beach State College only. Section 5.1 Exponents 311 5.1 Exponents OBJECTIVE OBJECTIVES 1 Evaluating Exponential Expressions 1 Evaluate Exponential As we reviewed in Section 1.4, an exponent is a shorthand notation for repeated factors. 5 5 Expressions. For example, 2 # 2 # 2 # 2 # 2 can be written as 2 . The expression 2 is called an exponen- tial expression. It is also called the fifth power of 2, or we say that 2 is raised to the fifth 2 Use the Product Rule for power. Exponents. 56 = 5 # 5 # 5 # 5 # 5 # 5 and -3 4 = -3 # -3 # -3 # -3 3 Use the Power Rule for (+1+)++1* (++++++)+1++++* Exponents. 6 factors; each factor is 5 1 2 41 factors;2 1 each2 1factor2 is1 -32 4 Use the Power Rules for The base of an exponential expression is the repeated factor. The exponent is the ­Products and Quotients. number of times that the base is used as a factor. Í 5 Use the Quotient Rule for Í 6 exponent 4 exponent ­Exponents, and Define a 5Í -3Í Number Raised to the base base 0 Power. 1 2 EXAMPLE 1 Evaluate each expression. 6 Decide Which Rule(s) to Use to 4 Simplify an Expression. 3 2 1 a. 2 b. 31 c. -4 2 d. -4 e. f. 0.5 3 g. 4 # 32 2 Solution 1 2 1 2 a. 23 = 2 # 2 # 2 = 8 b. To raise 3 to the first power means to use 3 as¢ a≤ factor only once. Therefore, 31 = 3. Also, when no exponent is shown, the exponent is assumed to be 1. c. -4 2 = -4 -4 = 16 d. -42 = - 4 # 4 = -16 1 4 1 1 1 1 1 f. 0.5 3 = 0.5 0.5 0.5 = 0.125 e. 1 2 = 1# #21# 2= 1 2 2 2 2 2 2 16 1 2 1 21 21 2 g. 4 # 32 = 4 # 9 = 36 PRACTICE 1 Evaluate each expression. ¢ ≤ a. 33 b. 41 c. -8 2 d. -82 3 3 e. f. 0.3 4 g. 13 # 522 4 1 2 Notice how similar -42 is to -4 2 in the example above. The difference between the two is the parentheses. In -4 2, the parentheses tell us that the base, or repeated ¢ ≤ 2 factor, is -4. In -4 , only 4 is the base.1 2 1 2 Helpful Hint Be careful when identifying the base of an exponential expression. Pay close attention to the use of parentheses. -3 2 -32 2 # 32 The base is -3. The base is 3. The base is 3. 1 2 -3 2 = -3 -3 = 9 -32 = - 3 # 3 = -9 2 # 32 = 2 # 3 # 3 = 18 1 2 1 21 2 1 2 An exponent has the same meaning whether the base is a number or a variable. If x is a real number and n is a positive integer, then xn is the product of n factors, each of which is x. xn = x # x # x # x # x # # x (++++)++++*c n factors of x Copyright Pearson. All rights reserved. M05_MART8978_07_AIE_C05.indd 311 11/9/15 5:45 PM For use by Palm Beach State College only. 312 CHAPTER 5 Exponents and Polynomials EXAMPLE 2 Evaluate each expression for the given value of x. 9 a. 2x3; x is 5 b. ; x is -3 x2 3 3 9 9 Solution a. If x is 5, 2x = 2 # 5 b. If x is -3, = x2 -3 2 = 2 # 5 # 5 # 5 1 2 9 = 2 # 125 = 1 2 1 2 -3 -3 250 = 9 = 1 21 2 9 = 1 PRACTICE 2 Evaluate each expression for the given value of x. 6 a. 3x4; x is 3 b. ; x is -4 x2 OBJECTIVE 2 Using the Product Rule Exponential expressions can be multiplied, divided, added, subtracted, and themselves raised to powers. By our definition of an exponent, 54 # 53 = 5 # 5 # 5 # 5 # 5 # 5 # 5 (11+)1111* (111)111* 41 factors of 25 31 factors2 of 5 = 5 # 5 # 5 # 5 # 5 # 5 # 5 (11+1+)++111* 7 factors of 5 = 57 Also, x2 # x3 = x # x # x # x # x x x x x x = 1 # # 2 # 1 # 2 = x5 In both cases, notice that the result is exactly the same if the exponents are added. 54 # 53 = 54 + 3 = 57 and x2 # x3 = x2 + 3 = x5 This suggests the following rule. Product Rule for Exponents If m and n are positive integers and a is a real number, then m n m + n d Add exponents. a # a = aÍ Keep common base. 5 7 5 + 7 12 d Add exponents. For example, 3 # 3 = 3 = 3Í Keep common base. Helpful Hint Don’t forget that 5 7 12 d Add exponents. 3 # 3 9Í Common base not kept. 35 # 37 = 3 # 3 # 3 # 3 # 3 3 # 3 # 3 # 3 # 3 # 3 # 3 (++)++* ~ (++++)++++* 5 factors of 3 7 factors of 3 = 312 12 factors of 3, not 9 Copyright Pearson. All rights reserved. M05_MART8978_07_AIE_C05.indd 312 11/9/15 5:46 PM For use by Palm Beach State College only. Section 5.1 Exponents 313 In other words, to multiply two exponential expressions with the same base, we keep the base and add the exponents. We call this simplifying the exponential expression. EXAMPLE 3 Use the product rule to simplify. a. 42 # 45 b. x4 # x6 c. y3 # y d. y3 # y2 # y7 e. -5 7 # -5 8 f. a2 # b2 Solution 1 2 1 2 2 5 2 + 5 7 d Add exponents. a. 4 # 4 = 4 = 4Í Keep common base. b. x4 # x6 = x4 + 6 = x10 Helpful Hint c. y3 # y = y3 # y1 Don’t forget that if no exponent is written, it is assumed to be 1. = y3 + 1 = y4 d. y3 # y2 # y7 = y3 + 2 + 7 = y12 e. -5 7 # -5 8 = -5 7 + 8 = -5 15 f. a2 b2 Cannot be simplified because a and b are different bases. 1 # 2 1 2 1 2 1 2 PRACTICE 3 Use the product rule to simplify. a. 34 # 36 b. y3 # y2 c. z # z4 d. x3 # x2 # x6 e. -2 5 # -2 3 f. b3 # t5 1 2 1 2 CONCEPT CHECK Where possible, use the product rule to simplify the expression. a. z2 # z14 b. x2 # y14 c. 98 # 93 d. 98 # 27 EXAMPLE 4 Use the product rule to simplify 2x 2 -3x 5 . 2 2 5 5 Solution Recall that 2x means 2 # x and -3x means1 -213 # x . 2 2x2 -3x5 = 2 # x2 # -3 # x5 Remove parentheses. 2 3 x2 x5 Group factors with common bases. 1 21 2 = # - # # = -6x7 Simplify. PRACTICE 4 Use the product rule to simplify -5y3 -3y4 . 1 21 2 EXAMPLE 5 Simplify. a. x2y x3y2 b. -a7b4 3ab9 Solution 1 21 2 1 21 2 a. x2y x3y2 = x2 # x3 # y1 # y2 Group like bases and write y as y1. x5 y3 or x5y3 Multiply. 1 21 2 = 1 # 2 1 2 b. -a7b4 3ab9 = -1 # 3 # a7 # a1 # b4 # b9 3a8b13 1 21 2 = 1- 2 1 2 1 2 PRACTICE Answers to Concept Check: 16 5 Simplify. a. z b. cannot be simplified 7 3 5 4 4 10 c. 911 d. cannot be simplified a. y z y z b. -m n 7mn Copyright1 Pearson.21 2All rights reserved. 1 21 2 M05_MART8978_07_AIE_C05.indd 313 11/9/15 5:46 PM For use by Palm Beach State College only. 314 CHAPTER 5 Exponents and Polynomials Helpful Hint These examples will remind you of the difference between adding and multiplying terms. Addition 5x3 + 3x3 = 5 + 3 x3 = 8x3 By the distributive property. 2 2 7x + 4x = 71x + 4x2 Cannot be combined. Multiplication 5x3 3x3 = 5 # 3 # x3 # x3 = 15x3 + 3 = 15x6 By the product rule. 2 2 1 + 2 3 1 7x214x 2 = 7 # 4 # x # x = 28x = 28x By the product rule.
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