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

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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 exponential 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

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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

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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

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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. 1 21 2 Objective 3 Using the Power Rule Exponential expressions can themselves be raised to powers. Let’s try to discover a rule that simplifies an expression like x2 3. By definition, 2 3 2 2 2 x =1 2x x x (11+)1+1* 2 1 2 13 factors21 21 of x2 which can be simplified by the product rule for exponents. 3 x2 = x2 x2 x2 = x2 + 2 + 2 = x6 Notice that the result is 1exactly2 the1 same21 21 if we2 multiply the exponents. 2 3 2 # 3 6 x = x = x The following property states1 this2 result.

Power Rule for Exponents If m and n are positive integers and a is a real number, then m n mn d Multiply exponents. a = aÍ Keep common base. 1 2

2 5 2 # 5 10 d Multiply exponents. For example, 7 = 7 = 7Í Keep common base. 1 2 To raise a power to a power, keep the base and multiply the exponents.

Example 6 Use the power rule to simplify. 2 7 a. y8 b. 84 5 c. -5 3 Solution 1 2 1 2 31 2 4 8 2 8 # 2 16 4 5 4#5 20 3 7 21 a. y = y = y b. 8 = 8 = 8 c. -5 = -5 Practice 1 2 1 2 31 2 4 1 2 6 Use the power rule to simplify. a. z3 7 b. 49 2 c. -2 3 5

Helpful Hint 1 2 1 2 31 2 4 Take a moment to make sure that you understand when to apply the product rule and when to apply the power rule.

Product Rule S Add Exponents Power Rule S Multiply Exponents 5 7 5 + 7 12 5 7 5 # 7 35 x # x = x = x x = x = x 6 2 6 + 2 8 6 2 6 # 2 12 y # y = y = y 1y 2 = y = y 1 2

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Objective 4 Using the Power Rules for Products and Quotients When the base of an exponential expression is a product, the definition of xn still applies. To simplify xy 3, for example,

3 3 xy 1= 2 xy xy xy xy means 3 factors of (xy). x # x # x # y # y # y Group factors with common bases. 1 2 = 1 21 21 2 1 2 = x3y3 Simplify. Notice that to simplify the expression xy 3, we raise each factor within the parentheses to a power of 3. 1 2 xy 3 = x3y3 In general, we have the following1 rule.2

Power of a Product Rule If n is a positive integer and a and b are real numbers, then ab n = anbn 1 2 For example, 3x 5 = 35x5. In other words, to raise a product to a power, we raise each factor to the power. 1 2 Example 7 Simplify each expression.

2 4 1 2 a. st b. 2a 3 c. mn3 d. -5x2y3z 3 Solution 1 2 1 2 a b 1 2 a. st 4 = s4 # t4 = s4t4 Use the power of a product rule. b. 2a 3 23 a3 8a3 Use the power of a product rule. 1 2 = # = 2 2 1 1 2 1 c. 1 mn2 3 = # m 2 # n3 = m2n6 Use the power of a product rule. 3 3 9 2 2 2 2 d. a-5x2y3bz =a b-5 12 # 2x2 1 # 2y3 # z1 Use the power of a product rule. 25x4y6z2 Use the power rule for exponents. 1 2 = 1 2 1 2 1 2 1 2 Practice 7 Simplify each expression. 3 2 1 2 a. pr 5 b. 6b c. x y d. -3a3b4c 4 4 1 2 1 2 a b 1 2 x 3 Let’s see what happens when we raise a quotient to a power. To simplify , y for example, a b x 3 x x x x 3 x = means 3 factors of y y y y y y a b xa # xb# ax b a b a b a b = Multiply fractions. y # y # y x3 = Simplify. y3 x 3 Notice that to simplify the expression , we raise both the numerator and the y denominator to a power of 3. a b x 3 x3 = y y3 a b Copyright Pearson. All rights reserved.

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In general, we have the following.

Power of a Quotient Rule If n is a positive integer and a and c are real numbers, then a n an = , c 0 c cn a b y 4 y4 For example, = . 7 74 In other words,a tob raise a quotient to a power, we raise both the numerator and the denominator to the power.

Example 8 Simplify each expression. m 7 x3 4 a. b. n 3y5 Solution a b a b m 7 m7 a. = , n 0 Use the power of a quotient rule. n n7 4 a x3b 4 x3 b. , y 0 Use the power of a product or quotient rule. 5 = 4 5 4 3y 3 1# y2 a b 12 x 1 2 = Use the power rule for exponents. 81y20

Practice 8 Simplify each expression. x 5 2a4 5 a. b. y2 b3 a b a b

Objective 5 Using the Quotient Rule and Defining the Zero Exponent Another pattern for simplifying exponential expressions involves quotients. x5 To simplify an expression like , in which the numerator and the denominator x3 have a common base, we can apply the fundamental principle of fractions and divide the numerator and the denominator by the common base factors. Assume for the remainder of this section that denominators are not 0.

x5 x # x # x # x # x = x3 x # x # x x # x # x # x # x = x # x # x = x # x = x2 Notice that the result is exactly the same if we subtract exponents of the common bases. x5 = x5 - 3 = x2 x3 The quotient rule for exponents states this result in a general way.

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Quotient Rule for Exponents If m and n are positive integers and a is a real number, then am = am - n an as long as a is not 0.

x6 For example, = x6 - 2 = x4. x2 In other words, to divide one exponential expression by another with a common base, keep the base and subtract exponents.

Example 9 Simplify each quotient. x5 47 -3 5 s2 2x5y2 a. b. c. d. e. x2 3 2 t3 xy 4 1 -32 Solution 1 2 x5 a. = x5 - 2 = x3 Use the quotient rule. x2 47 b. = 47 - 3 = 44 = 256 Use the quotient rule. 43 5 -3 3 c. 2 = -3 = -27 Use the quotient rule. 1 -32 s2 1 2 d. 1 2 Cannot be simplified because s and t are different bases. t3 e. Begin by grouping common bases. 2x5y2 x5 y2 = 2 # # xy x1 y1 = 2 # x5 - 1 # y2 - 1 Use the quotient rule. 2x4y1 or 2x4y = 1 2 1 2 Practice 9 Simplify each quotient. z8 -5 5 88 q5 6x3y7 a. b. c. d. e. z4 3 86 2 5 1 -52 t xy 1 2 Concept Check Suppose you are simplifying each expression. Tell whether you would add the exponents, subtract the exponents, multiply the exponents, divide the exponents, or none of these. y15 a. x63 21 b. c. z16 + z8 d. w45 # w9 y3 1 2 3 0 x Let’s now give meaning to an expression such as x . To do so, we will simplify 3 in two ways and compare the results. x x3 = x3 - 3 = x0 Apply the quotient rule. x3 Answers to Concept Check: x3 x # x # x a. multiply b. subtract 3 = = 1 Apply the fundamental principle for fractions. c. none of these d. add x x # x # x

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x3 x3 Since = x0 and = 1, we define that x0 = 1 as long as x is not 0. x3 x3

Zero Exponent a0 = 1, as long as a is not 0.

In other words, any base raised to the 0 power is 1 as long as the base is not 0.

Example 10 Simplify each expression. 0 0 3 2 0 0 0 3 a. 3 b. 5x y c. -5 d. -5 e. f. 4x0 100 Solution 1 2 1 2 a b a. 30 = 1 b. Assume that neither x nor y is zero. 5x3y2 0 = 1 c. 5 0 1 - = 1 2 d. 50 1 50 1 1 1 1- 2= - # = - # = - 3 0 e. = 1 100 f. 4ax0 =b4 # x0 = 4 # 1 = 4 Practice 10 Simplify the following expressions. a. -30 b. -3 0 c. 80 d. 0.2 0 e. 7a2y4 0 f. 7y0 1 2 1 2 1 2

Objective 6 Deciding Which Rule to Use Let’s practice deciding which rule(s) to use to simplify. We will continue this discussion with more examples in Section 5.5.

Example 11 Simplify each expression. t 4 a. x7 # x4 b. c. 9y5 2 2 Solution a b 1 2 a. Here we have a product, so we use the product rule to simplify. x7 # x4 = x7 + 4 = x11 b. This is a quotient raised to a power, so we use the power of a quotient rule. t 4 t4 t4 = = 2 24 16 c. Thisa b is a product raised to a power, so we use the power of a product rule. 9y5 2 = 92 y5 2 = 81y10

Practice 1 z 2 2 1 2 11 a. b. 4x6 3 c. y10 # y3 12 a b 1 2

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Example 12 Simplify each expression. 7 2 3 4 3 0 3 0 5 3y 2a b a. 42 - 40 b. x + 2 c. d. 6x5 a9b2 1-8 2 Solution 1 2 1 2 a b 2 0 a. 4 - 4 = 16 - 1 = 15 Remember that 40 = 1. b. x0 3 + 20 5 = 13 + 15 = 1 + 1 = 2 7 2 2 7 2 14 14 1 3y2 1 32 y 9 # y y c. 5 = 2 5 2 = 10 = 10 6x 6 1x 2 36 # x 4x a 3 b4 3 3 3 3 4 3 2a b 21 a2 b 8a9b12 d. = = = -1 # a9 - 9 # b12 - 2 a9b2 a9b2 a9b2 1-8 2 1-82 1 2 -8 = -1 # a0 # b10 = -1 # 1 # b10 = -b101 2 1 2

Practice 12 Simplify each expression. 5x3 2 2z8x5 4 a. 82 - 80 b. z0 6 + 40 5 c. d. 15y4 z2x20 -1 16 2 1 2 1 2 a b

Vocabulary, Readiness & Video Check Use the choices below to fill in each blank. Some choices may be used more than once. 0 base add 1 exponent multiply 1. Repeated multiplication of the same factor can be written using a(n) . 2. In 52, the 2 is called the and the 5 is called the . 3. To simplify x2 # x7, keep the base and the exponents. 4. To simplify x3 6, keep the base and the exponents. 5. The understood exponent on the term y is . 1 2 6. If x□ = 1, the exponent is .

Martin-Gay Interactive Videos Watch the section lecture video and answer the following questions. Objective 1 7. Examples 3 and 4 illustrate how to find the base of an exponential expression both with and without parentheses. Explain how identifying the base of Example 7 is similar to identifying the base of Example 4.

Objective 2 8. Why were the commutative and associative properties applied in Example 12? Were these properties used in another example?

Objective See Video 5.1 3 9. What point is made at the end of Example 15? Objective 4 10. Although it’s not especially emphasized in Example 20, what is helpful to remind yourself about the -2 in the problem? Objective 5 11. In Example 24, which exponent rule is used to show that any nonzero base raised to zero is 1?

Objective 6 12. When simplifying an exponential expression that’s a fraction, will you always use the quotient rule? Refer to Example 30 for this objective to support your answer.

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5.1 Exercise Set MyMathLab®

For each of the following expressions, state the exponent shown Mixed Practice and its corresponding base. Use the power rule and the power of a product or quotient rule to 1. 32 2. -3 6 simplify each expression. See Examples 6 through 8. 3. 42 4. 5 34 45. x9 4 46. y7 5 - 1 # 2 5. 5x2 6. 5x 2 8 6 47. 1pq2 48. 1ab2 5 3 6 2 Evaluate each expression. See Example1 1. 2 49. 12a 2 50. 14x2 2 2 2 3 5 4 7 7. 7 8. -3 51. 1x y2 52. 1a b2 1 2 2 5 2 7 2 3 9. -5 10. -3 53. 1 -7a2b c 54. 1 -3x2 yz 9 11 11. 24 12. 43 r q 1- 2 1- 2 55. 1 2 56. 1 2 s t 13. -2 4 14. -4 3 a b 5 a b 2 5 5 mp xy 15. 10.12 16. 10.22 57. 58. n 7 4 2 1 1 2 1 12 4 17. 18. - a -2xzb 2 a xyb 3 3 9 59. 60. y5 3z3 19. 7a # 2b5 20. 9a # 17 b - a b a 5b 3 3 61. The square shown has sides of length 8z decimeters. Find its 21. -2 # 5 22. -4 # 3 area. A = s2 Evaluate each expression for the replacement values given. See Example 2. 1 2 23. x2; x = -2 24. x3; x = -2 3 2 25. 5x ; x = 3 26. 4x ; x = -1 8z5 decimeters 27. 2xy2; x = 3 and y = 5 28. -4x2y3; x = 2 and y = -1 4 2z 10 62. Given the circle below with radius 5y centimeters, find its 29. ; z = -2 30. ; y = 5 5 3y3 area. Do not approximate p. A = pr2 Use the product rule to simplify each expression. Write the results 1 2 using exponents. See Examples 3 through 5 5y cm 31. x2 # x5 32. y2 # y 33. -3 3 # -3 9 34. -5 7 # -5 6 4 3 2 35. 15y 2 31y 2 36. 1 -2z2 1-2z2 63. The vault below is in the shape of a cube. If each side is 3y4 37. x9y x10y5 38. a2b a13b17 1 21 2 1 21 2 feet, find its volume. V = s3 39. -8mn6 9m2n2 40. -7a3b3 7a19b 1 21 2 1 21 2 1 2 41. 4z10 6z7 z3 42. 12x5 x6 x4 3y 4 feet 1 -21 2 1 -21 2 3y 4 feet 2 3 43. The1 rectangle21 21 below2 has width 4x1 feet21 and length21 2 5x feet. Find its area as an expression in x. A l w = # 3y 4 feet 1 2 4x2 feet

5x3 feet 64. The silo shown is in the shape of a cylinder. If its radius is 4x meters and its height is 5x3 meters, find its volume. Do not 2 44. The parallelogram below has base length 9y7 meters and approximate p. V = pr h height 2y10 meters. Find its area as an expression in y. 1 2 A = b # h 4x meters 1 2 5x 3 10 2y meters meters

9y7 meters

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Use the quotient rule and simplify each expression. See Example 9. Concept Extensions x3 y10 Solve. See the Concept Checks in this section. For Exercises 123 65. 66. x y9 through 126, match the expression with the operation needed to simplify each. A letter may be used more than once and a letter may 6 13 -4 -6 not be used at all. 67. 68. 3 11 1 -42 1 -62 123. x14 23 A. Add the exponents 7 20 8 6 p q x y 124. x14 x23 B. Subtract the exponents 69. 1 2 70. 1 2 1 # 2 15 5 pq xy 125. x14 + x23 C. Multiply the exponents 35 7x2y6 9a4b7 x D. Divide the exponents 71. 72. 126. 14x2y3 27ab2 x17 E. None of these Simplify each expression. See Example 10. Fill in the boxes so that each statement is true. (More than one answer is possible for each exercise.) 73. 70 74. 230 75. 2x 0 127. x□ # x□ = x12 128. x□ □ = x20 76. 4y 0 77. 7x0 78. 2x0 - 1- 2 □ 0 0 0 0 y 7 79. 5 + y 80. -3 + 4 129. = y 130. 1y□2□ # y□ □ = y30 1 2 y□ 131. The formula V x3 can be used 1to find2 the1 volume2 V of a Mixed Practice = cube with side length x. Find the volume of a cube with side Simplify each expression. See Examples 1 through 12. length 7 meters. (Volume is measured in cubic units.) 81. -92 82. -9 2 1 3 2 3 83. 84. 1 2 x 4 3 85. ba4bb2 86. ya4yb 87. a2a3a4 88. x2x15x9 132. The formula S = 6x2 can be used to find the surface area 89. 2x3 -8x4 90. 3y4 -5y S of a cube with side length x. Find the surface area of a cube with side length 5 meters. (Surface area is measured 91. a7b12 a4b8 92. y2z2 y15z13 1 21 2 1 21 2 in square units.) 93. 2mn6 13m8n 94. 3s5t 7st10 1- 21 - 2 1 - 21 - 2 133. To find the amount of water that a swimming pool in the shape 95. z4 10 96. t5 11 1 21 2 1 21 2 of a cube can hold, do we use the formula for volume of the 97. 4ab 3 98. 2ab 4 cube or surface area of the cube? (See Exercises 131 and 132.) 1 2 1 2 99. 6xyz3 2 100. 3xy2a3 3 134. To find the amount of material needed to cover an ottoman in 1 - 2 1 - 2 3x5 5x9 the shape of a cube, do we use the formula for volume of the 101. 1 2 102. 1 2 4 3 cube or surface area of the cube? (See Exercises 131 and 132.) x x 4 4 103. 9xy 2 104. 2ab 5 135. Explain why -5 = 625, while -5 = -625. 2 2 105. 23 20 106. 72 70 136. Explain why 5 # 4 = 80, while 5 # 4 = 400. 1 + 2 1 - 2 1 2 5 3 4 137. In your own words, explain why 50 1. 3y 2ab 1 =2 107. 108. n 6x4 6yz 138. In your own words, explain when -3 is positive and a 3 2b a12 13b when it is negative. 2x y z x y 1 2 109. 110. 5 7 xyz x y Simplify each expression. Assume that variables represent positive 111. 50 3 + y0 7 112. 90 4 + z0 5 integers. 5x9 2 3a4 2 139. x5ax4a 140. b9ab4a 141. ab 5 113. 1 2 1 2 114. 1 2 1 2 10y11 9b5 x9a y15b 142. 2a4b 4 143. 144. 1 2 a2a5b3 b4 a 2x6yb2 5 x4a y6b 115. 116. 20 7 20 10 1 2 -1 16a 2b -132x y2 Solve. Round money amounts to 2 decimal places. 145. Suppose you borrow money for 6 months. If the interest is Review and Preview r 6 compounded monthly, the formula A = P 1 + gives Simplify each expression by combining any like terms. Use the dis- 12 tributive property to remove any parentheses. See Section 2.1. the total amount A to be repaid at the end aof 6 months.b For 117. y - 10 + y a loan of P = $1000 and interest rate of 9% r = 0.09 , how much money is needed to pay off the loan? 118. -6z + 20 - 3z 1 2 146. Suppose you borrow money for 3 years. If the interest is com- 119. 7x 2 8x 6 + - - r 12 pounded quarterly, the formula A = P 1 + gives the 120. 10y - 14 - y - 14 4 121. 2 x - 5 + 3 5 - x total amount A to be repaid at the end ofa 3 years. bFor a loan of 122. -3 w + 7 + 5 w + 1 $10,000 and interest rate of 8% r = 0.08 , how much money 1 2 1 2 is needed to pay off the loan in 3 years? 1 2 1 2 Copyright Pearson. All rights reserved. 1 2

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5.2 Adding and Subtracting Polynomials

Objective Objectives 1 Defining Polynomial, Monomial, Binomial, Trinomial, and Degree 1 Define Polynomial, Monomial, In this section, we introduce a special algebraic expression called a polynomial. Let’s Binomial, Trinomial, and first review some definitions presented in Section 2.1. Degree. Recall that a term is a number or the product of a number and variables raised to powers. The terms of the expression 4x2 3x are 4x2 and 3x. 2 Find the Value of a Polynomial + The terms of the expression 9x4 7x 1 are 9x4, 7x, and 1. Given Replacement Values for - - - - the Variables. Expression Terms Simplify a Polynomial by 3 2 2 Combining Like Terms. 4x + 3x 4x , 3x 4 4 4 Add and Subtract 9x - 7x - 1 9x , -7x, -1 Polynomials. 7y3 7y3 5 5

The numerical coefficient of a term, or simply the coefficient, is the numerical factor of each term. If no numerical factor appears in the term, then the coefficient is understood to be 1. If the term is a number only, it is called a constant term or simply a constant.

Term Coefficient x5 1 3x2 3 -4x -4 -x2y -1 3 (constant) 3

Now we are ready to define a polynomial.

Polynomial A polynomial in x is a finite sum of terms of the form axn, where a is a real number and n is a whole number.

For example, x5 - 3x3 + 2x2 - 5x + 1 is a polynomial. Notice that this polynomial is written in descending powers of x because the powers of x decrease from left to right. (Recall that the term 1 can be thought of as 1x0. ) On the other hand, x-5 + 2x - 3 is not a polynomial because it contains an exponent, -5, that is not a whole number. (We study negative exponents in Section 5.5 of this chapter.) Some polynomials are given special names.

Types of Polynomials A monomial is a polynomial with exactly one term. A binomial is a polynomial with exactly two terms. A trinomial is a polynomial with exactly three terms.

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The following are examples of monomials, binomials, and trinomials. Each of these examples is also a polynomial.

Polynomials Monomials Binomials Trinomials More than Three Terms

ax2 x + y x2 + 4xy + y2 5x3 - 6x2 + 3x - 6 -3z 3p + 2 x5 + 7x2 - x -y5 + y4 - 3y3 - y2 + y 2 4 3 4 4x - 7 -q + q - 2q x6 + x4 - x3 + 1

Each term of a polynomial has a degree.

Degree of a Term The degree of a term is the sum of the exponents on the variables contained in the term.

Example 1 Find the degree of each term. a. -3x2 b. 5x3yz c. 2 Solution a. The exponent on x is 2, so the degree of the term is 2. b. 5x3yz can be written as 5x3y1z1. The degree of the term is the sum of the exponents on its variables, so the degree is 3 + 1 + 1 or 5. c. The constant, 2, can be written as 2x0 (since x0 = 1 ). The degree of 2 or 2x0 is 0.

Practice 1 Find the degree of each term. a. 5y3 b. -3a2b5c c. 8

From the preceding, we can say that the degree of a constant is 0. Each polynomial also has a degree.

Degree of a Polynomial The degree of a polynomial is the greatest degree of any term of the polynomial.

Example 2 Find the degree of each polynomial and tell whether the polynomial is a monomial, binomial, trinomial, or none of these. 2 3 2 a. -2t + 3t + 6 b. 15x - 10 c. 7x + 3x + 2x - 1 Solution a. The degree of the trinomial -2t2 + 3t + 6 is 2, the greatest degree of any of its terms. b. The degree of the binomial 15x - 10 or 15x1 - 10 is 1. c. The degree of the polynomial 7x + 3x3 + 2x2 - 1 is 3.

Practice 2 Find the degree of each polynomial and tell whether the polynomial is a monomial, binomial, trinomial, or none of these. a. 5b2 - 3b + 7 b. 7t + 3 c. 5x2 + 3x - 6x3 + 4

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Example 3 Complete the table for the polynomial

7x2y - 6xy + x2 - 3y + 7 Use the table to give the degree of the polynomial. Solution Term Numerical Coefficient Degree of Term 7x2y 7 3 -6xy -6 2 x2 1 2 -3y -3 1 7 7 0

The degree of the polynomial is 3. Í

Practice 3 Complete the table for the polynomial -3x3y2 + 4xy2 - y2 + 3x - 2.

Term Numerical Coefficient Degree of Term

-3x3y2 4xy2 -y2 3x -2

Objective 2 Evaluating Polynomials Polynomials have different values depending on replacement values for the variables.

Example 4 Find the value of each polynomial when x = -2. a. -5x + 6 b. 3x2 - 2x + 1 Solution a. -5x + 6 = -5 -2 + 6 Replace x with -2. 10 6 = 1+ 2 = 16 2 2 b. 3x - 2x + 1 = 3 -2 - 2 -2 + 1 Replace x with -2. 3 4 4 1 = 1 2+ + 1 2 12 4 1 = 1 +2 + = 17

Practice 4 Find the value of each polynomial when x = -3. a. -10x + 1 b. 2x2 - 5x + 3

Many physical phenomena can be modeled by polynomials.

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Example 5 Finding the Height of a Dropped Object The Swiss Re Building, in London, is a unique building. Londoners often refer to it as the “pickle building.” The building is 592.1 feet tall. An object is dropped from the high- est point of this building. Neglecting air resistance, the height in feet of the object above ground at time t seconds is given by the polynomial -16t2 + 592.1. Find the height of the object when t = 1 second and when t = 6 seconds. Solution To find each height, we evaluate the polynomial when t = 1 and when t = 6. t ϭ 1 -16t2 + 592.1 = -16 1 2 + 592.1 Replace t with 1. 16 1 592.1 = - 1 2 + 16 592.1 = - 1+2 = 576.1 592.1 ft The height of the object at 1 second is 576.1 feet. 576.1 ft -16t2 + 592.1 = -16 6 2 + 592.1 Replace t with 6. 16 36 592.1 = - 1 2 + = -576 + 592.1 = 16.1 t ϭ 6 1 2 The height of the object at 6 seconds is 16.1 feet.

16.1 ft Practice 5 The cliff divers of Acapulco dive 130 feet into La Quebrada several times a day for the entertainment of the tourists. If a tourist is standing near the diving platform and drops his camera off the cliff, the height of the camera above the water at time t seconds is given by the polynomial -16t2 + 130. Find the height of the camera when t = 1 second and when t = 2 seconds.

Objective 3 Simplifying Polynomials by Combining Like Terms Polynomials with like terms can be simplified by combining the like terms. Recall that like terms are terms that contain exactly the same variables raised to exactly the same powers.

Like Terms Unlike Terms

5x2, -7x2 3x, 3y y, 2y -2x2, -5x 1 a2b, -a2b 6st2, 4s2t 2

Only like terms can be combined. We combine like terms by applying the distributive property.

Example 6 Simplify each polynomial by combining any like terms. a. -3x + 7x b. x + 3x2 c. 9x3 + x3 d. 11x2 + 5 + 2x2 - 7 2 2 1 1 e. x4 + x3 - x2 + x4 - x3 5 3 10 6 Solution a. -3x + 7x = -3 + 7 x = 4x b. x 3x2 These terms cannot be combined because x and 3x2 are not like terms. + 1 2 c. 9x3 + x3 = 9x3 + 1x3 = 10x3 (Continued on next page)

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d. 11x2 + 5 + 2x2 - 7 = 11x2 + 2x2 + 5 - 7 = 13x2 - 2 Combine like terms. 2 2 1 1 e. x4 + x3 - x2 + x4 - x3 5 3 10 6 2 1 2 1 = + x4 + - x3 - x2 5 10 3 6 a 4 1b a 4 1b = + x4 + - x3 - x2 10 10 6 6 a5 3 b a b = x4 + x3 - x2 10 6 1 1 = x4 + x3 - x2 2 2

Practice 6 Simplify each polynomial by combining any like terms. a. -4y + 2y b. z + 5z3 c. 15x3 - x3 3 5 1 1 d. 7a2 - 5 - 3a2 - 7 e. x3 - x2 + x4 + x3 - x4 8 6 12 2

Concept Check When combining like terms in the expression 5x - 8x2 - 8x, which of the following is the proper result? a. -11x2 b. -8x2 - 3x c. -11x d. -11x4

Example 7 Combine like terms to simplify.

-9x2 + 3xy - 5y2 + 7yx Solution 9x2 3xy 5y2 7yx = 9x2 3 7 xy 5y2

- + - + Í - + + - 9x2 10xy 5y2 = - + 1 -2 Helpful Hint Practice 2 2 This term can be written as 7yx 7 Combine like terms to simplify: 9xy - 3x - 4yx + 5y or 7xy.

Example 8 Write a polynomial that describes the total area of the squares and rectangles shown below. Then simplify the polynomial. Solution x x 2x x 3 x 3 3 4 x

Area: x # x + 3 # x + 3 # 3 + 4 # x + x # 2x Recall that the area of a 2 2 rectangle is length times = x + 3x + 9 + 4x + 2x width. = 3x2 + 7x + 9 Combine like terms. Practice 8 Write a polynomial that describes the total area of the squares and rectangles shown below. Then simplify the polynomial. x x 3x

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Objective 4 Adding and Subtracting Polynomials We now practice adding and subtracting polynomials.

Adding Polynomials To add polynomials, combine all like terms.

Example 9 Add -2x 2 + 5x - 1 and -2x 2 + x + 3 . Solution 1 2 1 2 -2x2 + 5x - 1 + -2x2 + x + 3 = -2x2 + 5x - 1 - 2x2 + x + 3 2x2 2x2 5x 1x 1 3 1 2 1 2 = - - + + + - + 4x2 6x 2 = 1- + +2 1 2 1 2 Practice 9 Add -3x2 - 4x + 9 and 2x2 - 2x . 1 2 1 2 Example 10 Add: 4x 3 - 6x 2 + 2x + 7 + 5x 2 - 2x Solution 1 2 1 2 4x3 - 6x2 + 2x + 7 + 5x2 - 2x = 4x3 - 6x2 + 2x + 7 + 5x2 - 2x 4x3 6x2 5x2 2x 2x 7 1 2 1 2 = + - + + - + 4x3 x2 7 = - 1 + 2 1 2 Practice 10 Add: -3x3 + 7x2 + 3x - 4 + 3x2 - 9x 1 2 1 2 Polynomials can be added vertically if we line up like terms underneath one another.

3 2 2 Example 11 Add 7 y - 2 y + 7 an d 6 y + 1 using the vertical format. Solution Vertically line up like1 terms and add.2 1 2 7y3 - 2y2 + 7 6y2 + 1 7y3 + 4y2 + 8

Practice 11 Add 5z3 + 3z2 + 4z and 5z2 + 4z using the vertical format.

To subtract1 one polynomial2 from1 another,2 recall the definition of subtraction. To subtract a number, we add its opposite: a - b = a + -b . To subtract a polynomial, we also add its opposite. Just as -b is the opposite of b, - x2 + 5 is the opposite of x2 + 5 . 1 2 1 2 1 2 Example 12 Subtract: 5x - 3 - 2x - 11 Solution From the definition of 1subtraction,2 1we have 2 5x - 3 - 2x - 11 = 5x - 3 + - 2x - 11 Add the opposite. 5x 3 2x 11 Apply the distributive property. 1 2 1 2 = 1 - 2 + 3 -1 + 24 5x 2x 3 11 = 1 - 2 +1 - + 2 3x 8 = 1 + 2 1 2 Practice 12 Subtract 8x - 7 - 3x - 6 Copyright Pearson.1 All rights2 reserved.1 2

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Subtracting Polynomials To subtract two polynomials, change the signs of the terms of the polynomial being subtracted and then add.

Example 13 Subtract: 2x 3 + 8x 2 - 6x - 2x 3 - x 2 + 1 Helpful Hint Solution Change the sign of each1 term of the second2 1polynomial and2 then add. Notice the sign of each term is changed. 3 2 3 2 3 2 3 2 2x + 8x - 6x - 2x - x + 1 = 2x + 8x - 6x + -2x + x - 1 2x3 2x3 8x2 x2 6x 1 1 2 1 2 = 1 - + 2+ 1 - - 2 = 9x2 - 6x - 1 Combine like terms. Practice 13 Subtract: 3x3 - 5x2 + 4x - x3 - x2 + 6 1 2 1 2

Example 14 Subtract 5y 2 + 2y - 6 from -3y 2 - 2y + 11 using the vertical format. 1 2 1 2 Solution Arrange the polynomials in vertical format, lining up like terms. -3y2 - 2y + 11 -3y2 - 2y + 11 - 5y2 + 2y - 6 -5y2 - 2y + 6 8y2 4y 17 1 2 - - + Practice 14 Subtract 6z2 + 3z - 7 from -2z2 - 8z + 5 using the vertical format.

1 2 1 2

Example 15 Subtract 5z - 7 from the sum of 8z + 11 and 9z - 2 . Solution Notice that 5z - 7 is1 to be subtracted2 from 1a sum. The2 translation1 2is 8z + 11 +1 9z - 22 - 5z - 7 8z 11 9z 2 5z 7 Remove grouping symbols. 31 = 2+ 1+ -24 - 1 + 2 = 8z + 9z - 5z + 11 - 2 + 7 Group like terms. = 12z + 16 Combine like terms.

Practice 15 Subtract 3x + 5 from the sum of 8x - 11 and 2x + 5 . 1 2 1 2 1 2

Example 16 add or subtract as indicated. a. 3x2 - 6xy + 5y2 + -2x2 + 8xy - y2 b. 9a2b2 6ab 3ab2 5b2a 2ab 3 9b2 1 + - 2 1- + -2 - Solution 1 2 1 2 a. 3x2 - 6xy + 5y2 + -2x2 + 8xy - y2 3x2 6xy 5y2 2x2 8xy y2 1 = - + 2 -1 + - 2 = x2 + 2xy + 4y2 Combine like terms. b. 9a2b2 6ab 3ab2 5b2a 2ab 3 9b2 Change the sign of each + - - + - - term of the polynomial 9a2b2 6ab 3ab2 5b2a 2ab 3 9b2 1 = + - 2 -1 - + + 2 being subtracted. = 9a2b2 + 4ab - 8ab2 + 3 + 9b2 Combine like terms.

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Practice 16 Add or subtract as indicated. a. 3a2 - 4ab + 7b2 + -8a2 + 3ab - b2 b. 5x2y2 6xy 4xy2 2x2y2 4xy 5 6y2 1 - - 2 1- + - 2 +

1 2 1 2

Concept Check If possible, simplify each expression by performing the indicated operation. a. 2y + y b. 2y # y c. -2y - y d. -2y -y e. 2x + y 1 21 2 Answers to Concept Check: a. 3y b. 2y2 c. -3y d. 2y2 e. cannot be simplified

Vocabulary, Readiness & Video Check

Use the choices below to fill in each blank. Not all choices will be used. least monomial trinomial coefficient greatest binomial constant

1. A is a polynomial with exactly two terms. 2. A is a polynomial with exactly one term. 3. A is a polynomial with exactly three terms. 4. The numerical factor of a term is called the . 5. A number term is also called a . 6. The degree of a polynomial is the degree of any term of the polynomial.

Martin-Gay Interactive Videos Watch the section lecture video and answer the following questions. Objective 1 7. For Example 2, why is the degree of each term found when the example asks for the degree of the polynomial only? Objective 2 8. From Example 3, what does the value of a polynomial depend on? Objective 3 9. When combining any like terms in a polynomial, as in Examples 4–6, what are we doing to the polynomial? Objective 4 10. From Example 7, when we simply remove parentheses and combine See Video 5.2 the like terms of two polynomials, what operation do we perform? Is this true of Examples 9–11?

5.2 Exercise Set MyMathLab®

Find the degree of each of the following polynomials and deter- In the second column, write the degree of the polynomial in the first mine whether it is a monomial, binomial, trinomial, or none of column. See Examples 1 through 3. these. See Examples 1 through 3. Polynomial Degree 1. x + 2 2. -6y2 + 4 9. 3xy2 - 4 3. 9m3 - 5m2 + 4m - 8 10. 8x2y2 4. a + 5a2 + 3a3 - 4a4 2 4 2 2 2 4 11. 5a 2a 1 5. 12x y - x y - 12x y - + 2 2 5 12. 4z6 3z2 6. 7r s + 2rs - 3rs + 7. 3 - 5x8 8. 5y7 + 2 Copyright Pearson. All rights reserved.

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Find the value of each polynomial when (a) x = 0 and 32. 5y + 7y - 6y ( b) x 1. See Examples 4 and 5. 2 2 = - 33. 0.1y - 1.2y + 6.7 - 1.9 13. 5x - 6 34. 7.6y + 3.2y2 - 8y - 2.5y2 14. 2x 10 - 2 4 3 1 4 3 2 35. x + 12x + x - 19x - 19 15. x - 5x - 2 3 6 2 16. x 3x 4 2 4 2 1 4 2 + - 36. x - 23x + x + 5x - 5 3 2 5 15 17. -x + 4x - 15x + 1 2 2 1 3 2 1 3 18. -2x3 + 3x2 - 6x + 1 37. x - x + x - x + 6 5 3 4 The CN Tower in Toronto, Ontario, is 1821 feet tall and is the 1 4 1 2 1 4 3 2 38. x - x + 5 - x - x + 1 world’s tallest self-supporting structure. An object is dropped 6 7 2 7 from the Skypod of the Tower, which is at 1150 feet. Neglecting 39. 6a2 - 4ab + 7b2 - a2 - 5ab + 9b2 air resistance, the height of the object at time t seconds is given by 40. x2y + xy - y + 10x2y - 2y + xy the polynomial -16t2 + 1150. Find the height of the object at the given times. Perform the indicated operations. See Examples 9 through 13.

Time, t Height 41. 8 + 2a - -a - 3 2 42. 4 5a a 5 (in seconds) 16t 1150 1 + 2 - 1 - - 2 43. 2x2 5 3x2 9 19. 1 1 + 2 -1 - 2 44. 5x2 4 7x2 6 20. 7 1 + 2 - 1 - 2 45. 7x 5 3x2 7x 5 21. 3 1 - + 2 +1 - +2 + 46. 3x 8 4x2 3x 3 22. 6 1 - 2+ 1 - + 2 47. 3x 5x 9 1 - 2 - 1 2 Acadia 48. 4 - -12y - 4 National 1 2 49. 2x2 3x 9 4x 7 Park 1+ - 2- - + 50. 7x2 4x 7 8x 2 1 - + + 2 -1 - + 2 Perform1 the indicated 2operations.1 See2 Examples 11, 14, and 15. 3t2 + 4 7x3 + 3 51. 52. + 5t2 - 8 + 2x3 + 1

4z2 - 8z + 3 7a2 - 9a + 6 53. 54. - 6z2 + 8z - 3 - 11a2 - 4a + 2 Canyon De Chelly 1 3 2 2 1 5 2 2 National 5x - 4x + 6x - 2 5u - 4u + 3u - 7 55. 3 2 56. Monument - 3x - 2x - x - 4 - 3u5 + 6u2 - 8u + 2

23. The polynomial 7.5x2 103x 2000 models the yearly 1 2 1 2 - + + 57. Subtract 19x2 + 5 from 81x2 + 10 . number of visitors (in thousands) x years after 2006 at Acadia 58. Subtract 2x xy from 3x 9xy . National Park in Maine. Use this polynomial to estimate the 1 + 2 1 - 2 59. Subtract 2x 2 from the sum of 8x 1 and number of visitors to the park in 2016. 1 + 2 1 2 + x2 x 6x + 3 . 24. The polynomial -0.13 + + 827 models the yearly num- 1 2 1 2 x 60. Subtract 12x 3 from the sum of 5x 7 and ber of visitors (in thousand) years after 2006 at Canyon 1 2 - - - - De Chelly National Monument in Arizona. Use this poly- 12x + 3 . nomial to estimate the number of visitors to the park in 1 2 1 2 1 2 2010. Mixed Practice Perform the indicated operations. Simplify each of the following by combining like terms. See Examples 6 and 7. 61. 2y + 20 + 5y - 30 62. 14y 12 3y 5 25. 9x - 20x 1 + 2 +1 - - 2 63. x2 2x 1 3x2 6x 2 26. 14y - 30y 1 + +2 1- -2 + 2 2 64. 5y2 3y 1 2y2 y 1 27. 14x + 9x 1 - - 2 -1 + + 2 3 3 65. 3x2 5x 8 5x2 9x 12 8x2 14 28. 18x - 4x 1 + - 2 + 1 + + 2 - - 2 2 66. 2x2 7x 9 x2 x 10 3x2 30 29. 15x - 3x - y 1 + - 2 + 1 - + 2- 1 - 2 3 3 67. a2 1 a2 3 5a2 6a 7 30. 12k - 9k + 11 1 - + - 2 -1 + -2 1+ 2 68. m2 3 m2 13 6m2 m 1 31. 8s - 5s + 4s 1 - + 2 - 1 - 2 +1 - +2 1 2 1 2 1 2 Copyright Pearson. All rights reserved.

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3 10 Translating 93. 12x3 -x5 94. 6r 7r Perform each indicated operation. 95. 10x2 20xy2 96. z2y 11zy 1 21 2 - 1 2 69. Subtract 4x from 7x - 3 . Concept E1xtensi2 ons 1 2 70. Subtract y from y2 - 4y + 1 . 1 2 Recall that the perimeter of a figure is the sum of the lengths of 71. Add 4x2 6x 1 and 3x2 2x 1 . - 1+ 2 + + its sides. For Exercises 97 through 100, find the perimeter of each 72. Add 3x2 5x 2 and x2 6x 9 . 1- - + 2 1 - + 2 figure. Write the perimeter as a simplified polynomial. 73. Subtract 5x + 7 from 7x2 + 3x + 9 . 1 2 1 2 97. 9x 98. 5x 74. Subtract 5y2 8y 2 from 7y2 9y 8 . 1 + 2 + 1 + 2- 2 2 3 75. Subtract 4y - 6y - 3 from the sum of 8y + 7 and 7 1 2 1 2 10 6y + 9 . 4x 1 2 1 2 3 76. Subtract 4x2 2x 2 from the sum of x2 7x 1 2x 2x 1 2 - + + + 7x and 7x + 5 . 1 2 1 2 3x 1 2 6 Find the area of each figure. Write a polynomial that describes the 15 total area of the rectangles and squares shown in Exercises 77 and 78. 12 Then simplify the polynomial. See Example 8. 3x 77. 2x 7 4x 2xx 99. (2x2 ϩ 5) feet (Ϫx2 ϩ 3x) feet x 5 x x (4x Ϫ 1) feet 78. 2x 2x 7 100. (Ϫx ϩ 4) centimeters 5x centimeters 2 x 2x 2x x centimeters (x2 Ϫ 6x Ϫ 2) centimeters 6 4 101. A wooden beam is 4y2 4y 1 meters long. If a piece 34 + + y2 - 10 meters is cut, express the length of the remaining piece of beam as a polynomial1 in y.2 Add or subtract as indicated. See Example 16. 1 2 79. 9a + 6b - 5 + -11a - 7b + 6 2 ϩ 4y ϩ 1) meters 80. 3x 2 6y 7x 2 y (4y 1 - + 2 + 1 - - 2 81. 4x2 y2 3 x2 y2 2 1 + + 2 - 1 + - 2 82. 7a2 3b2 10 2a2 b2 12 1 - + 2 -1 - + 2 - ? 83. x2 2xy y2 5x2 4xy 20y2 1 + - 2+ 1 - + 2 84. a2 - ab + 4b2 + 6a2 + 8ab - b2 1 2 1 2 2 Ϫ 10) meters 85. 11r2s 16rs 3 2r2s2 3sr2 5 9r2s2 (y 1 + -2 -1 - 2+ - 86. 3x2y 6xy x2y2 5 11x2y2 1 5yx2 102. A piece of quarter-round molding is 13x - 7 inches 1 - + - -2 1 - + 2 long. If a piece 2x + 2 inches is removed, express the 1 2 1 2 length of the remaining piece of molding1 as a polynomial2 Simplify each polynomial by combining like terms. in x. 1 2 87. 7.75x 9.16x2 1.27 14.58x2 18.34 + - - - (2x ϩ 2) inches 88. 1.85x2 - 3.76x + 9.25x2 + 10.76 - 4.21x

Perform each indicated operation. ? 89. 7.9y4 - 6.8y3 + 3.3y + 6.1y3 - 5 - 4.2y4 + 1.1y - 1 90. 311.2x2 - 3x + 9.1 -2 7.81x2 - 3.1 +248 + 1.2x - 6 (13x Ϫ 7) inches 1 2 Review31 and Preview 2 1 24 1 2 Multiply. See Section 5.1. 91. 3x(2x) 92. -7x x 1 2 Copyright Pearson. All rights reserved.

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The number of worldwide Internet users (in millions) x years after Match each expression on the left with its simplification on the the year 2000 is given by the polynomial 4.8x2 + 104x + 431 for right. Not all letters on the right must be used, and a letter may be the years 1995 through 2015. Use this polynomial for Exercises 103 used more than once. and 104. 109. 10y - 6y2 - y A. 3y 2 110. 5x 5x B. 9y - 6y Worldwide Internet Users + C. 10x 3500 111. 5x - 3 + 5x - 3 D. 25x2 3000 112. 15x - 3 - 5x - 3 1 2 1 2 E. 10x - 6

(in milions) 2500 1 2 1 2 F. none of these

2000 Simplify each expression by performing the indicated operation. Explain how you arrived at each answer. See the last Concept 1500 Check in this section. 113. a. z + 3z b. z # 3z 1000 c. -z - 3z d. -z -3z 500 114. a. 5y y b. 5y y + 1 # 21 2 Number of Internet Users c. -5y - y d. -5y -y 0 1995 2000 2005 2010 2015 115. a. m # m # m b. m + m + m Year 1 21 2 c. -m -m -m d. -m - m - m Data from International Telecommunication Union and United Nations Population Division 1 21 21 2 116. a. x + x b. x # x 103. Estimate the number of Internet users in the world in c. -x - x d. -x -x 2015. Fill in the squares so that each is a1 true21 statement.2 104. Use the given polynomial to predict the number of Internet users in the world in 2020. 117. 3x□ + 4x2 = 7x□ 118. 9y7 + 3y□ = 12y7 4 3 Concept Extensions 119. 2x□ + 3x□ - 5x□ + 4x□ = 6x - 2x □ □ □ □ 5 2 105. Describe how to find the degree of a term. 120. 3y + 7y - 2y - y = 10y - 3y

106. Describe how to find the degree of a polynomial. Write a polynomial that describes the surface area of each figure. 107. Explain why xyz is a monomial while x + y + z is a trino- (Recall that the surface area of a solid is the sum of the areas of the mial. faces or sides of the solid.) 3 108. Explain why the degree of the term 5y is 3 and the degree 121. 122. of the polynomial 2y + y + 2y is 1. x 9

x y

x 5

5.3 Multiplying Polynomials

Objective Objectives 1 Multiplying Monomials 3 4 1 Multiply Monomials. Recall from Section 5.1 that to multiply two monomials such as -5x and -2x , we use the associative and commutative properties and regroup. Remember, also, that 2 Use the Distributive Property to to multiply exponential expressions with a common base, we use1 the product2 1rule for2 Multiply Polynomials. exponents and add exponents. 3 Multiply Polynomials Vertically. -5x3 -2x4 = -5 -2 x3 x4 = 10x7 1 21 2 1 21 21 21 2

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Examples Multiply. 1. 6x # 4x = 6 # 4 x # x Use the commutative and associative properties. 2 = 241 x 21 2 Multiply. 2. -7x2 # 0.2x5 = -7 # 0.2 x2 # x5 7 = 1-1.4x 21 2 1 2 1 2 3. - x5 - x = - # - # x5 # x 3 9 3 9 a b a b a2 b 1 2 = x6 27

Practice 1–3 Multiply. 1 7 1. 5y # 2y 2. 5z3 # -0.4z5 3. - b6 - b3 9 8 1 2 1 2 a b a b

Concept Check Simplify. a. 3x # 2x b. 3x + 2x

Objective 2 Using the Distributive Property to Multiply Polynomials To multiply polynomials that are not monomials, use the distributive property.

Example 4 Use the distributive property to find each product. a. 5x 2x3 + 6 b. -3x2 5x2 + 6x - 1 Solution 1 2 1 2 a. 5x 2x3 + 6 = 5x 2x3 + 5x 6 Use the distributive property. 4 1 2 = 10x1 +230x 1 2 Multiply.

b. -3x2 5x2 + 6x - 1 2 2 2 2 = -1 3x 5x + 2 -3x 6x + -3x -1 Use the distributive property. = 1-15x421- 18x23 + 13x2 21 2 1 21 2 Multiply.

Practice 4 Use the distributive property to find each product. a. 3x 9x5 + 11 b. -6x3 2x2 - 9x + 2 1 2 1 2 We also use the distributive property to multiply two binomials. To multiply x + 3 by x + 1 , distribute the factor x + 3 first. 1 2 1 2 1 2 x + 3 x + 1 = x x + 1 + 3 x + 1 Distribute x + 3 . x x x 1 3 x 3 1 Apply the distributive 1 21 2 = 1 + 2 +1 2+ 1 2 property a second time. = x21 +2 x + 13x2 + 3 1 2 1 2 Multiply. = x2 + 4x + 3 Combine like terms.

Answers to Concept Check: a. 6x2 b. 5x

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This idea can be expanded so that we can multiply any two polynomials.

To Multiply Two Polynomials Multiply each term of the first polynomial by each term of the second polynomial and then combine like terms.

Example 5 Multiply: 3x + 2 2x - 5 Solution Multiply each term of 1the first21 binomial2 by each term of the second. 3x + 2 2x - 5 = 3x 2x + 3x -5 + 2 2x + 2 -5 6x2 15x 4x 10 Multiply. 1 21 2 = 1 - 2 + 1 -2 1 2 1 2 = 6x2 - 11x - 10 Combine like terms.

Practice 5 Multiply: 5x - 2 2x + 3 1 21 2 Example 6 Multiply: 2x - y 2 2 2 Solution Recall that a = a # a, so1 2x -2 y = 2x - y 2x - y . Multiply each term of the first polynomial by each term of the second. 1 2 1 21 2 2x - y 2x - y = 2x 2x + 2x -y + -y 2x + -y -y 4x2 2xy 2xy y2 Multiply. 1 21 2 = 1 - 2 - 1 2+ 1 21 2 1 21 2 = 4x2 - 4xy + y2 Combine like terms.

Practice 6 Multiply: 5x - 3y 2 1 2 Concept Check Square where indicated. Simplify if possible. a. 4a 2 + 3b 2 b. 4a + 3b 2 1 2 1 2 1 2 Example 7 Multiply t + 2 by 3t 2 - 4t + 2 . Solution Multiply each term of1 the first2 polynomial1 by each2 term of the second. t + 2 3t2 - 4t + 2 = t 3t2 + t -4t + t 2 + 2 3t2 + 2 -4t + 2 2 3t3 4t2 2t 6t2 8t 4 1 21 2 = 1 -2 +1 2+ 1-2 + 1 2 1 2 1 2 = 3t3 + 2t2 - 6t + 4 Combine like terms.

Practice 7 Multiply y + 4 by 2y2 - 3y + 5 . 1 2 1 2 Example 8 Multiply: 3a + b 3 3 Solution Write 3a + b as 31a + b 23a + b 3a + b . 1 2 1 21 21 2 3a + b 3a + b 3a + b = 9a2 + 3ab + 3ab + b2 3a + b 9a2 6ab b2 3a b 1 21 21 2 = 1 + + +21 2 9a2 3a b 6ab 3a b b2 3a b = 1 + + 21 + 2 + + 27a3 9a2b 18a2b 6ab2 3ab2 b3 = 1 + 2 + 1 + 2 + 1 + 2 = 27a3 + 27a2b + 9ab2 + b3 Answers to Concept Check: a. 16a2 9b2 Practice + 3 b. 16a2 + 24ab + 9b2 8 Multiply: s + 2t Copyright1 Pearson.2 All rights reserved.

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Objective 3 Multiplying Polynomials Vertically Another convenient method for multiplying polynomials is to use a vertical format similar to the format used to multiply real numbers. We demonstrate this method by multiplying 3y2 - 4y + 1 by y + 2 . 1 2 1 2 Example 9 Multiply 3y 2 - 4y + 1 by y + 2 . Use a vertical format. Solution 1 2 1 2 3y2 - 4y + 1 Helpful Hint * y + 2 1st, multiply 3y2 - 4y + 1 by 2. 2 Make sure like terms are 6y - 8y + 2 2nd, multiply 3y2 - 4y + 1 by y. lined up. 3 2 3y - 4y + y Line up like terms. 3y3 + 2y2 - 7y + 2 3rd, combine like terms.

Thus, 3y2 - 4y + 1 y + 2 = 3y3 + 2y2 - 7y + 2 . Practice 1 21 2 9 Multiply 5x2 - 3x + 5 by x - 4 . Use a vertical format. 1 2 1 2 When multiplying vertically, be careful if a power is missing; you may want to leave space in the partial products and take care that like terms are lined up.

Example 10 Multiply 2x 3 - 3x + 4 by x 2 + 1 . Use a vertical format. Solution 1 2 1 2 2x3 - 3x + 4 * x2 + 1 2x3 - 3x + 4 Leave space for missing powers of x. 2x5 - 3x3 + 4x2 d Line up like terms. 2x5 - x3 + 4x2 - 3x + 4 Combine like terms.

Practice 10 Multiply x3 - 2x2 + 1 by x2 + 2 using a vertical format. 1 2 1 2 Example 11 Find the product of 2x 2 - 3x + 4 and x 2 + 5x - 2 using a vertical format. 1 2 1 2 Solution First, we arrange the polynomials in a vertical format. Then we multiply each term of the second polynomial by each term of the first polynomial. 2x2 - 3x + 4 * x2 + 5x - 2 2 -4x + 6x - 8 Multiply 2x2 - 3x + 4 by -2. 3 2 10x - 15x + 20x Multiply 2x2 - 3x + 4 by 5x. 4 3 2 2x - 3x + 4x Multiply 2x2 - 3x + 4 by x2. 2x4 + 7x3 - 15x2 + 26x - 8 Combine like terms.

Practice 11 Find the product of 5x2 + 2x - 2 and x2 - x + 3 using a vertical format. 1 2 1 2

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Vocabulary, Readiness & Video Check

Fill in each blank with the correct choice. 1. The expression 5x 3x + 2 equals 5x # 3x + 5x # 2 by the property. a. commutative b. associative c. distributive 1 2 2. The expression x + 4 7x - 1 equals x 7x - 1 + 4 7x - 1 by the property. a. commutative b. associative c. distributive 1 21 2 1 2 1 2 3. The expression 5y - 1 2 equals . a. 2 5y 1 b. 5y 1 5y 1 c. 5y 1 5y 1 - 1 2 - + - - 4. The1 expression2 9x # 3x equals1 21 2 . 1 21 2 a. 27x b. 27x2 c. 12x d. 12x2

Martin-Gay Interactive Videos Watch the section lecture video and answer the following questions. Objective 1 5. For Example 1, we use the product property to multiply the monomials. Is it possible to add the same two monomials? Why or why not? Objective 2 6. What property and what exponent rule is used in Examples 2–6?

Objective 3 7. Would you say the vertical format used in Example 7 also applies the distributive property? Explain.

See Video 5.3

5.3 Exercise Set MyMathLab®

Multiply. See Examples 1 through 3. 24. y - 10 y + 11 1. -4n3 # 7n7 2. 9t6 -3t5 2 1 25. 1 x + 21x - 2 3. 3.1x3 4x9 4. 5.2x4 3x4 3 3 - -1 2 1 2 3 1 a 3 b a 2 b 5. 1 - y2 21 y 2 6. 1 - y7 21 y42 26. x + x - 3 5 4 7 5 5 2 2 7. a2x -b3x a2 b4x5 8. ax 5x4b a -6xb7 27. a3x + b1 a 4x +b7 28. 5x2 2 6x2 2 Multiply.1 21 See Example21 2 4. 1 21 21 2 1 + 21 + 2 29. 2y 4 2 9. 3x 2x + 5 10. 2x 6x + 3 1 - 21 2 2 11. 2a a 4 12. 3a 2a 7 30. 6x - 7 - 1 + 2 - 1 + 2 1 2 13. 3x 2x2 3x 4 31. 4x - 3 3x - 5 1 - 2 + 1 2 1 2 14. 4x 5x2 6x 10 32. 8x - 3 2x - 4 1 - - 2 1 21 2 33. 3x2 1 2 15. -2a2 3a2 - 2a + 3 1 + 21 2 1 2 2 2 16. 4b2 3b3 12b2 6 34. x + 4 - 1 - -2 1 2 35. Perform the indicated operations. 17. -y 4x3 - 7x2y + xy2 + 3y3 1 2 1 2 2 2 18. x 6y3 5xy2 x2y 5x3 a. 4y -y - 1 - + - 2 b. 4y2 y2 1 1- 2 19. x12 8x2 - 6x + 1 2 2 c. Explain the difference between the two expressions. 1 21 2 2 20. y 9y - 6y + 1 36. 3 Perform the indicated operations. a. 9x2 -10x2 Multiply.1 See Examples 25 and 6. b. 9x2 - 10x2 21. x 4 x 3 1 2 + + c. Explain the difference between the two 22. x 2 x 9 1 + 21 + 2 expressions. 23. a 7 a 2 1 + 21 - 2 1 21 2 Copyright Pearson. All rights reserved.

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Multiply. See Example 7. 65. 2x - 5 3 37. x - 2 x2 - 3x + 7 66. 3y 1 3 1 - 2 38. x + 3 x2 + 5x - 8 67. 4x 5 8x2 2x 4 1 21 2 1 + 2 + - 39. x + 5 x3 - 3x + 4 68. 5x 4 x2 x 4 1 21 2 1 + 21 - + 2 40. a + 2 a3 - 3a2 + 7 69. 3x2 2x 4 2x2 4x 3 1 21 2 1 + 21 - -2 + 41. 2a - 3 5a2 - 6a + 4 70. a2 3a 2 2a2 5a 1 1 21 2 1 + - 21 - - 2 42. 3 + b 2 - 5b - 3b2 1 21 2 Express1 as the product21 of polynomials.2 Then multiply. Multiply.1 See21 Example 8. 2 71. Find the area of the rectangle. 43. x + 2 3 (2x ϩ 5) yards 44. y 1 3 1 - 2 45. 2y 3 3 1 - 2 (2x Ϫ 5) yards 46. 3x 4 3 1 + 2 Multiply1 vertically.2 See Examples 9 through 11. 72. Find the area of the square field. 47. 2x - 11 6x + 1 48. 4x 7 5x 1 1 - 21 + 2 49. 5x 1 2x2 4x 1 1 + 21 + 2 - 50. 4x 5 8x2 2x 4 1 - 21 + - 2 51. x2 5x 7 2x2 7x 9 1 + 21- - 2 - 52. 3x2 x 2 x2 2x 1 1 - + 21 + + 2 (x ϩ 4) feet 1 21 2 Mixed Practice Multiply. See Examples 1 through 11. 73. Find the area of the triangle. 6 1 53. -1.2y -7y (Triangle Area = # base # height ) 2 54. 4.2x 2x5 - 1 - 2 55. 3x x2 2x 8 - 1 + 2 - 56. 5x x2 3x 10 - 1 - + 2 57. x 19 2x 1 +1 + 2 58. 3y 4 y 11 4x inches 1 + 21 + 2 1 3 59. 1 x + 21x - 2 7 7 (3x Ϫ 2) inches a 2b a b1 60. m + m - 74. Find the volume of the cube. 9 9 2 (Volume = length # width # height ) 61. a3y + 5 b a b 62. 7y 2 2 1 + 2 63. a + 4 a2 - 6a + 6 1 2 (y Ϫ 1) meters 64. t 3 t2 5t 5 1 + 21 - + 2 1 21 2 Review and Preview In this section, we review operations on monomials. Study the box below, then proceed. See Sections 2.1, 5.1, and 5.2. (Continued on next page)

Operations on Monomials Multiply Review the product rule for exponents. Divide Review the quotient rule for exponents. Add or Subtract Remember, we may only combine like terms.

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Perform the operations on the monomials if possible. The first two rows have been completed for you.

Monomials Add Subtract Multiply Divide 6x 6x, 3x 6x + 3x = 9x 6x - 3x = 3x x x x2 = 2 6 # 3 = 18 3x

2 2 2 2 2 3 -12x -12x , 2x -12x + 2x, can’t be simplified -12x - 2x, can’t be simplified -12x # 2x = -24x = -6x 2x

75. 5a, 15a

7 3 76. 4y , 4y

5 4 77. -3y , 9y

78. 2 2 -14x , 2x

Concept Extensions 90. Write an expression for the area 3x 1 79. Perform each indicated operation. Explain the difference of the figure in two different ways. between the two expressions. 3x a. 3x + 5 + 3x + 7 b. 3x 5 3x 7 1 + 2 1+ 2 80. Perform each indicated operation. Explain the difference 1 1 21 2 between the two expressions. a. 8x - 3 - 5x - 2 Simplify. See the Concept Checks in this section. b. 8x 3 5x 2 91. 5a + 6a 92. 5a # 6a 1 - 2 1- 2 Square where indicated. Simplify if possible. Mixed P1ractice21 2 93. 5x 2 + 2y 2 94. 5x + 2y 2 Perform the indicated operations. See Sections 5.2 and 5.3. 1 2 1 2 1 2 81. 3x - 1 + 10x - 6 95. Multiply each of the following polynomials. 82. 2x 1 10x 7 a. a + b a - b 1 - 2 + 1 - 2 83. 3x 1 10x 6 b. 2x + 3y 2x - 3y 1 - 2 1 - 2 1 21 2 84. 2x 1 10x 7 c. 4x + 7 4x - 7 1 - 21 - 2 1 21 2 85. 3x 1 10x 6 d. Can you make a general statement about all products of 1 - 21- -2 1 21 2 x y x y 86. 2x 1 10x 7 the form + - ? 1 - 2 - 1 - 2 96. Evaluate each of the following. 1 21 2 1 2 1 2 2 2 2 Concept Extensions a. 2 + 3 ; 2 + 3 b. 8 10 2; 82 102 87. The area of the larger rectangle on 1 + 2 + the right is x x 3 . Find another c. Does a b 2 a2 b2 no matter what the values of + 1 2+ = + expression for this area by finding the x a and b are? Why or why not? 1 2 1 2 sum of the areas of the two smaller 97. Write a polynomial that de- rectangles. scribes the area of the shaded x 3 2 88. Write an expression for the area of the region. (Find the area of the larg- larger rectangle on the right in two dif- x er square minus the area of the 2 x ϩ 3 smaller square.) ferent ways. 12x 89. The area of the figure on the right x 3 x x is + 2 + 3 . Find another x ϩ 3 expression for this area by finding 98. Write a polynomial that describes the area of the shaded the1 sum of21 the areas2 of the four x regio . (See Exercise 97.) smaller rectangles. n x ϩ 4 2

x ϩ 1 x

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5.4 special Products

Objective Objectives 1 Using the FOIL Method 1 Multiply Two Binomials Using In this section, we multiply binomials using special products. First, a special order for the FOIL Method. multiplying binomials called the FOIL order or method is introduced. This method is demonstrated by multiplying 3x 1 by 2x 5 as shown below. 2 Square a Binomial. + + 3 Multiply the Sum and The FOIL Method 1 2 1 2 Difference of Two Terms. F stands for the product of the First terms. 3x + 1 2x + 5 4 Use Special Products to 3x 2x 6x2 F 1 21 2 = Multiply Binomials. O stands for the product of the Outer terms. 3x 1 2x 5 + 1+ 21 2 3x 5 15x O 1 21 2 = I stands for the product of the Inner terms. 3x 1 2x 5 + + 1 21 2 1 2x 2x I 1 21 2 = L stands for the product of the Last terms. 3x 1 2x 5 + + 1 21 2 1 5 = 5 L 1 21 2 F O I L 1 21 2 3x + 1 2x + 5 = 6x2 + 15x + 2x + 5 6x2 17x 5 Combine like terms. 1 21 2 = + +

Concept Check Multiply 3x + 1 2x + 5 using methods from the last section. Show that the product is still 6x2 + 17x + 5. 1 21 2

Example 1 Multiply x - 3 x + 4 by the FOIL method. Solution 1 21 2 L F

! ! ! ! F O I L x - 3 x + 4 = x x + x 4 + -3 x + -3 4 ! ! ! ! I 1 O21 2 1 21 2 1 21 2 1 21 2 1 21 2 Helpful Hint = x2 + 4x - 3x - 12 Remember that the FOIL order 2 for multiplying can be used only = x + x - 12 Combine like terms. for the product of two binomials. Practice 1 Multiply x + 2 x - 5 by the FOIL method. 1 21 2

Example 2 Multiply 5x - 7 x - 2 by the FOIL method. Solution L 1 21 2 F F O I L ! ! ! ! 5x - 7 x - 2 = 5x x + 5x -2 + -7 x + -7 -2 ! ! ! ! I 1 21O 2 1 2 1 2 1 21 2 1 21 2 = 5x2 - 10x - 7x + 14 = 5x2 - 17x + 14 Combine like terms. Answer to Concept Check: Multiply and simplify. Practice 3x 2x + 5 + 1 2x + 5 2 Multiply 4x - 9 x - 1 by the FOIL method. 1 2 1 2 Copyright Pearson.1 All rights21 reserved.2

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Example 3 Multiply: 2 y + 6 2y - 1 1F 21O I 2 L Solution 2 y + 6 2y - 1 = 2 2y2 - 1y + 12y - 6 2 2y2 11y 6 Simplify inside parentheses. 1 21 2 = 1 + - 2 4y2 22y 12 Now use the distributive property. = 1 + - 2 Practice 3 Multiply: 3 x + 5 3x - 1

Objective 1 21 2 2 Squaring Binomials Now, try squaring a binomial using the FOIL method.

Example 4 Multiply: 3y + 1 2 2 Solution 3y + 1 = 3y + 1 3y +2 1 1 2 1 F 21 O2 I L = 3y 3y + 3y 1 + 1 3y + 1 1 9y2 3y 3y 1 = 1 21+ 2 + 1 +21 2 1 2 1 2 = 9y2 + 6y + 1

Practice 4 Multiply: 4x - 1 2 1 2 Notice the pattern that appears in Example 4. 3y + 1 2 = 9y2 + 6y + 1 2 1 2 9y is the first term of the binomial squared. 3y 2 = 9y2. 6y is 2 times the product of both terms 1 2 of the binomial. 2 3y 1 = 6y. 1 is the second term of the binomial 1 21 21 2 squared. 1 2 = 1. 1 2 This pattern leads to the following, which can be used when squaring a binomial. We call these special products.

Squaring a Binomial A binomial squared is equal to the square of the first term plus or minus twice the product of both terms plus the square of the second term. a + b 2 = a2 + 2ab + b2 a b 2 a2 2ab b2 1 - 2 = - + 1 2 This product can be visualized geometrically.

The area of the large square is side # side. a a2 ab Area = a + b a + b = a + b 2 a ϩ b The area of the large1 square21 is also2 the 1sum of2 the areas b ab b2 of the smaller rectangles. 2 2 2 2 a b Area = a + ab + ab + b = a + 2ab + b 2 2 2 a ϩ b Thus, a + b = a + 2ab + b . 1 2 Copyright Pearson. All rights reserved.

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Example 5 Use a special product to square each binomial. a. t + 2 2 b. p - q 2 c. 2x + 5 2 d. x2 - 7y 2 Solution 1 2 1 2 1 2 1 2 first plus twice the second term or product of plus term squared minus the terms squared ∂ R T T b a. t + 2 2 = t2 + 2 t 2 + 22 = t2 + 4t + 4 b. p q 2 p2 2 p q q2 p2 2pq q2 1 - 2 = - 1 21 2 + = - + c. 2x 5 2 2x 2 2 2x 5 52 4x2 20x 25 1 + 2 = + 1 21 2 + = + + d. x2 7y 2 x2 2 2 x2 7y 7y2 x4 14x2y 49y2 1 - 2 = 1 2 - 1 21 2 + = - +

1 2 1 2 1 21 2 1 2 Practice 5 Use a special product to square each binomial. a. b + 3 2 b. x - y 2 c. 3y 2 2 d. a2 5b 2 1 + 2 1 - 2 1 2 1 2

Helpful Hint Notice that

a + b 2 a2 + b2 The middle term 2ab is missing. a b 2 a b a b a2 2ab b2 1 + 2 = + + = + + Likewise, 1 2 1 21 2 a - b 2 a2 - b2 a b 2 a b a b a2 2ab b2 1 - 2 = - - = - + 1 2 1 21 2

Objective 3 Multiplying the Sum and Difference of Two Terms Another special product is the product of the sum and difference of the same two terms, such as x + y x - y . Finding this product by the FOIL method, we see a pattern emerge. 1 21 2 L F

! ! ! ! F O I L x + y x - y = x2 - xy + xy - y2 ! ! ! ! I 1 21O 2 = x2 - y2 Notice that the middle two terms subtract out. This is because the Outer product is the opposite of the Inner product. Only the difference of squares remains.

Multiplying the Sum and Difference of Two Terms The product of the sum and difference of two terms is the square of the first term minus the square of the second term. a + b a - b = a2 - b2 1 21 2

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Example 6 Use a special product to multiply. 1 1 a. 4 x + 4 x - 4 b. 6t + 7 6t - 7 c. x - x + 4 4 d. 21p - q21 2p + 2q e. 13x2 - 215y 3x2 2+ 5y a b a b Solution 1 21 2 1 21 2 first second term minus term squared squared TT b a. 4 x + 4 x - 4 = 4 x2 - 42 = 4 x2 - 16 = 4x2 - 64 b. 6t 7 6t 7 6t 2 72 36t2 49 1 + 21 - 2 = 1 - 2 = 1 - 2 1 1 1 2 1 c. 1 x - 21 x + 2 = 1x2 2- = x2 - 4 4 4 16 d. a2p - qb a 2p + qb = 2p a2 -b q2 = 4p2 - q2 2 2 2 2 2 4 2 e. 13x - 521y 3x +25y 1= 23x - 5y = 9x - 25y Practice 1 21 2 1 2 1 2 6 Use a special product to multiply. a. 3 x + 5 x - 5 b. 4b - 3 4b + 3 2 2 c. 1x + 21x - 2 d. 15s + t 215s - t 2 3 3 e. a2y - 3bz2 a 2y +b 3z2 1 21 2 1 21 2

Concept Check Match expression number 1 and number 2 to the equivalent expression or expressions in the list below. 1. a + b 2 2. a + b a - b A. a + b a + b B. a2 - b2 C. a2 + b2 D. a2 - 2ab + b2 E. a2 + 2ab + b2 1 2 1 21 2 1 21 2 Objective 4 Using Special Products Let’s now practice multiplying polynomials in general. If possible, use a special product.

Example 7 Use a special product to multiply, if possible. a. x - 5 3x + 4 b. 7x + 4 2 c. y - 0.6 y + 0.6 2 1 d. 1 y4 + 21 3y2 -2 e. 1a - 3 2a2 + 2a - 11 21 2 5 5 Solution a b a b 1 21 2 a. x - 5 3x + 4 = 3x2 + 4x - 15x - 20 FOIL. 3x2 11x 20 1 21 2 = - - b. 7x + 4 2 = 7x 2 + 2 7x 4 + 42 Squaring a binomial. 49x2 56x 16 1 2 = 1 2 + 1 +21 2 c. y - 0.6 y + 0.6 = y2 - 0.6 2 = y2 - 0.36 Multiplying the sum 2 1 1 6 2 and difference of 2 terms. d. 1 y4 + 21 3y2 - 2 = 3y61- 2 y4 + y2 - FOIL. 5 5 5 5 25 e. I’vea insertedb a this productb as a reminder that since it is not a binomial times Answers to Concept Check: 1. A and E 2. B a binomial, the FOIL order may not be used.

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a - 3 a2 + 2a - 1 = a a2 + 2a - 1 - 3 a2 + 2a - 1 Multiplying each term of the binomial by each = a3 2a2 a 3a2 6a 3 1 21 2 1 + - -2 1- + 2 term of the trinomial. = a3 - a2 - 7a + 3

Practice 7 Use a special product to multiply, if possible. a. 4x + 3 x - 6 b. 7b - 2 2 3 2 c. 1x + 0.421 x - 0.42 d. 1 x2 - 2 3x4 + 7 7 e. 1x + 1 21x2 + 5x -2 2 a b a b 1 21 2 Helpful Hint • When multiplying two binomials, you may always use the FOIL order or method. • When multiplying any two polynomials, you may always use the distributive property to find the product.

Vocabulary, Readiness & Video Check Answer each exercise true or false. 1. x + 4 2 = x2 + 16 2. For x + 6 2x - 1 the product of the first terms is 2x2. x x x2 x x3 x 3. 1 + 42 - 4 = + 16 4. The 1product21 - 1 2 + 3 - 1 is a polynomial of degree 5. 1 21 2 1 21 2 Martin-Gay Interactive Videos Watch the section lecture video and answer the following questions. Objective 1 5. From Examples 1–3, for what type of multiplication problem is the FOIL order of multiplication used?

Objective 2 6. Name at least one other method you can use to multiply Example 4. Objective 3 7. From Example 5, why does multiplying the sum and difference of the same two terms always give you a binomial answer?

Objective See Video 5.4 4 8. Why was the FOIL method not used for Example 10?

5.4 Exercise Set MyMathLab®

Multiply using the FOIL method. See Examples 1 through 3. 1 2 11. x - x + 1. x + 3 x + 4 3 3 2. x + 5 x - 1 a 2 b a 1 b 1 21 2 12. x - x + 3. x 5 x 10 5 5 1 - 21 + 2 4. y 12 y 4 Multiply.a See bExamples a b 4 and 5. 1 - 21 + 2 5. 5x 6 x 2 2 2 1 - 21 + 2 13. x + 2 14. x + 7 6. 3y - 5 2y - 7 15. 2x 1 2 16. 7x 3 2 1 21 2 1 - 2 1 - 2 7. 5 y - 6 4y - 1 17. 3a 5 2 18. 5a 2 2 1 21 2 1 - 2 1 + 2 8. 2 x - 11 2x - 9 19. 5x 9 2 20. 6s 2 2 1 21 2 1 + 2 1 - 2 9. 2x + 5 3x - 1 1 21 2 1 2 1 2 10. 6x 2 x 2 1 + 21 - 2 1 21 2

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Multiply. See Example 6. 2 2 62. a - b2 a + b2 21. a - 7 a + 7 3 3 b b x2 x2 x 22. 1 + 321 - 32 63. 5 3 - + 2 23. 3x 1 3x 1 x3 x2 x 1 - 21 +2 64. 4 12 + 5 - 12 24. 4x 5 4x 5 ¢ r s ≤ ¢r s ≤ 1 - 21 + 2 65. 2 1- 3 2 + 32 1 1 66. 6r - 2x 6r + 2x 25. 1 3x - 21 3x + 2 1 21 2 2 2 x y 2 67. 13 - 7 21 2 2 2 68. 4s 2y 2 26. 10x + 10x - 1 - 2 7 7 x x 69. 14 + 5 2 4 - 5 27. 9x y 9x y ¢ + ≤ ¢ - ≤ x x 70. 13 + 5213 - 52 28. 2x y 2x y 1 - 21 + 2 x 2 71. 18 + 421 2 29. ¢2x 0.1≤ ¢2x 0.1 ≤ 1 + 21 - 2 x 2 72. 13 + 22 30. 5x 1.3 5x 1.3 1 - 21 + 2 1 1 73. 1 a - y2 a + y 1 21 2 2 2 Mixed Practice a a Multiply. See Example 7. 74. + 4y - 4y 2 2 31. a 5 a 4 + + ¢ 1 ≤ ¢ 1 ≤ a a 75. x - y x + y 32. 1 - 521 - 72 5 5 a 2 33. 1 + 721 2 ¢ y ≤ ¢y ≤ 76. 8 8 b 2 - + 34. 1 - 22 6 6 a a 2 35. 14 + 12 3 - 1 77. ¢a + 1 ≤3a ¢ - a +≤1 36. 6a 7 6a 5 b b2 b 1 + 21 + 2 78. 1 + 3212 + - 32 37. x + 2 x - 2 ¢ ≤ ¢ ≤ 1 21 2 Express1 each21 as a product of2 polynomials in x. Then multiply and x x 38. 1 - 1021 + 102 simplify. a 2 39. 13 + 121 2 79. Find the area of the square rug shown if its side is a 2 40. 14 - 22 2x + 1 feet. 41. x2 y 4x y4 1 + 2 - 1 2 x3 x y 42. 1 - 2215 + 2 x x2 x (2x ϩ 1) feet 43. 1 + 3 21 - 6 2+ 1 x x2 x 44. 1 - 221 - 4 + 22 (2x ϩ 1) feet a 2 45. 12 - 321 2 80. Find the area of the rectangular canvas if its length is 46. 5b - 4x 2 1 2 3x - 2 inches and its width is x - 4 inches. x z x z 47. 15 - 6 2 5 + 6 x y x y 1 2 1 2 48. 111 - 721 11 + 72 x5 x5 49. 1 - 3 21 - 5 2 a4 a4 50. 1 + 521 + 62 (x Ϫ 4) inches 51. x 0.8 x 0.8 1 + 21 - 2 (3x Ϫ 2) inches y y 52. 1 - 0.921 + 0.92 a3 a4 53. 1 + 1121 - 3 2 Review and preview x5 x2 54. 1 + 5 21 - 8 2 Simplify each expression. See Section 5.1. 55. 3 x - 2 2 50b10 x3y6 1 21 2 81. 82. b 2 5 2 56. 213 + 27 70b xy 8 57. 3b + 7 2b - 5 8a17b15 -6a y 1 2 83. 84. y y a7b10 3a4y 58. 13 - 1321 - 3 2 -4 4 12 6 59. 7p - 8 7p + 8 2x y -48ab 1 21 2 85. 86. s s 3x4y4 32ab3 60. 13 - 4 213 + 4 2 1 1 61. 1 a2 -217 a2 2+ 7 3 3

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Find the slope of each line. See Section 3.4. 99.

87. 88.

y y (5x Ϫ 3) 5 5 meters 4 4 3 3 (x ϩ 1) m 2 2 1 1

Ϫ Ϫ Ϫ Ϫ Ϫ x Ϫ Ϫ Ϫ Ϫ Ϫ x x Ϫ 53413 2 Ϫ1 1 245 53413 2 Ϫ1 1 245 (5 3) meters Ϫ2 Ϫ2 Ϫ3 Ϫ3 100. Ϫ4 Ϫ4 (3x Ϫ 4) Ϫ5 Ϫ5 xxxx centimeters x xxx 89. 90. (3x ϩ 4) centimeters y y For Exercises 101 and 102, find the area of the shaded figure. 5 5 4 4 101. x 5 3 3 2 2 1 1 x x Ϫ53Ϫ41Ϫ3Ϫ2Ϫ1 245 Ϫ Ϫ Ϫ Ϫ Ϫ x Ϫ1 53413 2 Ϫ1 1 245 Ϫ2 Ϫ2 5 Ϫ3 Ϫ3 Ϫ 4 Ϫ4 102. 2y 11 Ϫ5 Ϫ5 2y

Concept Extensions 11 Match each expression on the left to the equivalent expression on the right. See the second Concept Check in this section. 2 103. In your own words, describe the different methods that can 91. a - b A. a2 - b2 be used to find the product 2x - 5 3x + 1 . 92. a b a b a2 b2 1 - 2 + B. + 104. In your own words, describe the different methods that can 93. a b 2 2 2 1 212 2 1 + 21 2 C. a - 2ab + b be used to find the product 5x + 1 . 94. a b 2 a b 2 2 2 2 1 + 2 - D. a + 2ab + b 105. Suppose that a classmate asked you why 2x + 1 is not 2 1 2 E. none of these 4x + 1 . Write down your response to this classmate. 1 2 1 2 1 2 106. Suppose that a classmate asked you why 2x 1 2 is Fill in the squares so that a true statement forms. 1 2 + 4x2 + 4x + 1 . Write down your response to this class- 4 2 95. x□ + 7 x□ + 3 = x + 10x + 21 mate. 1 2 96. 5x□ 2 2 25x6 20x3 4 1 2 1 - 21 = 2 - + 107. Using your own words, explain how to square a binomial such as a + b 2. Find 1the area of2 the shaded figure. To do so, subtract the area of 108. Explain how to find the product of two binomials using the the smaller square(s) from the area of the larger geometric figure. 1 2 FOIL method. 97. Square Find each product. For example, 2 2 2 (x Ϫ 1) x meters a + b - 2 a + b + 2 = a + b - 2 meters a2 2ab b2 4 31 2 431 2 4 = 1 + 2 + - 109. x + y - 3 x + y + 3 110. a c 5 a c 5 31 + 2 - 431 + 2 + 4 111. a 3 b a 3 b 98. 2x ϩ 3 31 - 2 + 431 - 2 - 4 112. x 2 y x 2 y 31 - 2 + 431 - 2 - 4 x 2x Ϫ 3 x 31 2 431 2 4

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Integrated Review Exponents and Operations on Polynomials

Sections 5.1–5.4 Perform the indicated operations and simplify. 1. 5x2 7x3 2. 4y2 8y7 3. -42 4. 4 2 5. x 5 2x 1 6. 3x 2 x 5 1 - 21 2 1 -21 2 + - + 7x9y12 7. 1x -2 5 + 2x + 1 8. 13x - 221 + x +2 5 9. 1 21 2 x3y10 201 a2b8 2 1 2 1 2 1 2 10. 11. 12m7n6 2 12. 4y9z10 3 14a2b2 13. 3 4y - 3 4y + 3 14. 21 7x - 12 7x + 1 15. 1x7y5 92 16. 31x9 3 17. 7x2 2x 3 5x2 9 1 21 2 1 21 2 - + - 1 +2 18. 10x2 7x 9 4x2 6x 2 19. 0.7y2 1.2 1.8y2 6y 1 1 2+ - - - + 1 - + 2 1- + 2 20. 7.8x2 6.8x 3.3 0.6x2 9 21. x 4y 2 1 - + 2 +1 - 2 + 22. y 9z 2 23. x 4y x 4y - 1 + 2 + + 24. y 9z y 9z 25. 7x2 6xy 4 y2 xy 1 - 2 + - 1 - 2 + 1 - 2 26. 5a2 3ab 6 b2 a2 27. x 3 x2 5x 1 1 - 2 +1 - 2 - + 1 - 2 28. x 1 x2 3x 2 29. 2x3 7 3x2 10 + - 1 - 2 1 -21 + 2 30. 5x3 1 4x4 5 31. 2x 7 x2 6x 1 1 -21 + 2 1 - 21 - +2 32. 5x 1 x2 2x 3 1 - 21 + -2 1 21 2 Perform1 the indicated21 operations2 and simplify if possible. 33. 5x3 + 5y3 34. 5x3 5y3 35. 5x3 3 5x3 36. 37. x1 + 21x 2 38. x1 # x 2 5y3

5.5 negative Exponents and Scientific Notation

Objective Objectives 1 Simplifying Expressions Containing Negative Exponents 1 Simplify Expressions Contain- Our work with exponential expressions so far has been limited to exponents that are 3 ing Negative Exponents. positive integers or 0. Here we expand to give meaning to an expression like x- . x2 2 Use All the Rules and Defini- Suppose that we wish to simplify the expression 5. If we use the quotient rule tions for Exponents to Simplify for exponents, we subtract exponents: x Exponential Expressions. 2 x 2 5 3 3 Write Numbers in Scientific = x - = x- , x 0 x5 Notation. x2 4 Convert Numbers from Scientific But what does x-3 mean? Let’s simplify using the definition of xn. Notation to Standard Form. x5 5 Perform Operations on x2 x # x = Numbers Written in Scientific x5 x # x # x # x # x Notation. 1 1 x # x Divide numerator and denominator = x # x # x # x # x by common factors by applying the 1 1 fundamental principle for fractions. 1 = x3 -3 1 If the quotient rule is to hold true for negative exponents, then x must equal 3. From this example, we state the definition for negative exponents. x Copyright Pearson. All rights reserved.

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Negative Exponents If a is a real number other than 0 and n is an integer, then 1 a-n = an

1 For example, x-3 = . x3 In other words, another way to write a-n is to take its reciprocal and change the sign of its exponent.

Example 1 Simplify by writing each expression with positive exponents only. a. 3-2 b. 2x-3 c. 2-1 + 4-1 d. -2 -4 e. y-4 Solution 1 2 1 1 a. 3-2 = = Use the definition of negative exponents. 32 9 1 2 b. 2x-3 = 2 # = Use the definition of negative exponents. x3 x3

1 1 1 1 2 1 3 Helpful Hint c. 2- + 4- = + = + = 2 4 4 4 4 Don’t forget that since there are no parentheses, only x is the 4 1 1 1 d. -2 - = = = base for the exponent -3. -2 4 -2 -2 -2 -2 16 1 2 1 e. y-4 = 1 2 1 21 21 21 2 y4

Practice 1 Simplify by writing each expression with positive exponents only. a. 5-3 b. 3y-4 c. 3-1 + 2-1 d. -5 -2 e. x-5 1 2 Helpful Hint A negative exponent does not affect the sign of its base. Remember: Another way to write a-n is to take its reciprocal and change the sign of its 1 exponent: a-n = . For example, an 1 1 1 x-2 = , 2-3 = or x2 23 8 1 1 1 = = y4, = 52 or 25 y-4 1 5-2 y4

1 1 From the preceding Helpful Hint, we know that x-2 = and = y4. We can x2 y-4 use this to include another statement in our definition of negative exponents.

Negative Exponents If a is a real number other than 0 and n is an integer, then 1 1 a-n = and = an an a-n

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Example 2 Simplify each expression. Write results using positive exponents only. 1 1 p-4 5-3 a. b. c. d. x-3 3-4 q-9 2-5 Solution 1 x3 1 34 a. = = x3 b. = = 81 x-3 1 3-4 1 p-4 q9 5-3 25 32 c. = d. = = q-9 p4 2-5 53 125

Practice 2 Simplify each expression. Write results using positive exponents only. 1 1 x-7 4-3 a. b. c. d. s-5 2-3 y-5 3-2

Example 3 Simplify each expression. Write answers with positive exponents. y 3 x-5 2 -3 a. b. c. d. y-2 x-4 x7 3 Solution a b 1 m y y 1 2 3 a a. = = y - - = y Remember that = am - n. -2 -2 an y y 1 2 3 1 b. = 3 # = 3 # x4 or 3x4 x-4 x-4 x-5 1 c. = x-5 - 7 = x-12 = x7 x12 2 -3 2-3 33 27 d. = -3 = 3 = 3 3 2 8

Practice a b 3 Simplify each expression. Write answers with positive exponents. x-3 5 z 5 -2 a. b. c. d. x2 y-7 z-4 9 a b Objective 2 Simplifying Exponential Expressions All the previously stated rules for exponents apply for negative exponents also. Here is a summary of the rules and definitions for exponents.

Summary of Exponent Rules If m and n are integers and a, b, and c are real numbers, then: Product rule for exponents: am # an = am + n m n m # n Power rule for exponents: a = a Power of a product: ab n anbn 1= 2 a n an Power of a quotient: 1 2 = , c 0 c cn a b am Quotient rule for exponents: = am - n, a 0 an Zero exponent: a0 = 1, a 0 1 Negative exponent: a-n = , a 0 an

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Example 4 Simplify the following expressions. Write each result using positive exp onents only. 3 4 2x x 3a2 -3 4-1x-3y -2x3y 3 a. y-3z6 - 6 b. c. d. e. x7 b -3 2 -6 -1 1 2 4 x y xy Solution 1 2 a b a b y18 a. y-3z6 -6 = y18 # z-36 = z36 1 2 3 4 2x x 24 # x12 # x 16 # x12 + 1 16x13 Use the b. = = = = 16x13 - 7 = 16x6 x7 x7 x7 x7 power rule. 1 2 3a2 -3 3-3 a2 -3 Raise each factor in the numerator and c. = b b-3 the denominator to the -3 power. 1 2 a b 3-3a-6 = Use the power rule. b-3 b3 = Use the negative exponent rule. 33a6 b3 = Write 33 as 27. 27a6 4-1x-3y 42y7 16y7 d. 4-1 - -3 x-3 - 2y1 - -6 42x-5y7 -3 2 -6 = = = 5 = 5 4 x y 1 2 1 2 x x 3 9 3 -2x3y 3 -2 x y -8x9y3 e. = = = -8x9 - 3y3 - -3 = -8x6y6 xy-1 x3y-3 x3y-3 1 2 1 2 Practice a b 4 Simplify the following expressions. Write each result using positive exponents only. x2 x5 3 5p8 -2 a. a4b-3 -5 b. c. x7 q 1 2 61-2x-4y2-7 -3x4y 3 a b d. e. 6-3x3y-9 x2y-2 a b Objective 3 Writing Numbers in Scientific Notation Both very large and very small numbers frequently occur in many fields of science. For example, the distance between the Sun and the dwarf planet Pluto is approximately 5,906,000,000 kilometers, and the mass of a proton is approximately 0.00000000000000000000000165 gram. It can be tedious to write these numbers in this standard decimal notation, so scientific notation is used as a convenient shorthand for expressing very large and very small numbers.

proton

Sun 5,906,000,000 Pluto kilometers

Mass of proton is approximately 0.000 000 000 000 000 000 000 001 65 gram Scientific Notation A positive number is written in scientific notation if it is written as the product of a number a, where 1 … a 6 10, and an integer power r of 10: r a * 10 Copyright Pearson. All rights reserved.

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The numbers below are written in scientific notation. The * sign for multiplication is used as part of the notation. 6 2.03 102 7.362 107 5.906 109 (Distance between the Sun and Pluto) * * * 1 * 10 -3 8.1 * 10 - 5 1.65 * 10 - 24 (Mass of a proton) 5

The following steps are useful when writing numbers in scientific notation.

To Write a Number in Scientific Notation Step 1. Move the decimal point in the original number to the left or right so that the new number has a value between 1 and 10 (including 1). Step 2. Count the number of decimal places the decimal point is moved in Step 1. If the original number is 10 or greater, the count is positive. If the original number is less than 1, the count is negative. Step 3. Multiply the new number in Step 1 by 10 raised to an exponent equal to the count found in Step 2.

Example 5 Write each number in scientific notation. a. 367,000,000 b. 0.000003 c. 20,520,000,000 d. 0.00085 Solution a. Step 1. Move the decimal point until the number is between 1 and 10. 367,000,000. 8 places Step 2. The decimal point is moved 8 places, and the original number is 10 or greater, so the count is positive 8. Step 3. 367,000,000 = 3.67 * 108. b. Step 1. Move the decimal point until the number is between 1 and 10. 0.000003 6 places Step 2. The decimal point is moved 6 places, and the original number is less than 1, so the count is -6. Step 3. 0.000003 = 3.0 * 10-6 c. 20,520,000,000 = 2.052 * 1010 d. 0.00085 = 8.5 * 10-4

Practice 5 Write each number in scientific notation. a. 0.000007 b. 20,700,000 c. 0.0043 d. 812,000,000

Objective 4 Converting Numbers to Standard Form A number written in scientific notation can be rewritten in standard form. For example, to write 8.63 * 103 in standard form, recall that 103 = 1000. 8.63 * 103 = 8.63 1000 = 8630 Notice that the exponent on the 10 is positive1 3, and2 we moved the decimal point 3 places to the right.

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1 1 To write 7.29 * 10-3 in standard form, recall that 10-3 = = . 103 1000 1 7.29 7.29 * 10-3 = 7.29 = = 0.00729 1000 1000 The exponent on the 10 is negative 3, anda we movedb the decimal to the left 3 places. In general, to write a scientific notation number in standard form, move the decimal point the same number of places as the exponent on 10. If the exponent is positive, move the decimal point to the right; if the exponent is negative, move the decimal point to the left.

Example 6 Write each number in standard notation, without exponents. a. 1.02 * 105 b. 7.358 * 10-3 c. 8.4 * 107 d. 3.007 * 10-5 Solution a. Move the decimal point 5 places to the right. 1.02 * 105 = 102,000. b. Move the decimal point 3 places to the left. 7.358 * 10 - 3 = 0.007358 c. 8.4 * 107 = 84,000,000. 7 places to the right

d. 3.007 * 10 - 5 = 0.00003007 5 places to the left Practice 6 Write each number in standard notation, without exponents. a. 3.67 * 10-4 b. 8.954 * 106 c. 2.009 * 10-5 d. 4.054 * 103

Concept Check Which number in each pair is larger? a. 7.8 * 103 or 2.1 * 105 b. 9.2 * 10-2 or 2.7 * 104 c. 5.6 * 10-4 or 6.3 * 10-5

Objective 5 Performing Operations with Scientific Notation Performing operations on numbers written in scientific notation uses the rules and definitions for exponents.

Example 7 perform each indicated operation. Write each result in standard decimal notation. 12 * 102 a. 8 * 10-6 7 * 103 b. 6 * 10-3 Solution 1 21 2 a. 8 * 10-6 7 * 103 = 8 # 7 * 10-6 # 103 56 10-3 1 21 2 = 1 * 2 1 2 = 0.056 12 * 102 12 b. = * 102 - -3 = 2 * 105 = 200,000 -3 6 6 * 10 1 2 Practice 7 Perform each indicated operation. Write each result in standard decimal notation. Answers to Concept Check: 3 5 4 64 * 10 a. 2.1 * 10 b. 2.7 * 10 a. 5 10-4 8 106 b. 4 * * -7 c. 5.6 * 10- 32 * 10 Copyright1 Pearson.21 All rights reserved.2

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Calculator Explorations

Scientific Notation To enter a number written in scientific notation on a scientific calculator, locate the scientific notation key, which may be marked EE or EXP . To enter 3.1 * 107, press 3.1 EE 7 . The display should read 3.1 07 . Enter each number written in scientific notation on your calculator. 1. 5.31 * 103 2. -4.8 * 1014 3. 6.6 * 10-9 4. -9.9811 * 10-2 Multiply each of the following on your calculator. Notice the form of the result. 5. 3,000,000 * 5,000,000 6. 230,000 * 1000 Multiply each of the following on your calculator. Write the product in scientific notation. 7. 3.26 * 106 2.5 * 1013 8. 8.76 10-4 1.237 109 1 * 21 * 2 1 21 2

Vocabulary, Readiness & Video Check

Fill in each blank with the correct choice.

1. The expression x-3 equals . 2. The expression 5-4 equals .

3 1 -1 1 1 1 a. -x b. c. d. a. -20 b. -625 c. d. x3 x3 x-3 20 625 3. The number 3.021 * 10-3 is written in . 4. The number 0.0261 is written in . a. standard form b. expanded form a. standard form b. expanded form c. scientific notation c. scientific notation

Martin-Gay Interactive Videos Watch the section lecture video and answer the following questions. Objective 1 5. What important reminder is made at the end of Example 1? Objective 2 6. Name all the rules and definitions used to simplify Example 8. Objective 3 7. From Examples 9 and 10, explain how the movement of the decimal point in step 1 suggests the sign of the exponent on the number 10. Objective 4 8. From Example 11, what part of a number written in scientific notation is key in telling you how to write the number in standard form? See Video 5.5 Objective 5 9. For Example 13, what exponent rules were needed to evaluate?

5.5 Exercise Set MyMathLab®

Simplify each expression. Write each result using positive exponents 1 -5 1 -2 1 - 3 7. 8. 9. - only. See Examples 1 through 3. 2 8 4 1. 4-3 2. 6-2 3. 3 -4 a b -2 a b a b - 1 1 1 1 2 10. - 11. 3- + 5- 12. 4- + 4- 8 4. 3 -5 5. 7x-3 6. 17x 2-3 - a b

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1 1 -5 2 4 -7 -5 13. 14. p -8xa b a b -3 -5 15. 66. 67. p q q-4 5xa5b 2 -1 -2 - 15a b 2 5 2 r - x- y 6 -2 4 16. 17. 18. a b 1 2 s-2 x y-3 68. 4a-3b-3 3 -4 -4 1 2 z x -2 -1 19. 20. 21. 3 + 3 Write1 each number2 in scientific notation. See Example 5. z-7 x-1 69. 78,000 70. 9,300,000,000 -1 -1 22. 4-2 - 4-3 23. 24. 71. 0.00000167 72. 0.00000017 p-4 y-6 73. 0.00635 74. 0.00194 25. -20 - 30 26. 50 + -5 0 75. 1,160,000 76. 700,000 Mixed Practice 1 2 77. More than 2,000,000,000 pencils are manufactured in the Simplify each expression. Write each result using positive exponents United States annually. Write this number in scientific nota- only. See Examples 1 through 4. tion. (Source: AbsoluteTrivia.com) 4 5 78. The temperature at the interior of the Earth is 20,000,000 de- x2x5 y y p2p 27. 28. 29. grees Celsius. Write 20,000,000 in scientific notation. x3 y6 p-1 79. As of this writing, the world’s largest optical telescope is 3 5 4 2 8 y y m m x x the Gran Telescopio Canaris, located in La Palma, Canary 30. 31. 32. -2 10 9 Islands, Spain. The elevation of this telescope is 2400 meters y 1 m2 1 x2 above sea level. Write 2400 in scientific notation. r p 33. 34. 35. x5y3 -3 r -3r -2 p-3q-5 3 1 2 x2 y4 2 36. z5x5 -3 37. 38. 10 12 1x 2 1y 2 1 22 5 a5 x2 8k4 39. 40. 41. 3 4 4 3 2k 1a 2 1x 2 27r4 -6m4 42. 1 2 1 2 43. 3r6 -2m3 15a4 -24a6b 44. 45. 2 -15a5 6ab -5x4y5 46. 47. -2x3y-4 3x-1y 80. In March 2004, the European Space Agency launched the 15x4y2 Rosetta spacecraft, whose mission was to deliver the Philae 1 21 2 lander to explore comet 67P/Churyumov-Gerasimenko. The 48. -5a4b-7 -a-4b3 49. a-5b2 -6 lander finally arrived on the comet in late 2014. This comet 2 4 x- y 2 is currently more than 320,000,000 miles from Earth. Write 50. 14-1x5 -221 2 51. 1 2 x3y7 320,000,000 in scientific notation. (Source: European Space 1 2 Agency) a5b -3 4a2z-3 b 52. 53. a7b-2 43z-5 a b 3-1x4 2-3x-4 54. 55. 33x-7 22x 5-1z7 7ab-4 56. 57. 5-2z9 7-1a-3b2 6-5x-1y2 a-5b -4 58. 59. 6-2x-4y4 ab3 a 3 5b r -2s-3 -3 xy 60. 61. -4 -3 -4 r s 1xy 2 a -3 b -3 - 3 rs 1 -22xy 62. 63. 2 3 2 xy-1 -1 1r s2 1 2 Write each number in standard notation. See Example 6. 2 2 -2 2 3 -3x y 6x1 y 2 81. 8.673 * 10-10 64. 1 2 65. -2 5 -4 1 xyz 2 -7xy 82. 9.056 * 10 83. 3.3 10-2 1 2 *

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84. 4.8 * 10-6 99. 4 * 10-10 7 * 10-9 85. 2.032 104 100. 5 106 4 10-8 * 1 * 21* 2 86. 9.07 * 1010 8 * 10-1 25 * 10-4 101. 1 21 2 102. 9 87. Each second, the Sun converts 7.0 * 108 tons of hydrogen 16 * 105 5 * 10- into helium and energy in the form of gamma rays. Write this 1.4 * 10-2 0.4 * 105 number in standard notation. (Source: Students for the Explo- 103. 104. 7 10-8 0.2 1011 ration and Development of Space) * * 88. In chemistry, Avogadro’s number is the number of atoms in one Review and preview mole of an element. Avogadro’s number is 6.02214199 * 1023. Simplify the following. See Section 5.1. Write this number in standard notation. (Source: National 14 Institute of Standards and Technology) 5x7 27y 105. 106. 4 7 89. The distance light travels in 1 year is 9.46 * 1012 kilometers. 3x 3y Write this number in standard notation. 15z4y3 18a7b17 9 107. 108. 90. The population of the world is 7.3 * 10 . Write this number 21zy 30a7b in standard notation. (Source: UN World Population Clock) Use the distributive property and multiply. See Sections 5.3 and 5.5. Mixed Practice 1 109. 5y2 - 6y + 5 See Examples 5 and 6. Below are some interesting facts about se- y lected countries’ external debts at a certain time. These are public 2 1 2 110. 3x5 + x4 - 2 and private debts owed to nonresidents of that country. If a num- x ber is written in standard form, write it in scientific notation. If a 1 2 number is written in scientific notation, write it in standard form. Concept Extensions (Source: CIA World Factbook) 111. Find the volume of the cube. Selected Countries and Their External Debt 1000 3xϪ2 inches 800 z

600 112. Find the area of the triangle. (in billions of dollars) 400 4 m 200 x 5xϪ3 0 m Brazil Russia Mexico China Indonesia 7 Amount of Debt Countries Simplify. 91. The external debt of Russia at a certain time was 113. 2a3 3a4 + a5a8 114. 2a3 3a-3 + a11a-5 $714,000,000,000. Fill in 1the boxes2 so that each statement is true.1 (More2 than one answer 92. The amount by which Russia’s debt was greater than may be possible for these exercises.) Mexico’s debt was $359,000,000,000. 11 1 1 93. At a certain time, China’s external debt was $8.63 * 10 . 115. x□ = 116. 7□ = x5 49 94. At a certain time, the external debt of the United States was □ □ 10 □ □ 15 13 117. z # z z- 118. x x- $1.5 * 10 . = = 95. At a certain time, the estimated per person share of the United 1 2 119. Which is larger? See the Concept Check in this section. States external debt was $4.7 * 104. a. 9.7 10-2 or 1.3 101 96. The bar graph shows the external debt of five countries. * * 5 7 Estimate the height of the tallest bar and the shortest bar b. 8.6 * 10 or 4.4 * 10 in standard notation. Then write each number in scientific c. 6.1 * 10-2 or 5.6 * 10-4 notation. 120. Determine whether each statement is true or false. a. 5-1 6 5-2 1 2 Evaluate each expression using exponential rules. Write each result 1 - 1 - b. 6 in standard notation. See Example 7. 5 5 -1 -2 97. 1.2 * 10-3 3 * 10-2 c. a b6 a aforb all nonzero numbers. 98. 2.5 106 2 10-6 1 * 21 * 2 1 21 2

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121. It was stated earlier that for an integer n, Simplify each expression. Write each result in standard notation. 12 -10 -n 1 127. 2.63 * 10 -1.5 * 10 x = n, x 0 x 128. 6.785 10-4 4.68 1010 1 * 21 * 2 Explain why x may not equal 0. Light 1travels at a rate21 of 1.86 1025 miles per second. Use this in- 122. The quotient rule states that * formation and the distance formula d r t to answer Exercises m = # a m n 129 and 130. = a - , a 0. an 129. If the distance from the moon to Earth is 238,857 miles, find Explain why a may not equal 0. how long it takes the reflected light of the moon to reach Simplify each expression. Assume that variables represent positive Earth. (Round to the nearest tenth of a second.) integers. 130. If the distance from the Sun to Earth is 93,000,000 miles, find how long it takes the light of the Sun to reach Earth. 123. a-4m # a5m 124. x-3s 3 2z 3 4m 1 4 125. 3y 126. a1 + 2 # a 1 2 5.6 Dividing Polynomials

Objective Objectives 1 Dividing by a Monomial 1 Divide a Polynomial by a Now that we know how to add, subtract, and multiply polynomials, we practice dividing Monomial. polynomials. 2 Use Long Division to Divide To divide a polynomial by a monomial, recall addition of fractions. Fractions that a Polynomial by Another have a common denominator are added by adding the numerators: Polynomial. a b a + b = c + c c If we read this equation from right to left and let a, b, and c be monomials, c 0, we have the following:

Dividing a Polynomial by a Monomial Divide each term of the polynomial by the monomial. a + b a b = , c 0 c c + c

Throughout this section, we assume that denominators are not 0.

Example 1 Divide 6m2 + 2m by 2m. Solution We begin by writing the quotient in fraction form. Then we divide each term of the polynomial 6m2 + 2m by the monomial 2m. 6m2 + 2m 6m2 2m = + 2m 2m 2m = 3m + 1 Simplify. 6m2 + 2m Check: We know that if = 3m + 1, then 2m # 3m + 1 must equal 2m 2

6m + 2m. Thus, to check, we multiply. 1 2 Í Í 2m 3m + 1 = 2m 3m + 2m 1 = 6m2 + 2m The quotient 3m + 1 1checks. 2 1 2 1 2

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9x 5 12x 2 3x Example 2 Divide: - + 3x 2 Solution 9x5 - 12x2 + 3x 9x5 12x2 3x = - + Divide each term by 3x2. 3x2 3x2 3x2 3x2 1 = 3x3 4 Simplify. - + x 1 Notice that the quotient is not a polynomial because of the term . This expression is x called a rational expression—we will study rational expressions further in Chapter 7. Although the quotient of two polynomials is not always a polynomial, we may still check by multiplying. 1 1 Check: 3x2 3x3 4 = 3x2 3x3 3x2 4 3x2 - + x - + x 9x5 12x2 3x a b = 1- 2 + 1 2 a b Practice 16x6 + 20x3 - 12x 2 Divide: 4x2

8x 2y 2 - 16xy + 2x Example 3 Divide: 4xy Solution 8x2y2 - 16xy + 2x 8x2y2 16xy 2x = - + Divide each term by 4xy. 4xy 4xy 4xy 4xy 1 = 2xy - 4 + Simplify. 2y 1 1 Check: 4xy 2xy - 4 + = 4xy 2xy - 4xy 4 + 4xy 2y 2y 2 2 a b = 8x y1 - 216xy + 12x2 a b Practice 15x4y4 - 10xy + y 3 Divide: 5xy

Concept Check x + 5 In which of the following is simplified correctly? x 5 a. + 1 b. x c. x + 1 5

Objective 2 Using Long Division to Divide by a Polynomial To divide a polynomial by a polynomial other than a monomial, we use a process known as long division. Polynomial long division is similar to number long division, so we review long division by dividing 13 into 3660.

Helpful Hint 281 Recall that 3660 is called the 133660 dividend.

26 Í 2#13 = 26 106 Subtract and bring down the next digit in the dividend.

104 Í 8#13 = 104 20 Subtract and bring down the next digit in the dividend.

Answer to Concept Check: 13 1#13 = 13 a 7 Subtract. There are no more digits to bring down, so the remainder is 7.

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7 d remainder The quotient is 281 R 7, which can be written as 281 . 13 d divisor Recall that division can be checked by multiplication. To check a division problem such as this one, we see that 13 # 281 + 7 = 3660 Now we demonstrate long division of polynomials.

Example 4 Divide x 2 + 7x + 12 by x + 3 using long division. Solution

x 2 To subtract, change 2 x

x + 3x2 + 7x + 12 How many times does x divide x ? = x. the signs of these Í x -x2 - 3x Multiply: x x + 3 terms and add. + T Subtract and bring down the next term. 4x + 12 1 2 Now we repeat this process. x + 4 4x x + 3x2 7x 12 How many times does x divide 4x? = 4. + + x - 2 - To subtract, change x + 3x

the signs of these 4x + 12 terms and add. -Í - 4x + 12 Multiply: 4 x + 3 0 Subtract. The remainder is 0. 1 2 The quotient is x + 4. Check: We check by multiplying. divisor # quotient + remainder = dividend T T T T x + 3 # x + 4 + 0 = x2 + 7x + 12 The quotient checks. 1 2 1 2 Practice 4 Divide x2 + 5x + 6 by x + 2 using long division.

Example 5 Divide 6x 2 + 10x - 5 by 3x - 1 using long division.

Solution 2x + 4 6x2 2 = 2x, so 2x is a term of the quotient. 3x - 16x + 10x - 5 3x 2 + -6x - 2x T Multiply 2x 3x - 1 .

12x 5 Subtract and bring down the next term. - 1 2 -12x + 4 12x - = 4, multiply 4 3x - 1 x -1 3 Subtract. The remainder1 is -21. Thus 6x2 + 10x - 5 divided by 3x - 1 is 2x + 4 with a remainder of -1. This can be written as 1 2 1 2 1 2 6x2 + 10x - 5 -1 d remainder = 2x + 4 + 3x - 1 3x - 1 d divisor Check: To check, we multiply 3x - 1 2x + 4 . Then we add the remainder, -1, to this product. 1 21 2 3x - 1 2x + 4 + -1 = 6x2 + 12x - 2x - 4 - 1 6x2 10x 5 1 21 2 1 2 = 1 + - 2 The quotient checks.

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In Example 5, the degree of the divisor, 3x - 1, is 1 and the degree of the remainder, -1, is 0. The division process is continued until the degree of the remainder polynomial is less than the degree of the divisor polynomial. Writing the dividend and divisor in a form with descending order of powers and with no missing terms is helpful when dividing polynomials.

4x 2 + 7 + 8x 3 Example 6 Divide: 2x + 3 Solution Before we begin the division process, we rewrite 4x2 + 7 + 8x3 as 8x3 + 4x2 + 0x + 7 Notice that we have written the polynomial in descending order and have represented the missing x1-term by 0x. 4x2 - 4x + 6 3 2 2x + 38x + 4x + 0x + 7 - 3 - 2 8x + 12x -8x2 + 0x + 2 + - 8x - 12x 12x + 7 - - 12x + 18 -11 Remainder

4x2 + 7 + 8x3 -11 Thus, = 4x2 - 4x + 6 + . 2x + 3 2x + 3

Practice 11x 3 9x3 6 Divide: - + 3x + 2

2x 4 x 3 3x 2 x 1 Example 7 Divide: - + + - x 2 + 1 Solution Before dividing, rewrite the divisor polynomial x2 + 1 as x2 + 0x + 1 The 0x term represents the missing x1@term in the divisor. 2x2 - x + 1 2 x + 0x + 12x4 - x3 + 3x2 + x - 1 - 4 - 3 - 2 2x + 0x + 2x -x3 + x2 + x + 3 + 2 + - x - 0x - x x2 + 2x - 1 - 2 - - x + 0x + 1 2x - 2 Remainder

2x4 - x3 + 3x2 + x - 1 2x - 2 Thus, = 2x2 - x + 1 + . x2 + 1 x2 + 1

Practice 3x4 2x3 3x2 x 4 7 Divide: - - + + x2 + 2

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Example 8 Divide x 3 - 8 by x - 2. Solution: Notice that the polynomial x3 - 8 is missing an x2@term and an x-term. We’ll represent these terms by inserting 0x2 and 0x. x2 + 2x + 4 x - 2 x3 + 0x2 + 0x - 8 3 + 2 -x - 2x 2x2 + 0x - 2 + 2x - 4x 4x - 8 - + 4x - 8 0 x3 - 8 Thus, = x2 + 2x + 4. x - 2 Check: To check, see that x2 + 2x + 4 x - 2 = x3 - 8. Practice 1 21 2 8 Divide x3 + 27 by x + 3.

Vocabulary, Readiness & Video Check Use the choices below to fill in each blank. Choices may be used more than once. dividend divisor quotient 3 1. In , the 18 is the , the 3 is the , and the 6 is the . 618 x + 2 2. In , the x + 1 is the , the x2 + 3x + 2 is the , and the x + 2 is the . x + 1 x2 + 3x + 2 Simplify each expression mentally. a6 p8 y2 a3 3. 4. 5. 6. a4 p3 y a

Martin-Gay Interactive Videos Watch the section lecture video and answer the following questions. Objective 1 7. The lecture before Example 1 begins with adding two fractions with the same denominator. From there, the lecture continues to a method for dividing a polynomial by a monomial. What role does the monomial play in the fraction example? Objective 2 8. In Example 5, you’re told that although you don’t have to fill in missing powers in the divisor and the dividend, it really is a good idea to do so. Why? See Video 5.6

5.6 Exercise Set MyMathLab®

Perform each division. See Examples 1 through 3. 4 2 12x + 3x 20x3 - 30x2 + 5x + 5 1. 3. x 5 2 5 15x - 9x 8x3 - 4x2 + 6x + 2 2. 4. x 2

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15p3 + 18p2 -4b + 4b2 - 5 -3y + 2y2 - 15 5. 31. 32. 3p 2b - 1 2y + 5 14m2 - 27m3 6. 7m 4 5 Mixed Practice -9x + 18x 7. 6x5 Divide. If the divisor contains 2 or more terms, use long division. See Examples 1 through 8. 6x5 + 3x4 8. a2b2 - ab3 3x4 33. ab -9x5 + 3x4 - 12 9. m3n2 - mn4 3x3 34. mn 6a2 - 4a + 12 10. 8x2 + 6x - 27 -2a2 35. 2x - 3 4x4 - 6x3 + 7 11. 18w2 + 18w - 8 -4x4 36. 3w + 4 -12a3 + 36a - 15 12. 2x2y + 8x2y2 - xy2 3a 37. 2xy Find each quotient using long division. See Examples 4 and 5. 11x3y3 - 33xy + x2y2 x2 + 4x + 3 38. 13. 11xy x + 3 2b3 + 9b2 + 6b - 4 x2 + 7x + 10 39. 14. b + 4 x + 5 2x3 + 3x2 - 3x + 4 2x2 + 13x + 15 40. 15. x + 2 x + 5 5x2 + 28x - 10 3x2 + 8x + 4 41. 16. x + 6 x + 2 2x2 + x - 15 2x2 - 7x + 3 42. 17. x + 3 x - 4 10x3 - 24x2 - 10x 3x2 - x - 4 43. 18. 10x x - 1 2x3 + 12x2 + 16 9a3 - 3a2 - 3a + 4 44. 19. 4x2 3a + 2 6x2 + 17x - 4 4x3 + 12x2 + x - 14 45. 20. x + 3 2x + 3 2x2 - 9x + 15 8x2 + 10x + 1 46. 21. x - 6 2x + 1 30x2 - 17x + 2 3x2 + 17x + 7 47. 22. 5x - 2 3x + 2 4x2 - 13x - 12 2x3 + 2x2 - 17x + 8 48. 23. 4x + 3 x - 2 3x4 - 9x3 + 12 4x3 + 11x2 - 8x - 10 49. 24. -3x x + 3 8y6 - 3y2 - 4y 50. Find each quotient using long division. Don’t forget to write the 4y polynomials in descending order and fill in any missing terms. See 3 2 x + 6x + 18x + 27 Examples 6 through 8. 51. x + 3 x2 - 36 a2 - 49 25. 26. x3 - 8x2 + 32x - 64 x - 6 a - 7 52. x - 4 3 3 x - 27 x + 64 3 2 27. 28. y + 3y + 4 x - 3 x + 4 53. y - 2 1 - 3x2 7 - 5x2 29. 30. x + 2 x + 3 Copyright Pearson. All rights reserved.

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3x3 + 11x + 12 a + 7 54. 69. In which of the following is simplified correctly? See x + 4 7 the Concept Check in this section. 5 - 6x2 55. a a. a 1 b. a c. + 1 x - 2 + 7 3 - 7x2 5x + 15 56. 70. In which of the following is simplified correctly? x - 3 5 See the Concept Check in this section. Divide. a. x + 15 b. x + 3 c. x + 1 x5 + x2 71. Explain how to check a polynomial long division result when 57. x2 + x the remainder is 0. x6 - x4 72. Explain how to check a polynomial long division result when 58. the remainder is not 0. x3 + 1 73. The area of the following parallelogram is 10x2 + 31x 15 square meters. If its base is 5x 3 meters, find its Review and preview + + height. 1 Multiply each expression. See Section 5.3. 2 1 2 59. 2a a2 + 1 2 ? 60. -41a 3a -2 4 2 61. 2x x1 + 7x -2 5 (5x ϩ 3) meters 2 62. 4y1y - 8y - 42 74. The area of the top of the Ping-Pong table is 63. xy xy2 x2y 2 -31 + 7 2 + 8 49x + 70x - 200 square inches. If its length is 7x + 20 64. -9xy 4xyz + 7xy2z + 2 inches, find its width. 1 2 1 2 1 2 2 65. 9ab ab c + 4bc - 8 (7x ϩ 20) inches 1 2 ? 2 2 66. -7sr1 6s r + 9sr + 92rs + 8 1 2 Concept Extensions 67. The perimeter of a square is 12x3 + 4x - 16 feet. Find the length of its side. a a a a a 1 2 75. 18x10 - 12x8 + 14x5 - 2x3 , 2x3 b b b b b 76. 25y11 + 5y6 - 20y3 + 100y , 5y Perimeter is 1 2 3 (12x ϩ 4x Ϫ 16) feet 1 2

68. The volume of the swimming pool shown is 36x5 - 12x3 + 6x2 cubic feet. If its height is 2x feet and its width is 3x feet, find its length. 1 2

3x feet

2x feet

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Chapter 5 Vocabulary Check

Fill in each blank with one of the words or phrases listed below. term coefficient monomial binomial trinomial polynomials degree of a term distributive FOIL degree of a polynomial

1. A is a number or the product of numbers and variables raised to powers. 2. The method may be used when multiplying two binomials. 3. A polynomial with exactly three terms is called a . 4. The is the greatest degree of any term of the polynomial. 5. A polynomial with exactly two terms is called a . 6. The of a term is its numerical factor. 7. The is the sum of the exponents on the variables in the term. 8. A polynomial with exactly one term is called a . 9. Monomials, binomials, and trinomials are all examples of . 10. The property is used to multiply 2x x - 4 . 1 2

Chapter 5 Highlights

Definitions and Concepts Examples

Section 5.1 Exponents

n a means the product of n factors, each of which is a. 32 = 3 # 3 = 9 -5 3 = -5 -5 -5 = -125 1 4 1 1 1 1 1 1 2 = 1# #21# 21= 2 2 2 2 2 2 16 If m and n are integers and no denominators are 0, a b m n m n Product Rule: a # a = a + x2 # x7 = x2 + 7 = x9 m n mn 3 8 3#8 24 Power Rule: a = a 5 = 5 = 5 n n n 4 4 4 Power of a Product1 2 Rule: ab = a b 17y2 = 7 y n n 3 3 1 a2 a 1 x 2 x Power of a Quotient Rule: = n = b b 8 83 m a b a9 b a m n x Quotient Rule: = a - = x9 - 4 = x5 an x4 Zero Exponent: a0 = 1, a 0. 50 = 1, x0 = 1, x 0

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Definitions and Concepts Examples

Section 5.2 Adding and Subtracting Polynomials

A term is a number or the product of numbers and Terms variables raised to powers. 1 -5x, 7a2b, y4, 0.2 4 The numerical coefficient or coefficient of a term is its Term Coefficient numerical factor. 7x2 7 y 1 -a2b -1

A polynomial is a term or a finite sum of terms in which Polynomials variables may appear in the numerator raised to whole 3x2 2x 1 (Trinomial) number powers only. - + -0.2a2b - 5b2 (Binomial) A monomial is a polynomial with exactly 1 term. 5 A binomial is a polynomial with exactly 2 terms. y3 (Monomial) A trinomial is a polynomial with exactly 3 terms. 6 The degree of a term is the sum of the exponents on the Term Degree variables in the term. -5x3 3 3 (or 3x0) 0 2a2b2c 5 The degree of a polynomial is the greatest degree of any Polynomial Degree term of the polynomial. 5x2 - 3x + 2 2 7y + 8y2z3 - 12 2 + 3 = 5

To add polynomials, add or combine like terms. Add:

7x2 - 3x + 2 + -5x - 6 = 7x2 - 3x + 2 - 5x - 6 7x2 8x 4 1 2 1 2 = - - To subtract two polynomials, change the signs of the terms Subtract: of the second polynomial, then add. 17y2 - 2y + 1 - -3y3 + 5y - 6 17y2 2y 1 3y3 5y 6 1 = 2 -1 + + 2- + 17y2 2y 1 3y3 5y 6 = 1 - + +2 1 - + 2 = 3y3 + 17y2 - 7y + 7 Section 5.3 Multiplying polynomials

To multiply two polynomials, multiply each term of one Multiply: polynomial by each term of the other polynomial and then 2x 1 5x2 6x 2 combine like terms. + - + 1 21 " " 2 " " " " = 2x 5x2 - 6x + 2 + 1 5x2 - 6x + 2 10x3 12x2 4x 5x2 6x 2 = 1 - + 2 + 1 - + 2 = 10x3 - 7x2 - 2x + 2

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Definitions and Concepts Examples

Section 5.4 special Products

The FOIL method may be used when multiplying two Multiply: 5x - 3 2x + 3 binomials. Last S First S 1S S 21 2 F O I L 5x - 3 2x + 3 = 5x 2x + 5x 3 + -3 2x + -3 3

S OuterS S InnerS 2 1 21 2 = 110x21+ 152 x 1- 621x -2 9 1 21 2 1 21 2 = 10x2 + 9x - 9 Squaring a Binomial Square each binomial. 2 2 2 a + b = a + 2ab + b x + 5 2 = x2 + 2 x 5 + 52 2 1 2 = x + 10x + 25 2 2 2 1 2 1 21 2 a - b = a - 2ab + b 3x - 2y 2 = 3x 2 - 2 3x 2y + 2y 2 9x2 12xy 4y2 1 2 1 2 = 1 2- 1 +21 2 1 2 Multiplying the Sum and Difference of Two Terms Multiply: 2 2 a + b a - b = a - b 6y + 5 6y - 5 = 6y 2 - 52 36y2 25 1 21 2 1 21 2 = 1 2 - Section 5.5 negative Exponents and Scientific Notation

If a 0 and n is an integer, 1 1 5 3-2 = = ; 5x-2 = 1 32 9 x2 a-n = an x-2y -2 x4y-2 Rules for exponents are true for positive and negative Simplify: 5 = -10 integers. x x a b = x4 - -10 y-2 x14 1 2 = y2

A positive number is written in scientific notation if it Write each number in scientific notation.

is written as the product of a number a, 1 … a 6 10, 4 12,000" 1.2 10 and an integer power r of 10. = * 6

r - 0.00000568" 5.68 10 a * 10 = *

Section 5.6 Dividing Polynomials

To divide a polynomial by a monomial: Divide:

a + b a b 15x5 - 10x3 + 5x2 - 2x 15x5 10x3 5x2 2x = + = - + - c c c 5x2 5x2 5x2 5x2 5x2 2 = 3x3 - 2x + 1 - 5x -4 To divide a polynomial by a polynomial other than a 5x - 1 + 2x 3 monomial, use long division. + 2x + 310x2 + 13x - 7 - 2 - 10x + 15x -2x - 7 + + - 2x - 3 -4

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Chapter 5 Review

(5.1) State the base and the exponent for each expression. t 0 1 3 5 9 4 1. 7 2. -5 seconds second seconds seconds 4 6 3. 5 4. x 2 - 1 2 -16t + 4000

Evaluate each expression. 5. 83 6. -6 2 7. 62 8. 43 40 - 1- 2- 8b 9. 3b 0 10. 8b 1 2 Simplify each expression. 11. y2 # y7 12. x9 # x5 13. 2x5 -3x6 14. -5y3 4y4 15. x4 2 16. y3 5 1 21 2 1 21 2 17. 3y6 4 18. 2x3 3 1 2 1 2 x9 z12 19. 1 2 20. 1 2 x4 z5 a5b4 x4y6 21. 22. ab xy 40. The surface area of a box with a square base and a height of 3x4y10 2x7y8 5 units is given by the polynomial 2x2 + 20x. Fill in the table 23. 24. 2 12xy6 8xy2 below by evaluating 2x + 20x for the given values of x. 25. 5a7 2a4 3 26. 2x 2 9x x 1 3 5.1 10 27. -5a 0 + 70 + 80 28. 8x0 + 90 1 2 1 2 1 2 2x2 + 20x 1 2 Simplify the given expression and choose the correct result. 3x4 3 29. 4y a 27bx64 27x12 9x12 3x12 a. b. c. d. 5 64y3 64y3 12y3 4y3 5a6 2 30. x b3 x a 10ba12 25a36 25a12 a. b. c. d. 25a12b6 b6 b9 b6 Combine like terms in each expression. 41. 7a2 - 4a2 - a2 (5.2) Find the degree of each term. 42. 9y + y - 14y 2 2 31. -5x4y3 32. 10x3y2z 43. 6a + 4a + 9a 5 2 33. 35a bc 34. 95xyz 44. 21x2 + 3x + x2 + 6 45. 4a2b - 3b2 - 8q2 - 10a2b + 7q2 Find the degree of each polynomial. 46. 2s14 + 3s13 + 12s12 - s10 35. y5 + 7x - 8x4 36. 9y2 + 30y + 25 Add or subtract as indicated. 2 2 3 2 2 37. -14x y - 28x y - 42x y 47. 3x2 + 2x + 6 + 5x2 + x 38. 6x2y2z2 5x2y3 12xyz 5 4 3 2 2 + - 48. 12x + 3x + 42x +1 5x + 2 4x + 7x + 6 39. The Glass Bridge Skywalk is suspended 4000 feet over the 2 2 49. 1 -5y + 3 - 2y + 42 1 2 Colorado River at the very edge of the Grand Canyon. 50. 3x2 7xy 7y2 4x2 xy 9y2 Neglecting air resistance, the height of an object dropped 1 - 2 + 1 - 2 - + 2 2 from the Skywalk at time t seconds is given by the polynomial 51. 1 -9x + 6x + 2 2+ 41x - x - 1 2 16t2 4000. Find the height of the object at the given times. 6 2 6 2 - + 52. 8x - 5xy - 10y - 7x - 9xy - 12y (See top of next column.) 1 2 1 2 1 2 1 2 Copyright Pearson. All rights reserved.

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Translating (5.5) Simplify each expression. -2 -2 -4 -4 Perform the indicated operations. 87. 7 88. -7 89. 2x 90. 2x 53. Subtract 3x y from 7x 14y . 1 -3 -2 -2 - - 91. 92. 1 2 54. Subtract 4x2 + 8x - 7 from the sum of x2 + 7x + 9 5 3 2 1 2 1 2 and x + 4 . 93. 2a0 b 2-4 94. 6a-1 b 7-1 1 2 1 2 + - (5.3)Multiply1 each2 expression. Simplify each expression. Write each answer using positive expo- 55. 4 2a + 7 nents only. 5 4 56. 9 6a 3 x z 1 - 2 95. 96. -3 -4 57. -7x x2 + 5 x z 1 2 2 58. 8y 4y2 6 r -3 y- - 1 - 2 97. 98. -4 -5 59. 3a3 - 4a + 1 -2a r y 1 2 3 4 60. 6b3 4b 2 7b bc-2 4 x- y- -3 1 - + 21 2 99. 100. 61. 2x 2 x 7 bc-3 x-2y-5 1 + - 21 2 a 4 6b a5 5 b 62. 2x - 5 3x + 2 x- y- a b- 1 21 2 101. 102. 63. x 9 2 x2y7 a-5b5 1 - 21 2 64. x - 12 2 x5 + h 3 1 2 103. a6ma5m 104. 65. 4a 1 a 7 5 1 - 2 + 1 x 2 66. 6a 1 7a 3 2z 3 m + 2 m + 3 1 - 21 + 2 105. 3xy 106. a a 67. 5x + 2 2 1 21 2 Write 1each number2 in scientific notation. 68. 3x 5 2 1 + 2 107. 0.00027 108. 0.8868 69. x 7 x3 4x 5 1 + 2 + - 109. 80,800,000 110. 868,000 70. x 2 x5 x 1 1 + 21 + + 2 111. Google.com is an Internet search engine that handles 71. x2 2x 4 x2 2x 4 1 + 21 + +2 - 2,500,000,000 searches every day. Write 2,500,000,000 in 72. x3 4x 4 x3 4x 4 scientific notation. (Source: Google, Inc.) 1 + + 21 + - 2 73. x + 7 3 112. The approximate diameter of the Milky Way galaxy is 1 21 2 150,000 light years. Write this number in scientific notation. 74. 2x - 5 3 1 2 (Source: NASA IMAGE/POETRY Education and Public (5.4)1 Use special2 products to multiply each of the following. Outreach Program) 75. x + 7 2 76. x - 5 2 1 2 150,000 light years 77. 3x 7 2 1 - 2 78. 4x 2 2 1 + 2 79. 5x 9 2 1 - 2 80. 5x 1 5x 1 1 + 2 - 81. 7x 4 7x 4 1 + 21 - 2 82. a 2b a 2b 1 + 21 - 2 83. 2x 6 2x 6 1 - 21 + 2 84. 4a2 2b 4a2 2b 1 - 21 +2 Express1 each as21 a product of2 polynomials in x. Then multiply and Write each number in standard form. simplify. 113. 8.67 * 105 114. 3.86 * 10-3 4 5 85. Find the area of the square if its side is 3x - 1 meters. 115. 8.6 * 10- 116. 8.936 * 10 117. The volume of the planet Jupiter is 1.43128 1015 cubic 1 2 * kilometers. Write this number in standard notation. (Source: National Space Science Data Center)

(3x Ϫ 1) meters 86. Find the area of the rectangle.

(x Ϫ 1) miles (5x ϩ 2) miles

M05_MART8978_07_AIE_C05.indd 366 11/9/15 5:49 PM For use by Palm Beach State College only. Chapter 5 Getting Ready for the Test 367

118. An angstrom is a unit of measure, Mixed Review -10 equal to 1 * 10 meter, used Evaluate. for measuring wavelengths or the diameters of atoms. Write this num- 1 3 131. - ber in standard notation. (Source: 2 National Institute of Standards and a b Technology) Simplify each expression. Write each answer using positive Simplify. Express each result in standard form. exponents only. 2 3 5 8 * 104 132. 4xy x y 119. 8 * 104 2 * 10-7 120. 2 * 10-7 18x9 133. 1 21 2 1 21 2 27x3 (5.6) Divide. 2 3a4 3 x + 21x + 49 134. 121. 2 7x2 b a b 5a3b - 15ab2 + 20ab 135. 2x-4y3 -4 122. -5ab 3 6 1 a- b 2 a2 a a 136. 123. - + 4 , - 2 9-1a-5b-2 124. 4x2 + 20x + 7 , x + 5 1 2 1 2 Perform the indicated operations and simplify. a3 + a2 + 2a + 6 125. 1 2 1 2 137. 6x + 2 + 5x - 7 a - 2 138. y2 4 3y2 6 9b3 - 18b2 + 8b - 1 1 - - 2 +1 -2 126. 2 2 3b - 2 139. 18y - 3y2 + 11 - 3y 2 + 2 4 3 2 4x - 4x + x + 4x - 3 140. 5x2 2x 6 x 4 127. 1 + - 2 - 1 - - 2 2x - 1 2 141. 41x 7x + 3 2 1 2 -10x2 - x3 - 21x + 18 128. 142. 2x1 + 5 32x - 2 x - 6 2 3 2 143. x 3 x 4x 6 129. The area of the rectangle below is 15x - 3x + 60 1 - 21 + 2- 2 square feet. If its length is 3x feet, find its width. 144. 7x - 2 4x - 9 1 2 1 21 2 1 21 2 Use special products to multiply. 145. 5x + 4 2 Area is (15x3 Ϫ 3x2 ϩ 60) sq feet 146. 6x 3 6x 3 1 + 2 - 130. The perimeter of the equilateral triangle below is 1 21 2 21a3b6 + 3a - 3 units. Find the length of a side. Divide. 8a4 - 2a3 + 4a - 5 1 2 147. 2a3 x2 + 2x + 10 148. x + 5 4x3 + 8x2 - 11x + 4 Perimeter is 149. (21a3b6 ϩ 3a Ϫ 3) units 2x - 3

Chapter 5 Getting Ready for the Test

Matching Match the expression with the exponent operation needed to simplify. Letters may be used more than once or not at all. 1. x2 # x5 A. multiply the exponents 2. x2 5 B. divide the exponents 3. x2 x5 C. add the exponents 1 +2 x5 D. subtract the exponents 4. x2 E. this expression will not simplify

M05_MART8978_07_AIE_C05.indd 367 11/9/15 5:50 PM For use by Palm Beach State College only. 368 Chapter 5 Exponents and Polynomials

Matching Match the operation with the result when the operation is performed on the given terms. Letters may be used more than once or not at all. Given Terms: 20y and 4y 5. Add the terms A. 80y E. 80y2 6. Subtract the terms B. 24y2 F. 24y 7. Multiply the terms C. 16y G. 16y2 8. Divide the terms. D. 16 H. 5y I. 5 9. Multiple Choice The expression 5-1 is equivalent to 1 1 A. -5 B. 4 C. D. - 5 5

10. Multiple Choice The expression 2-3 is equivalent to 1 1 A. -6 B. -1 C. - D. 6 8

Matching Match each expression with its simplified form. Letters may be used more than once or not at all. 3 3 11. y + y + y A. 3y E. -3y 12. y # y # y B. y3 F. -y3 13. -y -y -y C. 3y 14. y y y D. 3y 1- -21 -21 2 -

Chapter 5 Test MyMathLab®

Evaluate each expression.

5 4 4 -3 16. Simplify by combining like terms. 1. 2 2. -3 3. -3 4. 4 2 2 Simplify each exponential1 2 expression. Write the result using 5x + 4xy - 7x + 11 + 8xy only positive exponents. y7 5. 3x2 5x9 6. Perform each indicated operation. - 2 y 3 2 3 17. 8x + 7x + 4x - 7 + 8x - 7x - 6 2 3 r1-8 21 2 x y 2 3 2 3 2 7. 8. 18. 5x + x + 5x - 2 - 8x - 4x + x - 7 -3 3 -4 1 2 1 2 r x y 19. Subtract 4x 2 from the sum of 8x2 7x 5 and 2 4 1 + 1 +2 + 6 x- y- a b x3 8 . 9. - 63x-3y7 1 2 1 2 1 2 Express each number in scientific notation. Multiply. 10. 563,000 11. 0.0000863 20. 3x + 7 x2 + 5x + 2 Write each number in standard form. 21. 3x2 2x2 3x 7 1 21- + 2 12. 1.5 10-3 13. 6.23 104 22. x 7 3x 5 * * 1+ - 2 14. Simplify. Write the answer in standard form. 1 1 23. 1 3x - 21 3x +2 1.2 * 105 3 * 10-7 5 5 24. 4x - 2 2 15. a. Complete1 the table21 for the polynomial2 2 3 25. 8x 3 2 4xy + 7xyz + x y - 2. 1 + 2 26. ¢x2 - 9b≤ ¢x2 + 9b≤ Term Numerical Degree of 1 2 Coefficient Term Solve.1 21 2 4xy2 27. The height of the Bank of China in Hong Kong is 1001 7xyz feet. Neglecting air resistance, the height of an object dropped from this building at time t seconds is given by x3y -2

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the polynomial -16t2 + 1001. Find the height of the ob- Divide. ject at the given times below. 4x2 + 24xy - 7x 29. 8xy t 0 1 3 5 2 seconds second seconds seconds 30. x + 7x + 10 , x + 5 16t2 1001 27x3 - 8 - + 31. 1 2 1 2 3x + 2 28. Find the area of the top of the table. Express the area as a product, then multiply and simplify. (2x ϩ 3) inches (2x Ϫ 3) inches

Chapter 5 Cumulative Review

1. Tell whether each statement is true or false. 15. Simplify each expression. a. 8 Ú 8 b. 8 … 8 a. 10 + x + 12 c. 23 … 0 d. 23 Ú 0 b. 3 7x - 1 2 2. Find the absolute value of each number. 1 16. Use the1 distributive2 property to write -2 x + 3y - z with- a. -7.2 b. 0 c. - out parentheses. 2 1 2 17. Find each product by using the distributive property to 3. Divide. Simplify all quotients if possible. ` ` 0 0 0 0 remove parentheses. 4 5 7 3 3 a. , b. , 14 c. , a. 5 3x + 2 5 16 10 8 10 b. 2 y 0.3z 1 -1 + 2 - 4. Multiply. Write products in lowest terms. c. - 9x + y - 2z + 6 3 7 1 5 1 2 a. # b. # 4 4 21 2 6 18. Simplify:1 2 6x - 1 - 2x - 7 19. Solve x 7 10 for x. 5. Evaluate the following. - 1 = 2 1 2 a. 32 b. 53 c. 24 20. Write the phrase as an algebraic expression: double a number, subtracted from the sum of the number and seven 3 2 d. 71 e. 5 7 21. Solve: x = 15 2 2x - 7y 6. x y Evaluate 2 for = 5 and = 1. 1 3 x 22. Solve: 2x + = x - 8 8 7. Add. ¢ ≤ x x a. -3 + -7 b. -1 + -20 23. Solve: 4 2 - 3 + 7 = 3 + 5 c. 2 10 24. Solve: 10 = 5j - 2 - + 1 - 2 1 2 1 2 25. Twice a number, added to seven, is the same as three subtract- 8. Simplify: 8 3 2 6 1 1 + 2 # - ed from the number. Find the number. 9. Subtract 8 from -4. 1 2 7x + 5 2 26. Solve: = x + 3 10. Is x = 1 a solution of 5x + 2 = x - 8? 3 11. Find the reciprocal of each number. 27. The length of a rectangular road sign is 2 feet less than 3 a. 22 b. three times its width. Find the dimensions if the perimeter 16 is 28 feet. 9 c. -10 d. - 28. Graph x 6 5. Then write the solutions in interval notation. 13 9 12. Subtract. 29. Solve F = C + 32 for C. 5 a. 7 40 b. 5 10 - - - - 30. Find the slope of each line. 13. Use an associative property to complete 1each 2statement. a. x = -1 a. 5 + 4 + 6 = b. y = 7 b. 1 2 5 - 1# # 2= 31. Graph 2 6 x … 4. 4 -3 + -8 14. Simplify:1 2 5 5 1 +2 -1 2 Copyright Pearson. All rights reserved. 1 2

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32. Recall that the grade of a road is its slope written as a per- 39. Find the slope of the line x = 5. 3 cent. Find the grade of the road shown. z2 # z7 40. Simplify: 9 1 2z 2 feet 41. Graph: x + y 6 7 2 2 20 feet 42. Subtract: 5y - 6 - y + 2 43. Use the product rule to simplify 2x2 3x5 . 1 2 1 2 - 33. Complete the following ordered pair solutions for the 44. Find the value of x2 when - 1 21 2 equation 3x + y = 12. a. x = 2 a. 0, b. x = -2 b. , 6 2 2 1 2 45. Add -2x + 5x - 1 and -2x + x + 3 . c. -1, 46. Multiply: 10x2 3 10x2 3 1 2 1 - 2 +1 2 3x + 2y = -8 47. Multiply: 2x - y 2 34. Solve1 the2 system: 1 21 2 2x - 6y = -9 48. Multiply: 10x2 3 2 1 +2 35. Graph the linear equatione 2x + y = 5. 49. Divide 6m2 2m by 2m. 1 + 2 x = -3y + 3 50. Evaluate. 36. Solve the system: 2x + 9y = 5 a. 5-1 37. Graph the linear equatione x = 2. b. 7-2 38. Evaluate. a. -5 2 b. 52 1- 2 c. 2 # 52

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