# Practise Qixereisas

et dr Section P.1 Algebraic Expressions, Mathematical Models, and Real Numbers 17 Practice Exercises In Exercises 1-16, evaluate each algebraic expression for the given value or values of the variable(s). 1. 7 + 5x, for x = 10 2, 8 + 6x, for x = 5 3, 6x — y,forx = 3andy = 8 4. 8x — y, for x = 3 and y = 4 5, x? + Зх, Юг х = & 6. x? + Sx, for x = 6 7, х? — 6x + 3,forx = 7 8. x? — Tx + 4,forx = 8 9, 4 + 5(x — T°, for x = 9 10. 6 + 5(x — 6)°, forx = 8 11. x* — 3(x — y), for x = 8 and y = 2 12. x? — 4(x — y), for x = 8 and y = 3 S(x + 2) 13. 3x 014 ja > for x = 10 14. rn 2) for x =9 15, — > ,forx = —2andy = 4 x+1 2х + у xy — 2x The formula 16. ‚for x = —2andy = 4 C = SE 32) expresses the relationship between Fahrenheit temperature, F, and Celsius temperature, C. In Exercises 17-18, use the formula to convert the given Fahrenheit temperature to its equivalent temperature on the Celsius scale. 17. 50°F 18. 86°F A football was kicked vertically upward from a height of 4 feet with an initial speed of 60 feet per second. The formula h = 4 + 60t — 162 describes the ball's height above the ground, h, in feet, t seconds after it was kicked. Use this formula to solve Exercises 19-20. 19. What was the ball’s height 2 seconds after it was kicked? 20. What was the ball’s height 3 seconds after it was kicked? In Exercises 21-28, find the intersection of the sets. 21. {1, 2, 3, 4} N {2, 4, 5} 22. {1,3,7} N {2, 3, 8) 23. {s,e,t} N {t, es] 24. {r,e,a, 1} N {L e, a, r} 25. (1,3, 5,7} N {2, 4, 6, 8, 10} 26. {0, 1,3, 5} N {-5, —3, —1} 27. {a, b,c, dy ND 28. {w,y,z) NY In Exercises 29-34, find the union of the sets. 29. {1, 2,3, 4} U {2,4,5) 30. {1, 3, 7, 8} U {2, 3, 8} 31. {1,3, 5,7} U {2, 4, 6, 8, 10} 32. (0, 1, 3, 5} U (2, 4, 6} 33, la,e,i,o, ul UJ 34. lem, p, 1 y) UD In Exercises 35-38, list all numbers from the given set that are a. natural numbers, b. whole numbers, e. integers, d. rational numbers, e. irrational numbers, E. real numbers. 35. {-9,-4,0,0.25, V3 ‚9.2, V100} 36. {-7,—0.6,0, 49, V50) 37. (-11,-2;0,0.75, V5, п, V64) 38. {-5,-03,0, V2, V4} 39. Give an example of a whole number that is not a natural number. | 40. Give an example of a rational number that is not an integer. 41. Give an example of a number that is an integer, a whole number, and a natural number. 42. Give an example of a number that is a rational number, an integer, and a real number. Determine whether each statement in Exercises 43-50 is true or false. 43. —13 = —2 44, —6 > 2 45. 4 = —7 46, —13 < —5 47. —п = —т 48. —3 > —13 49. 0 = —6 | 50. 0 = —13 In Exercises 51-60, rewrite each expression without absolute value bars. 51. 1300| | 52. |-203| 53. [12 — al 54. |7 — | 55. | V2 — 5| 56. | VS — 13) —3 —7 57, — 58, — -3| -7| 59. |-3| — |--7| 60. 1-5] — |--13] In Exercises 61-66, evaluate each algebraic expression for x = 2 andy = —5. 61. |x + у) 62. |x — yl 63. |x| + |y| 64. |x| — |y| 65. — ве. |+ М! vl X In Exercises 67-74, express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. 67. 2 and 17 68. 4 and 15 69. —2 and 5 70. —6 and 8 74. —19 and —4 | 72. —26 and —3 73. —3.6 and —1.4 74. —5.4 and —1.2 In Exercises 75-84, state the name of the property illustrated. 75. 6 + (+4) = —4) +6 76. 11-(7 + 4) = 11:7 + 11:4 77. 6 + (2 + 7) = (6 + 2) + 7 78. 6. (2:3) = 6. (3.2) 79. (2 + 3) + (4 + 5) = (4 + 5) + (2 + 3) 80. 7. (11:8) = (11: 8).7 | 81. 2(-8 + 6) = —16 + 12 82. —8(3 + 11) = —24 + (-88) 18 Chapter P Prerequisites: Fundamental Concepts of Algebra 1 . + 3) = 1,x = —3 3 (x + 3) (x ) X 84. (х + 4) + [-(х + 4)] = 0 In Exercises 85-96, simplify each algebraic expression, 85. 5(3x + 4) — 4 86. 2(5x + 4) — 3 87. 5(3x — 2) + 12x 88. 2(5x — 1) + 14x 89. 7(3y — 5) + 2(4y + 3) 90. 42y — 6) + 3(5y + 10) 91. 5(3y — 2) — (Ty + 2) 92. 4(5y — 3) — (6y + 3) 93. 7 — 4/3 — (4y - 3 95. 18x% + 4 — [6(x* — 2) + 5] 96. 14x* + 5 — [7(x* = 2) + 4] 94. 6 — 5/8 — (2y — 4)] In Exercises 97-102, write each algebraic expression without parentheses. 97. —(—14x) 99, —(2x — 3y — 6) 101. 3(3x) + [(4y) + (—4y)] 98. —(—17y) 100. —(5x — 13y — 1) 102. 5(2y) + [(-7x) + 7x] Practice Plus In Exercises 103-110, insert either <, >, or = in the shaded area to make a true statement. 103. |-6| |-3| 104. |-20| |-50] 105. 2 0.6 106. Е |-2.5| 109. то |-1| 110. |-2| e In Exercises 111-120, use the order of operations to simplify each expression. 111. 82 — 16 + 22.4 —3 112. 10° — 100 = 52.2 — 3 .7 — 22 “= . 113, 92-3 114. Maria [32 — (-2)F (12 — 3-2) 115. 8 — 3[-2(2 — 5) — 4(8 — 6)] 116. 8 — 3[-2(5 — 7) — 5(4 — 2)] 2(—2) — 4(=3) 6(-4) — 5(-3) 117. о 118. 9-10 (5 — 6)* — 213 — 7| 12 + 3.512? + 32| 119. —— 120. = 89 — 3:35 7+3-—6 In Exercises 121-128, write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. 121. A number decreased by the sum of the number and four 122. A number decreased by the difference between eight and the number 123. Six times the product of negative five and a number 124. Ten times the product of negative four and a number 125. The difference between the product of five and a number and twice the number 126. The difference between the product of six and a number and negative two times the number 127. The difference between eight times a number and six more than three times the number 128. Eight decreased by three times the sum of a number and six Application Exercises The maximum heart rate, in beats per minute, that you should achieve during exercise is 220 minus your age: 220 — a. — This algebraic expression gives maximum heart rate in terms of age, a. The following bar graph shows the target heart rate ranges for four types of exercise goals. The lower and upper limits of these ranges are fractions of the maximum heart rate, 220 — a. Exercises 129-130 are based on the information in the graph. Target Heart Rate Ranges for Exercise Goals Exercise Goal Boost performance ней уе! 1 | as a competitive athlete - Improve cardiovascular - ...... Sonditioning Lose weight | Improve overall health and - ‚reduce risk of heart attack - 2 1 3 7 4 39 5 2 5 10 5 1 1 Fraction of Maximum Heart Rate, 220 — a 129. Ifyour exercise goalis to improve cardiovascular conditioning, the graph shows the following range for target heart rate, H, in beats per minute: “ Lower limit of range > i, = 15(220 — a) ВЕ у À : Upper limit of range > H = = (220 — a) a. What is the lower limit of the heart range, in beats per minute, for a 20-year-old with this exercise goal? b. What is the upper limit of the heart range, in beats per minute, for a 20-year-old with this exercise goal? 130. If your exercise goal is to improve overall health, the graph shows the following range for target heart rate, H, in beats per minute: Upper limit of range. = H = =(220 — a) a. What is the lower limit of the heart range, in beats per minute, for a 30-year-old with this exercise goal? b. What 1s the upper limit of the heart range, in beats per minute, for a 30-year-old with this exercise goal? The privé "Tuition and Fees Sc The mot coll infc 131 132 13: The bar graph shows the average cost of tuition and fees at private four-year colleges in the United States. Average Cost of Tuition and Fees at Private Four-Year United States Colleges 27 pane ena 26273 eee 26 - 25 24 23 eee eas 22 -- 2 20 + 19 ee a 18 EEE , № 17 Ш 16 3 HN 15] Tuition and Fees (in thousands of dollars) 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Ending Year in the School Year Source: The College Board | The formula T = 26x* + 819x + 15,527 models the average cost of tuition and fees, T, at private U.S. colleges for the school year ending x years after 2000. Use this information to solve Exercises 131-132. 131. a. Use the formula to find the average cost of tuition and fees at private U.S. colleges for the school year ending in 2010. b. By how much does the formula underestimate or overestimate the actual cost shown by the graph for the school year ending in 2010? Use the formula to project the average cost of tuition and fees at private U.S. colleges for the school year ending in 2013. | 132. a. Use the formula to find the average cost of tuition and fees at private U.S. colleges for the school year ending in 2009. b. By how much does the formula underestimate or overestimate the actual cost shown by the graph for the school year ending in 2009? | c. Use the formula to project the average cost of tuition and fees at private U.S. colleges for the school year ending in 2012. 133. You had $10,000 to invest. You put x dollars in a safe, government-insured certificate of deposit paying 5% per year. You invested the remainder of the money in noninsured corporate bonds paying 12% per year. Your total interest earned at the end of the year is given by the algebraic expression | с + 0.05x + 0.12(10,000 — x). a. Simplify the algebraic expression. b. Use each form of the algebraic expression to determine your total interest earned at the end of the year if you invested $6000 in the safe, government-insured certificate of deposit. Section P.1 Algebraic Expressions, Mathematical Models, and Real Numbers 19 | 134. It takes you 50 minutes to get to campus. You spend t minutes walking to the bus stop and the rest of the time riding the bus. Your walking rate is 0.06 mile per minute and the bus travels at a rate of 0.5 mile per minute. The total distance walking and traveling by bus is given by the algebraic expression 0.067 + 0.5(50 — 7). a. Simplify the algebraic expression. b. Use each form of the algebraic expression to determine the total distance that you travel if you spend 20 minutes walking to the bus stop. 135. Read the Blitzer Bonus beginning on page 15. Use the formula 600n BAC=—— —— AC = TO.6n + 169) and replace w with your body weight. Using this formula and a calculator, compute your BAC for integers fromn = 1 to n = 10. Round to three decimal places. According to this model, how many drinks can you consume in an hour without exceeding the legal measure of drunk driving? Writing in Mathematics Writing about mathematics will help you learn mathematics. For all writing exercises in this book, use complete sentences to respond to the question. Some writing exercises can be answered in a sentence, others require a paragraph or two. You can decide how much you need to write as long as your writing clearly and directly answers the question in the exercise. Standard references such as a dictionary and a thesaurus should be helpful. 136. What is an algebraic expression? Give an example with your explanation. 137. If n is a natural number, what does b” mean? Give an example with your explanation. 138. What does it mean when we say that a formula models real-world phenomena? 139. What is the intersection of sets À and В? 140. What is the union of sets A and B? 141. How do the whole numbers differ from the natural numbers? 142. Can a real number be both rational and irrational? Explain © your answer, 143. If you are given two real numbers, explain how to determine ~ which is the lesser. | | Critical Thinking Exercises Make Sense? In Exercises 144-1 47, determine whether each statement makes sense or does not make sense, and explain your reasoning. 144. My mathematical model describes the data for tuition and fees at public four-year colleges for the past ten years extremely well, so it will serve as an accurate prediction for the cost of public colleges in 2050. 145. A model that describes the average cost of tuition and fees at private U.S. colleges for the school year ending x years after 2000 cannot be used to estimate the cost of private education for the school year ending in 2000. 20 Chapter P Prerequisites: Fundamental Concepts of Algebra 146. The humor in this cartoon is based on the fact that the football will never be hiked. HUTT ONE... HUTT Two... HUTY THREE... POINT 141592653589 79323846204 358327050288419710939937510582 OTTHIMEI230 7810 002562089980 280248253421170679821480B65132 || B23000470938U4G09550SB2231725 |! 35408128 48111745028410270193 lil BEZ110855964462294,, OM... 4, UM... [1 WHOSE IDEA WAS BALL ON PI, AGAIN? м} 7 To HIKE THE YOURS, Now KEEP GOING. Foxtrot © 2003, 2009 by Bill Amend/Used by permission of Universal Uclick. All rights reserved. 147. Just as the commutative properties change groupings, the associative properties change order. In Exercises 148-155, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 148. Every rational number is an integer. 149. Some whole number not int 150. Some rational numbers are not positive. 151. Irrational numbers cannot be negative. 152. The term x has no coefficient. 153. 5 + 3(x — 4) = 8(x — 4) = 8x - 32 154. —х — х = —х + (-x) = 0 155. x — 0.02(x + 200) = 0.98x — 4 In Exercises 156—158, insert either < or > in the shaded area between the numbers to make the statement true. 156. V2 15 | 157. — 3.14 Tr —3,5 Proview Exercises Exercises 159-161 will help you prepare for the material covered in the next section. 159. In parts (a) and (b), complete each statement. a. b* b> = (b-b-b-bY(b-b-b) = b. b> b> =(bb-b-b-b)b-b-b-b+b) =p’ c¢. Generalizing from parts (a) and (b), what should be done with the exponents when multiplying exponential expressions with the same base? 160. In parts (a) and (b), complete each statement. Ь’ Be B-Wbbbb p? “6 BU DE BBb-bbbbb bh. — ES — b’ bh? By ¢. Generalizing from parts (a) and (b), what should be done with the exponents when dividing exponential expressions with the same base? 161. If 6.2 is multiplied by 10°, what does this multiplication do to the decimal point in 6.2? Use the product rule. Use the quotient rule. - Use the zero-exponent rule. Use the negative- exponent rule. Use the power rule. Find the power of a product. Find the power of a quotient. Simplify exponential expressions. Use scientific. notation. x Bigger than the biggest thing ever and then some. Much bigger than that in fact, | really amazingly immense, a totally stunning size, real wow, that's big”, time Gigantic multiplied by colossal multiplied by staggeringly huge is the sort of concept we're trying to get across here. Douglas Adams, The Restaurant at the End of the Universe Although Adams’s description may not quite apply to this $15.2 trillion national debt, exponents can be used to explore the meaning of this “staggeringly huge” number. In this section, you will learn to use exponents to provide a way of putting large and small numbers in perspective. The Product and Quotient Rules We have seen that exponents are used to indicate repeated multiplication. Now consider the multiplication of two exponential expressions, such as b* - b*. We are multiplying 4 factors of b and 3 factors of b. We have a total of 7 factors of b: i" practice Exercises wa wg A Bas Eh fes Fa # Evaluate each exponential expression in Exercises 1-22. 1. 5-2 2. 6-2 3. (-2)° 4. Gr 5, —2° 6. —2* 7. (-3)° 8. (-9)° 9, —3° 10. —9° 11. 47° 12. 27° 13, 2-7 14. 33.3? 15. (2) 16. (3%) 28 3% 17. 7 18. a 19. 372-3 20. 273-2 2° 34 21.75 22. 37 Simplify each exponential expression in Exercises 23-04. 23. x”y 24. ху” 25. x%° 26. x'y 27. xx’ 28. xx’ 29. x >. x 30. xx” 31. (x) 32. (xy 33. (x7) 34. (> x14 y 35. 7 36. i 14 30 X X 37. = 38. io 39. (83 40. (6x7 De I 6 3 a. (4) a. (-9) X У 43. (-32Y 44. (-3х“уб) 45. (3х“)(2х”) 46. (11x°)(9x'%) 47. (-9x yX-2x y") 48.. (-Sx*y)(—6x'y'1) 8x” 20x% 49. — 50. —— 2x* 10х° 13,4 141.6 SL. 25a”b 82, Ba —5a*b) —7a'b 53 14b” 54 205*9 764 " 105% 55. (43) 56. (10) 5 24x°y° 10x*y° 7. 12755 58. E X y xy 3X-2 &\73 59, E 60. (=) y y — 41,2 \3 _ 147.8\3 а. ( 15% 6 ( 30a 2) Sap 1007” 63. Section P.2 Exponents and Scientific Notation 33 ( 305b? Y A) 6 120 > In Exercises 65-76, write each number in decimal notation without the use of exponents. 65. 67. 69. 71. 73. 75. 3.8 x 10° 6 x 1074 —7,16 x 10° 79x10" —4,15 x 10° —6.00001 x 10% 66. 68. 70. 72. 74. 76. 9.2 x 10? 7 x 105 —8.17 x 10% 6.8 x 107 —3.14 x 10% —7.00001 x 109 In Exercises 77-86, write each number in scientific notation. 77. 79. 81. 83. 85. 32,000 638,000,000,000,000,000 —5716 0.6027 —0.00000000504 78. 80. 82. 84. 86. 64,000 579,000,000,000,000,000 —3829 0.6083 —0.00000000405 In Exercises 87-106, perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. 87. 89. 91. 93. 95. 97. 99. 101. 103. 105. (3 xX 1092.1 x 10% (1.6 x 105(4 x 10714) (6.1 x 1082 x 10) (4.3 x 10%)(6.2 x 10% 8.4 x 108 4 X 10° 3.6 x 10% 9 x 107 4.8 x 1072 2.4 x 106 24 x 107 4.8 X 10% 480,000,000,000 0.00012 0.00072 x 0.003 0.00024 Practice Plus 98. 100. 102. 104. 106. . (2 X 104.1 x 10°) . (14 x 105)(3 x 10715) . (5.1 x 1053 x 1079 . (8.2 X 10%)(4.6 x 10% 6.9 x 10° 3 x 10° 1.2 x 10° 2 x 107 7.5 х 107 2,5 x 10° 1.5 x 1072 3 x 1076 282,000,000,000 0.00141 66,000 x 0.001 0.003 x 0.002 In Exercises 107-114, simplify each exponential expression. Assume that variables represent nonzero real numbers. 107. (xy) (yy 109. (2х yz 5) (2x) 34,5 2 xy Z 111. | ————— (= ~4 =) y © 108. (ay y” (xy) 110. 3x*y7 3)” x*y°7° —4 132. \ ~——— х yz 34 Chapter P Prerequisites: Fundamental Concepts of Algebra 1 13 (21х2у (2x*y)*(16x3 —3 42 (2x7y”) 1 4 CE 0 A AZ (2x*y5) Application Exercises The bar graph shows the total amount Americans paid in federal taxes, in trillions of dollars, and the U.S. population, in millions, from 2007 through 2010. Exercises 115-116 are based on the numbers displayed by the graph. Federal Taxes and the United States Population Federal Taxes Collected Population $3.00 350 Federal Taxes Collected (trillions of dollars) Sources: 115. a. с. 116. а. с $2.50 ---- $2.00 $1.50 $1.00 $0.50 2.57 303 306 308 309 250 200 1 150 Population (millions) I 100 2007 2008 2009 2010 Year Internal Revenue Service and U.S. Census Bureau In 2010, the United States government collected $2.17 trillion in taxes. Express this number in scientific notation. In 2010, the population of the United States was approximately 309 million. Express this number in scientific notation. Use your scientific notation answers from parts (a) and (b) to answer this question: If the total 2010 tax collections were evenly divided among all Americans, how much would each citizen pay? Express the answer in decimal notation, rounded to the nearest dollar. In 2009, the United States government collected $2.20 trillion in taxes. Express this number in scientific notation. In 2009, the population of the United States was approximately 308 million. Express this number in scientific notation. Use your scientific notation answers from parts (a) and (b) to answer this question: If the total 2009 tax collections were evenly divided among all Americans, how much would each citizen pay? Express the answer in decimal notation, rounded to the nearest dollar. In the dramatic arts, ours is the era of the movies. As individuals and as a nation, we've grown up with them. Our images of love, war, family, country — even of things that terrify us—owe much to what we’ve seen on screen. The bar graph at the top of the next column quantifies our love for movies by showing the number of tickets sold, in millions, and the average price per ticket for five selected years. Exercises 117-118 are based on the numbers displayed by the graph. United States Film Admissions and Admission Charges 1600 1400 1190 1210 E 2 8 ? 1200 mg earns E e AA aE $6 = Е © 1000 |- ren 40 0 29= = $5 5 Q “ Ë chs |. mn 9 << 800 $4 E gg 600 еее ее $3 > = 8 = < = 400 FE $2 £ = > = < 200 1990 1995 2000 2005 2010 Year Source: Motion Picture Association of America 117. Use scientific notation to compute the amount of money that = the motion picture industry made from box-office receipts - in 2010. Express the answer in scientific notation. 118. Use scientific notation to compute the amount of money that the motion picture industry made from box office receipts in 2005. Express the answer in scientific notation. 119. The mass of one oxygen molecule is 5.3 X 10% gram. Find ; the mass of 20,000 molecules of oxygen. Express the answer in scientific notation. 120. The mass of one hydrogen atom is 1.67 x 107% gram. Find - the mass of 80,000 hydrogen atoms. Express the answer in | ; scientific notation. 121. There are approximately 3.2 X 107 seconds in a year. 3 According to the United States Department of Agriculture, Americans consume 127 chickens per second. How many chickens are eaten per year in the United States? Express the answer in scientific notation. 122 » year. Express the answer in scientific notation. Writing in Mathematics 123. Describe what it means to raise a number to a power. In … your description, include a discussion of the difference between —5* and (-5. 124. Explain the product rule for exponents. Use 2° - 2° in your explanation. 125. Explain the power rule for exponents. Use (3° in your explanation. 8 126. Explain the quotient rule for exponents. Use — in your Е explanation. 127. Why is (—3x")(2x*) not simplified? What must be done to simplify the expression? 128. How do you know if a number is written in scientific notation? 129. Explain how to convert from scientific to decimal notation and give an example. 130. Explain how to convert from decimal to scientific notation ; | and give an example. 131. Refer to the Blitzer Bonus on page 32. Use scientific notation | | to verify any three of the bulleted items on ways to spend $1 trillion. Convert 365 days (one year) to hours, to minutes, and, finally, to seconds, to determine how many seconds there are ina - Ci sia ret 13 13 13 13 In or pr 13 13 14 14 14 14 14 14 Average Frice per licket Critical Thinking Exercises Make Sense? In Exercises 132-135, determine whether each statement makes sense or does not make sense, and explain your reasoning. 132. There are many exponential expressions that are equal to 36x1, such as (6x, (6x>(6x), 3603), and A. 133. If 572 is raised to the third power, the result is a number between 0 and 1. 134. The population of Colorado is approximately 4.6 x 10%, 135. I just finished reading a book that contained approximately 1.04 x 10° words. In Exercises 136-143, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 136. 472 < 4% 137. 5? > 25 138. (-2)* = 27° 139. 52.57? > 25.27 8 х 10% 140. 534.7 = 5.347 x 10° 141 = 2 x 10% 4 x 107 142. (7 x 10°) + (2 x 10°) = 9 x 10° 143. (4 x 10%) + (3 X 10%) = 4.3 x 10° 144. The mad Dr. Frankenstein has gathered enough bits and pieces (so to speak) for 27 + 27 of his creature-to-be. Write a fraction that represents the amount of his creature that must still be obtained. 145. If b* = MN, b“ = M, and b? = N, what is the relationship among À, C, and D? 146. Our hearts beat approximately 70 times per minute. Express in scientific notation how many times the heart beats over —a lifetime of 80 years. Round the decimal factor in your scientific notation answer to two decimal places. Evaluate square roots. Simplify expressions of the form Va. Use the product rule to simplify square roots. Use the quotient rule to simplify square roots. Add and subtract square Rationalize denominators. Evaluate and perform operations with higher roots. Understand and use rational exponents. This photograph mathematical models used by Albert Einstein at a lecture on relativity. Notice the radicals that appear in many of the formulas. Among these models, there 1s one describing how an astronaut in a moving spaceship ages more slowly than friends who remain on Earth. No description of your world can be complete without roots and — Tadicals. In this section, in addition roots. to reviewing the basics of radical — expressions and the use of rational exponents to indicate radicals, you will see how radicals model time dilation for a futuristic high- speed trip to a nearby star. Section P.3 Radicals and Rational Exponents 35 Group Exercise 147. Putting Numbers into Perspective. A large number can be put into perspective by comparing it with another number. For example, we put the $15.2 trillion national debt (Example 12) and the $2.17 trillion the government collected in taxes (Exercise 115) by comparing these numbers to the number of U.S. citizens. For this project, each group member should consult an almanac, a newspaper, or the Internet to find a number greater than one million. Explain to other members of the group the context in which the large number is used, Express the number in scientific notation. Then put the number into perspective by comparing it with another number. Preview Exercises Exercises 148-150 will help you prepare for the material covered in the next section. 148. a. Find V16- V4. b. Find V16-4. с. Based on your answers to parts (a) and (b), what can you conclude? 149. a. Use a calculator to approximate V300 to two decimal places. b. Use a calculator to approximate 10V3 to two decimal places. €. Based on your answers to parts (a) and (b), what can you conclude? 150. a. Simplify: 21x + 10x. b. Simplify: 21V2 + 10V2. shows 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Fill in each blank so that the resulting statement is true. 1. The symbol V is used to denote the nonnegative, or, Square root of a number. 2, \/64 = 8 because _.. = 64. 3 Va= 4. The product rule for square roots states that if à and b are nonnegative, then Vab = 5. The quotient rule for square roots states that if à and b - are nonnegative and b # 0,then „= 6. 8V3 + 10V3 = 7. V3 + V5 = V3 + V95-3 = V3+_ V3 = 8. The conjugate of 7 + V3is —. a San E=Xercisoes Evaluate each expression in Exercises 1-12, or indicate that the root is not a real number. 1. V36 2. V25 3, —V36 4 —V25 5. V-36 6. V-25 7. V25 — 16 8. \/144 + 25 9. V25 — V16 10. V144 + V25 11. V(-13Y 12. V(-17Y Use the product rule to simplify the expressions in Exercises 13-22. In Exercises 17-22, assume that variables represent nonnegative real numbers. 13. V50 14. V27 15. V 45x? 16. \/125х? 17. \2х‹ \У/бх 18. V10x- V 8x 19. Vx3 20. Vy? 21. V2x?- V6r 22. V6x- Vx? Use the quotient rule to simplify the expressions in Exercises 23-32. Assume that x > O. 1 1 23. 31 24. 29 49 121 APTE 26. ./— 25 16 9 | 3 73 m. 48x 28. 72x V3x V8x 4 4 29, 150x 30. 24x V3x V3x V200x° V500x° 3, —— 32. | 10x” V 10x” In Exercises 33-44, add or subtract terms whenever possible. 33. 7V3 + 6V3 34. 8V5 + 11V5 35, 6V 17x — 8V 17x 36. 4V 13x — 6V 13x 9. 10. 11. 12 + 13. 14. 37, 39. 41. 43. 44. We rationalize the denominator of by 5 vid - V2 multiplying the numerator and denominator by : In the expression V64, the number 3 is called the and the number 64 1s called the Y-32 = —2 because = —32, If n is odd, Va” = ___. ln iseven, Va? =__. a" = 16 = (V16)* = (ys = . V20+6V5 \/8 + 3\/2 38 V50x — V8x 40. V63x — V28x 3/18 + 5V50 42. 4V12 - 2V75 3V8 — V32 + 3V72 — V5 3V54 — 224 — V96 + 463 In Exercises 45-54, rationalize the denominator. 45. 47. 49. 51. 53. 1 2 —— 46. —— V7 10 V2 48, L V5 V3 3 50, — — 3 + V11 3 + V7 7 5 — 52, ——— V5 —2 V3 —1 6 sa 1 V5 + V3 V7 — V3 Evaluate each expression in Exercises 55—66, or indicate that the root is not a real number. 55. 58. 61. 64. V125 56. Y8 57. Y-8 Y-125 59. Y-16 60. Y-81 V(-3)* 62. V(-2)* 63. V(-3) 2) 65. \Y_ 66. A; Simplify the radical expressions in Exercises 67-74 if possible. I DS al N Ÿ32 68. Y150 Vt 70. Yx5 Vo. Ve - 72. V12-V4 Y 64x* \162х5 74. In E 75. 77. 79. 80. 81. mE calc 83. 85. 67, 89, In! 91 93 9¢ in. 101 10: 10: 10 Pr In 10 11 In va Ц 11 uu In Exercises 75-82, add or subtract terms whenever possible, „в, 4N/2 + 3V/2 76. 6V/3 + 2V/3 77. 5/16 + №54 78. 3V/24 + V81 79. Ys4xy? — y V128x во. Y24xy? — y V8lx 81. V2 + 8 82. V3 + V15 In Exercises 83-90, evaluate each expression without using a calculator. 1 1 83. 36 84. 121? 1 1 85. 8° 86. 27° 2 2 87. 125° 88. 8° 4 LS 89. 32 ° 90. 16 ? In Exercises 91-100, simplify using properties of exponents. 1 1 2 3 91. (7x3) (2x4) 92. (3x3) (4x*) 1 3 20x? 72x* 93. — 94, UT Sx* Ox 3 Z\3 4 \ 5 0s. (x3) 96. (x5) 1 1 97. (25х“у“)? 98. (125x°y%)3 Log 14 зу“) (2y5) 99. 73 100. — 5 y? - 10 In Exercises 101-108, simplify by reducing the index of the radical. 101, №5? 102. №7? 103. Vx* 104. Vx 105, Vx* 106. Vx 107. Vx? 108. Vx%y® Practice Plus In Exercises 109-110, evaluate each expression. 109. V/16 + V625 110. VV V169 + V9 + V 1000 + Y216 In Exercises 111-114, simplify each expression. Assume that all variables represent positive numbers. Lol 3 34° 111. (49x 2y4) 2 xy 2) 112, (8x 6y3)3 (xy 75) _5 175 78 x “у 1 1 _7 q 2 113. ’ xy 4 114. _3 _2 x 4 y 4 Application Exercises 115, The popular comic strip Fox Trot follows the off-the-wall lives of the Fox family. Youngest son Jason is forever obsessed by his love of math. In the math-themed strip shown at the top of the next column, Jason shares his opinion in a coded message about the mathematical abilities of his sister Paige. Section P.3 Radicals and Rational Exponents 49 by Bill Amend t-{1-13-5-10-2-15-18-13-23-6-11-17-11-12-22-11-12-10 Key: * HAL De VUN R=" TE 5: (542x2)» cla VE = *Зсо5 = (9х+9х) + 3х Ves? Mz (9xi1)-(7x11) № 2079 м Y400 X= 920% 512 THe +3 or ve + ++ т = 89 +9 = tf a 7: АНН SE 9x dx Te La Foxtrot O 2003, 2009 by Bill Amend/Used by permission of Universal Uclick. All rights reserved. Solve problems A through Z in the left panel. Then decode Jason Fox’s message involving his opinion about the mathematical abilities of his sister Paige shown on the first line. | Hints: Here 1s the solution for problem C and partial solutions for problems Q and U. These are: = ain 7% — a o. from С = sin > = sin 90° = 1 trigonometry. 2 2 | = [9x2dx = 3 | = 3:22 — 3:0? = This ВО 0 = -3 сов т = —3 сов 180° = -3(-1) = rom calculos. 116. Note: “The comic strip FoxTrot is now printed in more than one thousand newspapers. What made cartoonist Bill Amend, a college physics major, put math in the comic? “I always try to use math in the strip to make the joke accessible to anyone,” he said. “But if you understand math, hopefully youll like it that much more!” We highly recommend the math humor in Amend's FoxTrot collection Math, Science, and Unix Underpants (Andrews McMeel Publishing, 2009). America is getting older. The graph shows the projected elderly U.S. population for ages 65-84 and for ages 85 and older. Projected Elderly United States Population Ages 65-84 Ages 85+ 80 Projected Population (millions) 2010 2020 2030 Year 2040 2050 Source; U.S. Census Bureau 50 Chapter P Prerequisites: Fundamental Concepts of Algebra 117 ” 118. The formula Е = 5Vx + 34.1 models the projected number of elderly Americans ages 65-84, E, in millions, x years after 2010. a. Use the formula to find the projected increase in the number of Americans ages 65-84, in millions, from 2020 to 2050. Express this difference in simplified radical form. ©. Use a calculator and write your answer in part (a) to the nearest tenth. Does this rounded decimal overestimate or underestimate the difference in the projected data shown by the bar graph on the previous page? By how much? The early Greeks believed that the most pleasing of all rectangles were golden rectangles, whose ratio of width to height 1s W 2 h 5-1 The Parthenon at Athens fits into a golden rectangle once the triangular pediment is reconstructed. Rationalize the denominator of the golden ratio. Then use a calculator and find the ratio of width to height, correct to the nearest hundredth, in golden rectangles. Use Einstein's special-relativity equation 2 R,= Rp /1 — (2). described in the Blitzer Bonus on page 47, to solve this exercise. You are moving at 90% of the speed of light. Substitute 0.9c for v, your velocity, in the equation. What is your aging rate, correct to two decimal places, relative to a friend on Earth? If you are gone for 44 weeks, approximately how many weeks have passed for your friend? The perimeter, P, of a rectangle with length 1 and width w is given by the formula P = 21 + Qw. The area, À, is given by the formula À = lw. In Exercises 119-120, use these formulas to find the perimeter and area of each rectangle. Express answers in simplified radical form. Remember that perimeter is measured in linear units, such as feet or meters, and area is measured in square units, 119. T1 2 such as square feet, ft”, or square meters, m”. 120. | 175 feet 4/20 feet | 2/20 feet рр Writing in Mathematics 121. Explain how to simplify V10- V5. 122. Explain how to add V3 + V1. 123. 124. 125. 126. 127. 128. Describe what it means to rationalize a denominator. Use 1 1 both 7 and 5 7 What difference is there in simplifying V(-5Y a VS? What does a” mean? Describe the kinds of numbers that have rational fifth roots. in your explanation. Why must a and b represent nonnegative numbers when we write Va- Vb = restriction in the case of Ya- Yb = Vab? Explain. Read the Blitzer Bonus on page 47. The future is now: You ê have the opportunity to explore the cosmos in a starship traveling near the speed of light. The experience will enable | you to understand the mysteries of the universe in deeply personal ways, transporting you to unimagined levels of knowing and being. The downside: You return from your two-year journey to a futuristic world in which friends and loved ones are long gone. Do you explore space or stay here on Earth? What are the reasons for your choice? | aT e i ay Fr pd + M à ge # & цей LE Wat wing E y ea do pep ss Biar AP ott NT E ge 0 we DUO HEHE ESSEN - Ro . Make Sense? In Exercises 129-132, determine whether each statement makes sense or does not make sense, and explain your reasoning. 129. 130. 131. 132. The joke in this Peanuts cartoon would be more effective if … Woodstock had rationalized the denominator correctly in the last frame. YOURE LUCKY, DO YOU | [YOU DON'T HAVE TO KNOW KNOW THAT BIRD ? YOU'RE | | ABOUT RATIONALIZING THE LUCKY BECAUSE YoU DONT | | DENOMINATOR AND DUMB HAVE TO STUDY MATH! THINGS LIKE THAT 77 | Beau В © 1979 United Feature Syndicate, Ine, Peanuts © 1978 Peanuts Worldwide LLC, Used by permission of Universal Uclick. All rights reserved. Using my calculator, I determined that 67 = 279,936, so 6 must be a seventh root of 279,936. I simplified the terms of 220 + 4\/75, and then I was able to add the like radicals. m When I use the definition for a”, I usually prefer to first raise a to the m power because smaller numbers are involved. In Exercises 133-136, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 133. 1 1 72. 7 = = 49 134. 8 3 =—2 ab? Is it necessary to use this ; 135. 136. ml 137 138 139 140 141 and f ; ots. ; hen - this - You ; hip. ble ply ur nd ère La Wd Use. : | of - 135. The cube root of —8 is not a real number. vo V10 136.5 74° 4 In Exercises 137-1 38, fil in each box to make the statement true. 137. (5+ Y )5-V )=2 138, V x = 5x’ Section P.4 Polynomials 51 b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, “Hold on! It is your birthday, so why 4 8 3 + 27° › › not take — —— of the cake? I'll eat half of what’s left 16 4 + 27! over.” How much of the cake did the professor eat? 139. Find the exact value of J 13 + \/2 + — — — the use of a calculator. "140. Place the correct symbol, > or <, in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. b. V7 + V18 V7 + 18 141. a. A mathematics professor recently purchased a birthday cake for her son with the inscription 11 а. 32 33 5 3 1 Happy(22-24 + 24)th Birthday. How old is the son? Understand t the he vocabulary of polynomials. о Add and subtract polynomials. Multiply polynomials. Use FOIL in polynomial multiplication. Use special products in polynomial multiplication. Perform operations with ~ polynomials in several — variables. Understand the vocabulary of polynomials. without SR Preview Praercisas Exercises 142-144 will help you prepare for the material covered In the next section. 142. Multiply: (2x*y")(5x*y7). 143. Use the distributive property to multiply: 2x*(8x* + 3x). 144. Simplify and express the answer in descending powers of x: 2x(x* + 4x + 5) + 3 + 4х + 5). Can that be Axl, your author’s yellow lab, sharing a special moment with a baby chick? And if it is (it is), what possible relevance can this have to polynomials? An answer 1s promised before you reach the Exercise Set. For now, we open the section by defining and describing polynomials. How We Define Polynor Y: More education results in a higher income. The mathematical models Old Dog...New Chicks M = 0.6x> + 285x? — 2256x + 15,112 and W = —1.2x" + 367x° — 4900x + 26,561 describe the median, or middlemost, annual income for men, M, and women, W, who have completed x years of education. Well be working with these models and the data upon which they are based in the Exercise Set. The algebraic expressions that appear on the right sides of the models are examples of polynomials. A polynomial is a single term or the sum of two or more terms containing variables with whole-number exponents. The polynomials above each contain four terms. Equations containing polynomials are used in such diverse areas as science, business, medicine, psychology, and sociology. In this section, we review basic ideas about polynomials and their operations. How We Describe Polynomials Consider the polynomial 7х° — 9х? + 13х — 6. Tited at ; ; cessive, lylabs ts. ng of ; | of = we al a gra вы Ÿ A Practice слег ее In Exercises 1-4, is the algebraic expression a polynomial? If it is, write the polynomial in standard form. 1. 2x + 3x* — 5 2. 2x + 3x * — 5 X . ‘In Exercises 5-8, find the degree of the polynomial. 6. — 4x) + 7x? — 11 8. x? — 8x7 + 15x* + 91 5. 3х? — 5х + 4 7, х? — 4x + 9х — 12х" + 63 In Exercises 9-14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. 9. (—6x> + 5x? — 8x + 9) + (17x + 2x* — 4x — 13) 10. (—7x* + 6x? — 11x + 13) + (19x? — 11x* + 7x — 17) 1. (17x — 5x? + 4x — 3) — (5x? — 9x? — 8x + 11) 12. (18x* — 2x* — 7x + 8) — (9x* — 6x* — Sx + 7) 13. (5x? — 7x — 8) + (2x* — 3x + 7) — (x* — 4x — 3) 14. (8x? + Tx — 5) — (3x* — 4x) - (—6x? — 5х? + 3) In Exercises 15-58, find each product. 15. (x + 1)(x* — x + 1) 16. (x + 5)(x* — 5x + 25) 17. (2x — 3)(x* — 3x + 5) 18. (2x — 1(x* — 4x + 3) 19. (x + 7)(x + 3) 20. (x + 8)(x + 5) 21. (x — 5)(x + 3) 22. (x — D(x + 2) 23. (3x + 5)(2x + 1) 24. (7х + 4)(3x + 1) 25. (2х — 3)(5х + 3) 26. (2х — 5)(7х + 2) 27. (5х — 4)(3x* — 7) 28. (7х? — 2)(3х? — 5) 29. (8x* + 3)(х” — 5) 30. (7x + 5)(x* — 2) 31. (x + 3)x — 3) | 32. (x + 5)(x — 5) 33. (3x + 2)(3x — 2) 34. (2x + 5)(2x — 5) 35. (5 — 7х)(5 + 7х) 36. (4 — 3x)(4 + 3x) 37. (4x? + 5xY(4x* — 5х) 38. (3х? + 4x)(3x* — 4x) 39. (1 — y)A + y”) 40. 2 — y) + y) 41, (x + 2)? 42. (x + 5) 43. (2x + 3)? 44. (3х + 2)* 45, (x — 3)? 46. (x — 4)“ 47. (4х? — 1)“ 48. (5х? — 3)? 49. (7 — 2х)? 50. (9 — 5х)” 51, (x + 1)° 52. (x +2)” 53. (2x + 3)? 54. (3x + 4) 55. (x — 39 56. (x — 1} 57. (3x — 4° 58. (2x — 3)° In Exercises 59-66, perform the indicated operations. Indicate the degree of the resulting polynomial. 59, (5x?y — 3xy) + (2x*y — xy) 60. (—2x%y + xy) + (4x”y + 7xy) Section P.4 Polynomials 61 61. (4xy + 8xy + 11) + (-2x”y + 5ху + 2) 62. (7x*y? — 5x%y? + 3xy) + (—18x%y? — 6x*y* — xy) 63. (х° + 7ху — 5y) — (6x>— xy + 4у”) 64. (x* — Txy — 5y*) — (6x* — 3xy + 4y”) 65. (3x1y? + Sx*y — 3y) — (2x'y” — 3x3y — 4y + 61) 66. (5x*y? +'6x*y — Ту) — (3x*y? — 5x"y — 6y + 8x) In Exercises 67-82, find each product. 67. (x + S5yX7x + 3y — 68. (x + 9y)(6x + 7y) 69. (x — 3y)(2x + Ty) 70. (3x — у)(2х + 5у) 71. (3xy — 1)(Sxy + 2) 72. (Tx%y + 1)(2x%y — 3) 73. (Tx + Sy)? 74, (9x + Ty)” 75. (x2y? — 3) 76. (xy? - 5Y 77. (х — у)? + ху + у”) 78. (x + y)” — xy + y” 79. (3x + 5y)(3x — 5y) 80. (7x + 3y)(7x — 3y) 81. (7ху? — 10у)(7ху? + 10у) — 82. (3xy” — 4y)(3xy* + 4y) Practice Plus In Exercises 83-90, perform the indicated operation or operations. 83. (3x + 4yY — (3x — 4y)? 84. (5х + 2yY — (5x — 2yY 85. (5х — 7)(3х — 2) — (4x — 5(6x — 1) 86. (3x + 5)(2x — 9) — (7x — 2)(x — 1) 87. (2x + 5)(2x — 5)(4x* + 25) 88. (3x + 4)(3x — 4(9x* + 16) (2х — 7) 89, ———— 90 (2x = 7) (5х — 3)° ° (5х — 3)" Application Exercises As you complete more years of education, you can count on a greater income. The bar graph shows the median, or middlemost, annual income for Americans, by level of education, in 2009. Median Annual Income, by Level of Education, 2009 Men % Women Median Annual Income (thousands of dollars) 8 10 12 13 14 16 18 20 Years of School Completed Source: Bureau of the Census 62 Chapter P Prerequisites: Fundamental Concepts of Algebra Here are polynomial models that describe the median annual income for men, M, and for women, W, who have completed x years of education: M = 312х? — 2615х + 16,615 W = 316x* — 4224x + 23,730 М = 0.6x* + 285x” — 2256x + 15,112 И’ = —1.2х° + 367х? — 4900х + 26,561 Exercises 91-92 are based on these models and the data displayed by the graph at the bottom of the previous page. 91. a. Use the equation defined by a polynomial of degree 2 to find the median annual income for a man with 16 years of education. Does this underestimate or overestimate the median income shown by the bar graph? By how much? b. Use the equations defined by polynomials of degree 3 to find a mathematical model for M — W. According to the model in part (b), what is the difference, rounded to the nearest dollar, in the median annual income between men and women with 14 years of education? Tao. + d. According to the data displayed by the graph, what is the actual difference in the median annual income between men and women with 14 years of education? Did the result of part (c) underestimate or overestimate this difference? By how much? Use the equation defined by a polynomial of degree 2 to find the median annual income for a woman with 18 years of education. Does this underestimate or overestimate the median income shown by the bar graph? By how much? b. Use the equations defined by polynomials of degree 3 to find a mathematical model for M — W. ce. According to the model in part (b), what is the difference, rounded to the nearest dollar, in the median annual income between men and women with 16 years of education? According to the data displayed by the graph, what is the actual difference in the median annual income between _ men and women with 16 years of education? Did the result of part (¢) underestimate or overestimate this difference? By how much? | с - 92. a + Se The volume, V, of a rectangular solid with length 1, width w, and height h is given by the formula V = Iwh. In Exercises 93-94, use this formula to write a polynomial in standard form that models, or represents, the volume of the open box. In Exercises 95-96, write a polynomial in standard form that models, or represents, the area of the shaded region. 95. — x+9 я + 70 e —— x + +5 — ff 0 96. ft x+4 mo x+3| x+1 a ~ ul Li С Writing in Mathematics 97. What is a polynomial in x? 98. Explain how to subtract polynomials. 99. Explain how to multiply two binomials using the FOIL method. Give an example with your explanation. 100. Explain how to find the product of the sum and difference of two terms. Give an example with your explanation. 101. Explain how to square a binomial difference. Give an example with your explanation. 102. Explain how to find the degree of a polynomial in two variables. Critical Thinking Exercises Make Sense? In Exercises 103-106, determine whether each statement makes sense or does not make sense, and explain your reasoning. 103. Knowing the difference between factors and terms is important: In (3x 2yY, I can distribute the exponent 2 on each factor, but in (3x? + y, I cannot do the same thing on each term. 104, 1 ‚used the FOIL method to find the product of x + 5 and Xx + 2x +1. 105. Many English words have prefixes with meanings similar to those used to describe polynomials, such as monologue, binocular, and tricuspid. — 106. Special-product formulas have patterns that make their multiplications quicker than using the FOIL method. In Exercises 107-110, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 107. (3x> + 2)Gx? — 2) = 9x? — 4 108. (х — 5)? = x? — 5х + 25 109. (x +12 =x +1 110. Suppose a square garden has an area represented by 9x” “square feet. If one side is made 7 feet longer and the other side is made 2 feet shorter, then the trinomial that models the area of the larger garden is 9x* + 15x — 14 square feet. In Exercises 111-113, perform the indicated operations. 111. [(7x + 5) + 4yl[(7x + 5) — 4y] 112. [(3x + у) + 1}? 113. (x" + 2)(x" — 2) — (x? — 3) ! 11: IL Ë | lice dll 15 lg 114. Express the area of the plane figure shown as a polynomial in standard form. a x+3 WHAT YOU KNOW: We defined the real numbers [{x|x is rational} U {x|x is irrational}] and graphed them as points on a number line. We reviewed the basic rules of algebra, using these properties to simplify algebraic expressions. We expanded our knowledge of exponents to include exponents other than natural numbers: 1 b9 = 1; pr = +. = b = V b; bh" bp” m 1 pt = (Vp) = Y". 57 =. b We used properties of exponents to simplify exponential expressions and properties of radicals to simplify radical expressions. Finally, we performed operations with polynomials. We used a number of fast methods for finding products of polynomials, including the FOIL method for multiplying binomials, a special-product formula for the product of the sum and difference of two terms [(А + В)(А — В) = A> — B%],andspecial-productformulas for squaring binomials [(A + BY = А? + 2АВ + B*; (А - В}? = А? — 2АВ + В”). “In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify, if possible), whichever is appropriate. 1. (3x + 5X4x — 7) 2. (3x + 5) - (4x - 7) 3. V6+9V6 4. 3V12 — V27 5. 7х + 3[9 — (2x — 6)! 6. (8x — 3)” 7 3 8. (2) _ 375 9. (2x — 5) — (х? — 3x +1) 10. (2x — 5(x? - 3x +1) 11.3 +x7-—.X 12. (9a — 10b)(2a + b) 13. (a, c, d, e) U (c, d, f, h) 14. (a, c, d, e) A fc, d, f, h) 15. (3xy> — ху + 4у?) — (-2Xy> — 3xy + 5y”) 24 2,13 16, 17. (Zu 5 y as y) —2x° y? 3 18. Хх“ 24 x 10° 19 “4 x 10 (Express the answer in scientific notation. ) 2 X 10° \ 32, 3 3 20. ; 21. (x° + 2)(х” — 2) 23. V50- V6 22. (x* + 2) Mid-Chapter Check Point 63 TE as ge ар ER i, © AE plat uefa eh a ini) iL x us y Hi A PSI qu PNG PE dg 1 i fs Yi e i if be Le El Hn Fo és ny Exercises 115-117 will help you prepare for the material covered in the next section, In each exercise, replace the boxed question mark with an integer that results in the given product. Some trial and error may be necessary. 115. (x + 3)(x + |? ]) = 22 + Tx + 12 116. (x — [2 ])(x — 12) = x? — 14x + 24 117. (4x + 1)@x — [2 |) = 8x2 — 10x — 3 7-3 26. List all the rational numbers in this set: La -5,0, 0.45, V23,V25 } In Exercises 27-28, rewrite each expression without absolute value bars. V13| 28. |x| if x<0 29. the population of the United States is approximately 3.0 x 10% and each person produces about 4.6 pounds of garbage per day, express the total number of pounds of garbage produced in the United States in one day in scientific notation. 30. A human brain contains 3 X 10'° neurons and a gorilla brain contains 7.5 X 10° neurons. How many times as many neurons are in the brain of a human as in the brain of a gorilla? 31. TVs keep getting fancier and bigger, but prices do not. The bar graph shows the average price of a TV in the United States from 2007 through 2012. Average Price of a TV $1050 935 Average TV Price 2007 2008 2009 2010 2011 2012 Year Source: Consumer Electronics Association Here are two mathematical models for the data shown by the graph. In each formula, P represents the average price of a TV x years after 2007. P = —86x + 890 - P = 18x% — 175x + 950 a. Which model better describes the data for 2007? b. Does the polynomial model of degree 2 underestimate or overestimate the average TV price for 20127 By how much? 74 Chapter P Prerequisites: Fundamental Concepts of Algebra Here is a list of the factoring techniques that we have discussed. | Factoring out the GCF Factoring by grouping Factoring trinomials by trial and error Factoring the difference of two squares А’ — В? = (А + В)(А - В) e. Factoring perfect square trinomials A* + 2AB + B* = (A + BY A — 2AB + B” = (A — BY “ff Factoring the sum of two cubes A + B® = (A + B)(A> — AB + B”) RO g. Factoring the difference of two cubes Æ — B° = (A — В)(А? + АВ + В?) Ts 2 ea ve 2 Ls Час 3 sg e Ff ОНО MARCIO In Exercises 1-10, factor out the greatest common factor. 1. 18x + 27 2. 16x — 24 3. 3x" + 6x 4. 4х? — 8x 5, 9x* — 18 + 27x“ 6. 6x* — 18x? + 12x? 7. x(x + 5) + 3(x + 5) 8. x(2x + 1) + 4(2x + 1) 9. Xx(x — 3) + 12(x ~ 3) 10. x(2x + 5) + 17(2x + 5) In Exercises 11-16, factor by grouping. 11. x — 2x + 5x — 10 12. x* — 3х? + 4х — 12 13. х° = х? + 2x — 2 14. х° + 6x7 — 2x — 12 15. 3х? — 2х? — 6х + 4 16. х° — х* — 5х + 5 In Exercises 17-38, factor each trinomial, or state that the - trinomial is prime. 17. х? + 5х + 6 18. х? + 8x + 15 19. х* — 2х — 15 20. х* — 4х — 5 21. x? — 8x + 15 22. x? — 14x + 45 23. 3x* —x — 2 24. 2x* + 5x — 3 25. 3x* — 25x — 28 26. 3x“ — 2x — 5 27.6% —1x+4 — 28. 6x7 — 17x + 12 29. 4х? + 16х + 15 30. 8х? + 33x + 4 31. 9х” — 9х + 2 32. 9х? + 5х — 4 33. 20х? + 27x — 8 34, 15x“ — 19x + 6 35. 2x7 + 3xy + у? 36. 3x" + 4ху + у? 37. 6x* — 5ху — бу? 38. 6х? — 7ху — 5у? In Exercises 39-48, factor the difference of two squares. 39, x“ — 100 40. x“ — 144 41. 36x? — 49 42. 64х? — 81 43. 9х? — 25)? 44. 36х? — 49у? Fill in each blank by writing the letter of the technique (a through g) for factoring the polynomial. 1. 16x? — 25 . 2x3 — 1 Xx + Tx + xy + 7y 4x” + 8x + 3 Ox? + 24x + 16 5x% + 10x x* + 1000 » о AMES 1 1 412 The algebraic expression (x + 1)2 — Fx + 1)2 can be factored using as the greatest common factor. 45. x* — 16 46. x* — 1 47. 16x* — 81 48. 81x* — 1 In Exercises 49-56, factor each perfect square trinomial. 49, x +2x +1 $0. х? + 4х + 4 51. x“ — 14x + 49 52. x* — 10x + 25 53, 4x* + 4x + 1 54. 25х? + 10x + 1 55. 9х” — 6х + 1 56. 64x* — 16x + 1 In Exercises 57-64, factor using the formula for the sum or difference of two cubes. 57. х° + 27 58. х° + 64 59, x> — 64 60. x* — 27 61. 8x* — 1 62. 27x — 1 63. 64x) + 27 64. 8x) + 125 In Exercises 65-92, factor completely, or state that the polynomial is prime. 65. 3x° — 3x 66. 5x3 — 45x 67. 4х? — 4х — 24 68. 6х? — 18x — 60 69. 2x* — 162 70. Tx* — 7 71. х° + 2х? — 9х — 18 72. х° + 3x" — 25x — 75 73. 2x" — 2x — 112 74. 6x7 — 6x — 12 75, х° — 4x 76. 9х” — 9х 77. х° + 64 78. х? + 36 79, х° + 2х? — 4х — 8 80. х° + 2х? — х — 2 81. y — 81y 82. y” — 16y 83. 20y* — 457? —. 84. 48y* — 37? 85, x” — 12x + 36 — 49y* 86. x — 10x + 25 — 36)? 87. 9b”x — 16y — 16x + 9b”y 88. 16а%х — 25y — 25x + 16a”y Л 89. 91. xy — 16y +32 - 2% 90. 12% — 27y — 4х + 9 2х3 — 8а?х + 24х° + 7дх 92. 2x7 — 98а?х + 28x” + 98x Jn Exercises 93-102, factor and simplify each algebraic expression. 93. 93. 97. 99. 100. 101. 102. 3 1 3 1 2 1 _3 1 4х 3 + ar 96. 12x 4 + 6x4 3 (x + ” — (х + 3)? 98. (x + 4} + (x? + 4) 03-0457 | 2 (2 + y + (2 +3)%3 1 3 (4x — 1)? — M4x — 1)? —8(4x + 3)7? + 10(5x + 1)(4x + 3)" Practice Plus In Exercises 103-114, factor completely. 103. 104. 105. 107. 109. 111. 113. 10x*(x + 1) — 7x(x + 1) — 6(x + 1) 12x*(x — 1) — 4x(x - 1) — 5(x — 1) 6x* + 35x* — 6 106. 7x* + 34x* — 5 y +y 108. (y + 1? +1 x* — 5x2? + 4y* 110. x* — 10x?y" + 9y* (x — y)* — 4(x = у)? 112. (x + y)* — 100(x + y)” 2x? — Txy? + 3y* 114. 3х? + Sxy? + 2y* Application Exercises 115. 116. Your computer store is having an incredible sale. The price on one model is reduced by 40%. Then the sale price is reduced by another 40%. If x is the computer's original price, the sale price can be modeled by (x — 0.4x) — 0.4(x — 0.4x). a. Factor out (x — 0.4x) from each term. Then simplify the resulting expression. b. Use the simplified expression from part (a) to answer these questions. With a 40% reduction followed by a 40% reduction, is the computer selling at 20% of its original price? If not, at what percentage of the original price is it selling? Your local electronics store is having an end-of-the-year sale. The price on a plasma television had been reduced by 30%. Now the sale price is reduced by another 30%. If x is the television’s original price, the sale price can be modeled by (x — 03x) — 0.3(x — 0.3x). a. Factor out (x — 0.3x) from each term. Then simplify the resulting expression. b. Use the simplified expression from part (a) to answer these questions. With a 30% reduction followed by a 30% reduction, is the television selling at 40% of its original price? If not, at what percentage of the original price 1s it selling? Section P.5 Factoring Polynomials 75 In Exercises 117-120, 117. 119. a. Write an expression for the area of the shaded region. b. Write the expression in factored form. LIA 118. I 3x >| |) — 120. x xy In Exercises 121-122, find the formula for the volume of the region outside the smaller rectangular solid and inside the larger rectangular solid. Then express the volume in factored form. 121. 122. Writing in Mathematics 123. 124. 125, 126. 127. 128. 129. Using an example, explain how to factor out the greatest common factor of a polynomial. Suppose that a polynomial contains four terms. Explain how to use factoring by grouping to factor the polynomial. Explain how to factor 3x* + 10x + 8. Explain how to factor the difference of two squares. Provide an example with your explanation. What is a perfect square trinomial and how is it factored? Explain how to factor x* + 1. What does it mean to factor completely? 132. I factored 4x* — 100 76 Chapter P Prerequisites: Fundamental Concepts of Algebra Critical Thinking Exercises Make Sense? In Exercises 130-133, determine whether each statement makes sense or does not make sense, and explain your reasoning. 130. Although 20x> appears in both 20x? + 8x* and 20x* + 10x, Pil need to factor 20x” in different ways to obtain each polynomial’s factorization. 131. You grouped the polynomial’s terms using different groupings than I did, yet we both obtained the same factorization. completely and obtained (2x + 10)(2x — 10). 133. First factoring out the greatest common factor makes it easier for me to determine how to factor the remaining factor, assuming that it is not prime. In Exercises 134-137, determine whether each statement is true —or false. If the statement is false, make the necessary change(s) to produce a true statement. + 4)(х? — 4). e polynomial. 134. x* — 16 is factored completely as © 135. The trinomial x* — 4х — 4 is a prim 136. x* + 36 = (x + 6)? 137. х° — 64 = (x + 4)(02 + 4x — 16) Specify umbers tl that -Must be excluded from “the domain of a rational expression. , Sir mplify Y expression y ratio nal expressions. Миру г ‘ Divide rational” expressions. В Add and: su btract: rational expressions. Simplify complex | rational expressions, possible, reduce the answer to its lowest terms. How do we describe the costs of reducing environmental pollution? We often use algebraic expressions involving quotients of polynomials. For example, the algebraic 250x 100 — x describes the cost, in millions of dollars, to remove Ш ation al | xpercent of the pollutants that are discharged into a | river. Removing a modest percentage of pollutants, say 40%, 1s far less costly than removing a substantially greater percentage, such as 95%. We see this by evaluating the algebraic expression forx = 40 and x = 95. 250x \ Evaluating 100 — + for 100 — x In Exercises 138—141, factor completely. 138. x2" + 6x" + 8 139, — 140. x* — y* — 2 y + 2xy* 141. (x — 5 + sy — (x + e — sy? In Exercises 142-143, find all integers b so that the trinomial can be factored. 142. x* + bx + 15 * — 4x +5 143. x? + 4x + b LS Preview Exercis Exercises 144-146 will helo you prepare for the material covered in the next section. 144. Factor the numerator and the denominator. Then simplify by dividing out the common factor in the numerator and the denominator. x” + 6x + 5 ? — 25 In Exercises 145-146, perform the indicated operation. Where 5 8 1 2 Cs 4. 15 16.373 = 40: x = 95: el ‚ 250(40) ‚ 250(95) What happens if you try Costis — = 167. Costis — —— = 4750. substituting 100 for x in 100 — 40 100 — 95 250x , The cost increases from approximately $167 million to a possibly prohibitive 100 — x What does this tell you about the cost of cleaning up all of the river’s pollutants? $4750 million, or $4.75 billion. Costs spiral upward as the percentage of removed pollutants increases. 86 Chapter P Prerequisites: Fundamental Concepts of Algebra . 3x + . 8. An equivalent expression for with a denominator of (3x + 4)(x — 5) can be obtained by multiplying the numerator and denominator by 9. A rational expression whose numerator or denominator or both contain rational expressions is called a/an rational expression or fraction. a/an In Exercises 1-6, find all numbers that must be excluded from the domain of each rational expression. 7 13 1. - 2. - x — 3 x + 9 х + 5 х + 7 Jo 2_ a 2_ X 25 X 49 x — 3 © x2 + Ах — 45 x — 1 SL STILL x“ + 11x + 10 In Exercises 7-14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 3x — 9 4x — 8 * 2 641 à 8. — II, x“ — 6x + 9 x“ — 4х + 4 2 + 2 + 9. X 12x + 36 10. X 8x + 16 dx — 24 3x — 12 y? +7y — 18 2 — 4у — 5 п, —— 12. —— у” — Зу + 2 у” + Sy +4 x + 12x + 2 — 14x + qa, Y +12 +36 д 2 — 1x +49 x“ — 36 x“ — 49 In Exercises 15-32, multiply or divide as indicated. ‚| х— 2 2х + 6 16 6x +9 х- 5 ° 3х + 9 2х — 4 TU 3x — 15 4х + 6 2 2 2 _ _ 17, * —. X 3x 18. À 4 2x 4 x x“ +x — 12 x” — dx +4 x +2 2. 5x +6 x2 — 19. ©. 5х E 1 х” — 2х — 3 х“ — 4 + 5х + 6х? — 9 20. © X — X +x—-6 XxX —x—Ó 1, x 8 x 2 27, Ÿ 6x +9 1 Xx —4 3x x +27 x+3 Хх +] 3x + 3 x+5 4x +20 23. + 24, + 3 3 7 7 9 2— 4 +2 2—4 + 25, Кн Ля 26. —. 12 x x-2 x-2 4x — 8 1 1 (25-5) 10.3 x XX +3) +3 x/ ——( 3 x(x + 3) 3 3x(x + 3) — 3x(x +3) 2 2 э7, ® + 10 | бх + 15 х — 3 х? — 9 2 4 2 1 28. © X X 1 x“ — 4 x“ + 5x +6 9 2-25, Y +10 + 25 "2-2 x +4x-5 x” — 4 х? + 5х + 6 30. — + x” + 3x — 10 х“ + 8x + 15 х* + х — 12 х* + 5х + 6 x + 3 31. 5 5 += x* +x-— 30 х” — 2х = 3 x‘ +7xc +8 32 x — 25x 2x* — 2 „X“ + 5x AX xX-6x+5 Tx +7 In Exercises 33-58, add or subtract as indicated. + + 33, Y 1 8x-+9 34, 3x +2 3x +6 6x +5 6x +5 3x +4 3x +4 2.7 2. 5 2 a x“ + 3x Xx +3x 2-4 4х — 4 36, pg x—-x-6 x -x-6 | 4x — 10 х- 4 2x + 3 3—x 7. — . — 3 x — 2 x — 2 38 3x -6 3x —6 2 +3 *— 12 39, — X — х“ + Хх — 12 x +x- 12 Yo x* — 4x _ 2-6 x -x-6 x2-x-6 3 6 8 2 41. + 42. x +4 x+5 x—2 x—3 43. 33 a 2 3 - x+1 x x x+3 2X x + 2 3x x + 4 Biri rec х+2 x +3 x-5 x+3 x—3 . + . 4. 3 x +5 * ya x + 3 3 2 5 7 49, + , + 2x + 4 3x + 6 > 2x +8 3x+12 4 4 3 5x 51. 52. + x*+6x+9 x+3 5x +2 25х? — 4 La р 6 6 Ti Xx A 53.24 3x — 10 Xx +x—6 x X 4. 7 ax 24 x? 7x +6 x+3 x+2 5.271 x — 1 4 +x-6 3 3 12 + 3х + 2 х +1 x+2 6x7 + 17x — 40 | 3 _ Sx 21,20 x—-4 x+5 x+5 x+1 x —4 x-2 56. 57. 58. x? + x — 20 In Exercises 59-72, simplify each complex rational expression. * 1 д 3 | 9. — 60. — 1 +7 8 + — x 62 - 61. 1 al 3-7 x 1,1 1 xX y 64. 63. Fy ху X х — x + 3 x — 3 65. — — —— 66. 65. х + 2 к — 3 x — 2 oh 5 73 +1 (7, 2 68. > 7 3 + 1 х* — 4 х? — 4 | х + 1 69, 1 1 Xx —2x-3 x-3 6 1 70 Xx +2x—-15 x-3 | ! + 1 х + 5 ! LL x + h X +4) x? x ne ) x 7 2h x +1 Practice Plus In Exercises 73-80, perform the indicated operations. Simplify the result, if possible. (ZEA ros __2 х + 1 2х% + х —3 x +2 m5— => ( 1 1 ) Cox =x -8 \x-4 x+2 (0-3 х + 1 2 Section P.6 Rational Expressions 87 3 5 76. | 4 — +) = (у +5)! 4-9 +21 7? (y +5) 78, ? (y + 2) 5 2 m. ( 1 Arad m ber ba) c—d a? — 53 1 a’ + ab + b? 80 ab * ac — ad — bc + bd a> =P a+ ab + Ъ? ac — ad + be — bd a +8 Application Exercises 81. The rational expression 130x 100 — x describes the cost, in millions of dollars, to inoculate x percent of the population against a particular strain of flu. a. Evaluate the expression for x = 40, x = 80, and x = 90. Describe the meaning of each evaluation in terms of percentage inoculated and cost. b. For what value of x is the expression undefined? ¢. What happens to the cost as x approaches 100%? How can you interpret this observation? 82. The average rate on a round-trip commute having a one-way distance d is given by the complex rational expression 2d d а’ F1 Го in which 7, and 7, are the average rates on the outgoing and return trips, respectively. Simplify the expression. Then find your average rate if you drive to campus averaging 40 miles per hour and return home on the same route averaging 30 miles per hour. Explain why the answer is not 35 miles per hour. 83. The bar graph shows the estimated number of calories per day needed to maintain energy balance for various gender and age groups for moderately active lifestyles. (Moderately active means a lifestyle that includes physical activity equivalent to walking 1.5 to 3 miles per day at 3 to 4 miles per hour, in addition to the light physical activity associated with typical day-to-day life.) Calories Needed to Maintain Energy Balance for Moderately Active Lifestyles 3200 2800 + 2400 2000 + - 1600 1200 800 400 - Gps Group $ - Group 4 о | nan Group 6 … Calories per Day 4-8 9-13 14-18 19-30 31-50 51+ Age Range Source: US.D.A. 88 Chapter P Prerequisites: Fundamental Concepts of Algebra (Be sure to refer to the graph at the bottom of the previous page.) a. The mathematical model W = —66x* + 526x + 1030 describes the number of calories needed per day, W, by women in age group x with moderately active lifestyles. According to the model, how many calories per day are needed by women between the ages of 19 and 30, inclusive, with this lifestyle? Does this underestimate or overestimate the number shown by the graph? By how much? b. The mathematical model М = —120x* + 998x + 590 ‘describes the number of calories needed per day, M, by men in age group x with moderately active lifestyles. According to the model, how many calories per day are needed by men between the ages of 19 and 30, inclusive, with this lifestyle? Does this underestimate or overestimate the number shown by the graph? By how much? ce. Write a simplified rational expression that describes the ratio of the number of calories needed per day by women in age group x to the number of calories needed per day by men in age group x for people with moderately active lifestyles. In Exercises 84-85, express the perimeter of each rectangle as a single rational expression. 84. X | 85. X X +3 х +5 Writing in Mathematics 86. What is a rational expression? 87. Explain how to determine which numbers must be excluded | from the domain of a rational expression. 88. Explain how to simplify a rational expression. 89. Explain how to multiply rational expressions. 90. Explain how to divide rational expressions. 91. Explain how to add or subtract rational expressions with the same denominators. 92. Explain how to add rational expressions having no common factors in their denominators. Use + in your x+5 x+2 explanation. 93. Explain how to find the least common denominator for denominators of x” — 100 and x? — 20x + 100. 3 2 — + metre | | XX Describe two ways to simplify 1 > —_— + — x* x 94 Explain the error in Exercises 95-97. Then rewrite the right side of the equation to correct the error that now exists. , 1 1 1 1 1 ST += 9. — + 7 = 9 a+b 6. 15777 а b ` N | E 97: - + a. x + b Preview E Critical Thinking Exercises Make Sense? In Exercises 98-101, determine whether each statement makes sense or does not make sense, and explain your reasoning. 3x — 3 4x(x — 1) 99. The rational expressions 7 7 14x and 14 + x E can both be simplified by dividing each numerator and each _ denominator by 7. 98. I evaluated for x = 1 and obtained 0. 100. When performing the division Dis Tx (x + 3)? Ra R—5" I began by dividing the numerator and the denominator by- the common factor, x + 3. 101. Isubtracted 3x X x —3 , from Tq and obtained a constant. — 1 In Exercises 102-105, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 2 x“ — 25 102, ——— =x—3 x — 5 —Зу — 6 103. The expression — 5 simplifies to the consecutive . integer that follows —4. 2х — 1 3x—1 5х — 2 ° + — = С 104 х — 7 х — 7 x — 7 0 ; Pr 105. 6 + 1-7 x x In Exercises 106-108, perform the indicated operations. 1 1 1 106 7 a] x” —1 1 1 1 1 — — 1 — re 197 (1 DU x + т) —)( +=) 108. (x — yY + (x — y)? 109. In one short sentence, five words or less, explain what Y 2. X X Ti ; 4 5 х° does to each number x. HEF rcises Exercises 110-112 will help you prepare for the material covered in the first section of the next chapter. 110. If y = 4 — x? find the value of y that corresponds to values of x for each integer starting with —3 and ending with 3. 111. Ify = 1 — x”, find the value of y that corresponds to values of x for each integer starting with —3 and ending with 3. 112. В у = |x + 1], find the value of y that corresponds to values of x for each integer starting with —4 and ending with 2. ur Summary, Review, and Test 89 ach Distance between Points a and b on a Number Line ‚|a—b| or |b-a y Properties of Real Numbers Commutative a+b=b-+a ab = ba | Associative (а + Б) + с = а + (Б + с) | (ab)c = a(bc) | Distributive a(b + c) = ab + ac Identity а + 0 = а а*1 = а е Inverse а + (-а) = 0 a} = 1,a #0 a Properties of Exponents 1 BR = Ь° = 1, pm pt = БР, bp” Me mn bo" mTR й == дб АЙ ay a" (BY = pm, gr = В , (ab) - a, (4) DR Product and Quotient Rules for nth Roots Bf JT AR „Ja _ Va Vab = Va Vb, b= о; Rational Exponents 1 11 1 а” = Va, а "= — = › a” Va n n A 1 a” = (Va) = Ма”, а " = —щ # Special Products (A + B)(A — B) = À — B” (А + В)? = А? + 2АВ + В? (A — BP = À - 24B + B? | (A + BY = 43 + ЗА?В + 3AB” + B* (А — B) = A’ - 3A’B + 34B* — B® Factoring Formulas A — B* = (A + B)(A — B) A + 2AB + B? = (A + BY A — 2AB + B? = (A — BY A? + B* = (A + BYIA — AB + BY) A’ — B® = (A - B\(4 + AB + B”) You can use these review exercises, like the review exercises at the end of each chapter, to test your understanding of the chapter’s topics. However, you can also use these exercises as a prerequisite test to check your mastery of the fundamental algebra skills needed in this book. Р.1 In Exercises 1-2, evaluate each algebraic expression for the given value or values of the variable(s). L 34 6(x —2)°forx = 4 2. x* — 5(х — y) forx = 6and y = 2 3. You are riding along an expressway traveling x miles per hour. The formula S = 0.015x* + x + 10 models the recommended safe distance, $, in feet, between your car and other cars on the expressway. What is the recom- mended safe distance when your speed is 60 miles per hour? In Exercises 4-7, let À = (а, |, с}, В = {а, с, а, е}, and С = la, d, f, e). Find the indicated set. 4 AMB 5 AUB 6. AUC TL CNA 8. Consider the set: [-17,—15,0,075,V2, 7, V81 ). List all numbers from the set that are a. natural numbers, b. whole numbers, e, integers, d. rational numbers, e. irrational numbers, f. real numbers. In Exercises 9-11, rewrite each expression without absolute value bars. V2 - 1 11. [3 — V17| 9. |—103| 10. 12. Express the distance between the numbers —17 and 4 using absolute value. Then evaluate the absolute value. 80 Chapter P Prerequisites: Fundamental Concepts of Algebra In Exercises 13-18, state the name of the property illustrated. 13. 3 + 17 = 17 + 3 14. (6:3)-9 = 6-(3-9) 15. V3(V5 + V3) = V15 + 3 16. (6-9)-2 = 2-(6-9) 17. Vil Vs + V3) = (V5 + УЗ) УЗ 18. (3-7) + (4-7) = (4-7) + (3-7) In Exercises 19-22, simplify each algebraic expression. 19. 52x — 3) + 7x 20. $(5х) + [(Зу) + (-Зу) | - (-х) 21. 3(4y — 5) — (7у + 2) 22. 8— 2/3 — (5x — 1) 23. The diversity index, from 0 (no diversity) to 100, measures the chance that two randomly selected people are a different race or ethnicity. The diversity index in the United States varies widely from region to region, from as high as 81 in Hawaii to as low as 11 in Vermont. The bar graph shows the national diversity index for the United States for four years in the period from 1980 through 2010. | Chance That Two Randomly Selected Americans Are a Different Race or Ethnicity 47... There: is isa 155% | “ol chance that two 1 randomly selected - Americans: differin: | race or ethniolty. 50 AO ber 30 20 |- (0-100 scale) 10- Diversity Index: Chance of Different Race or Ethnicity 2000 2010 Year 1980 1990 Source: USA Today The data in the graph can be modeled by the formula D = 0.005x* + 0.55x + 34, where D is the national diversity index in the United States x years after 1980. According to the formula, what was the U.S. diversity index in 2010? How does this compare with the index displayed by the bar graph? P.2 Evaluate each exponential expression in Exercises 24-27. 24. (-3(-2Y 25. 27% + 47! 33 26. 57:5 27. — 3 Simplify each exponential expression in Exercises 28-31. 28. (—2x*y*) 29. (-5x*y")(-2x}y7?) TX yS 28x15 y 2 In Exercises 32-33, write each number in decimal notation. 32. 3.74 x 10° 33. 7.45 x 10° 30. (2x°)* 31. In Exercises 34-35, write each number in scientific notation. 34. 3,590,000 35. 0.00725 In Exercises 36-37, perform the indicated operation and write t answer in decimal notation. 6.9 x 10° 3 x 10° In 2009, the United States government spent more than it had collected in taxes, resulting in a budget deficit of $1.35 trillion. Exercises 38~40, you will use scientific notation to put a number like 1.35 trillion in perspective. Use 102 for 1 trillion. 36. (3 x 101.3 x 103) 37. 38. Express 1.35 trillion in scientific notation. 39. There are approximately 32,000,000 seconds in a year. Express this number in scientific notation. 40. Use your scientific notation answers from Exercises 38 and 39 to answer this question: How many years is 1.35 trillion seconds? Round to the nearest year. (Note: 1.35 trillion seconds would take us back in time to a period when Neanderthals were using stones to make tools.) P.3 Use the product rule to simplify the expressions in Exercises 41-44, In Exercises 43-44, assume that variables represent nonnegative real numbers. V 300 42. V12x” V10x- V2x 44. Vr° Use the quotient rule to simplify the expressions in Exercises 45-46, 121 V 96x 45. |— — 46. 4 Vx In Exercises 47-49, add or subtract terms whenever possible. 47. 7V5 + 13V5 48. 2V50 + 3V8 49. 4V72 — 2V48 In Exercises 50-53, rationalize the denominator. 30 V2 50. — 51, — VS V3 5 s3.— 4 6 + V3 7-5 Evaluate each expression in Exercises 54-57 or indicate that the root Is not a real number. 55, V—32 54. V125 /—125 57. Y (-5Y Simplify the radical expressions in Exercises 58-62. 58. V/s1 59. Yy5 60. Ÿa- 10 61. 4V/16 + 5V2 4/25 .5 2 a (Assume that x > 0.) V 16x In Exercises 63-68, evaluate each expression. (Assume that x > 0.) 52. 62. 1 À 1 63. 162 64. 25 2 65. 125° 1 2 4 66. 27 3 67. 645 68. 27 3 In Exercises 69-71, simplify using properties of exponents. 3 2 1 15x* 4 69. (5x3) (4x4) 70. > su 2 71. (125х°)3 72. Simplify by reducing the index of the radical: V Y. p4 In Exercises 73-74 I resulting polynomial i 73. (6x7 + 7х2 — 9х + 3) + (14x) + 3x* — 11x — 7) па, (13x* — 8? + 2х”) — (Sx* — 3x% + 2x“ — 6) 74, perform the indicated operations. Write the n standard form and indicate its degree. mn Exercises 75-81, find each product. 75, (3x — 2)(4х? + 3x — 5) 76. (3x — 5)(2x + 1) T7, (4x + 5)(4x — 5) 78. (2x + 5Y 79, (3х — 4)? 80. (2х + 1Y 81. (5х — 2) In Exercises 82-83, perform the indicated operations. Indicate the degree of the resulting polynomial. 82, (7x” — 8xy + у?) + (-8х? — 9ху — 4у”) 83. (137 — 5х’ у — 9х”) — (~11x3y? — бх?у + 3x* — 4) In Exercises 84-88, find each product. 84. (x + 7y)(3x — 5y) 85. (3x — Sy)” 86. (3x7 + 2y7 87. (7х + 4у)(7х — 4у) 88. (a — b)(a® + ab + b?) Tn Exercises 89-105, factor completely, or state that the polynomial “is prime. 89, 15x° + 3х” 90. х? — 11x + 28 91. 15x —x—2 92. 64 — x* 93. x? + 16 94. 3x* — 9x — 30x“ 95, 20x” — 36x° 96. x — 3х? — 9х + 27 97, 16x* — 40х + 25 98. x* — 16 99. y — 8 100. x +64 27x? — 125 101. 3x* — 12x“ 102. In Exercises 1-18, simplify the given expression or perform the indicated operation (and simplify, if possible), whichever is appropriate. 1. 5(2x? — 6x) — (4x2 — 3x) 2, 7 + 2[3(х + 1) — 23x — 1)] 3. [1,2, 5} п 15, а) 4. (1,2, 5} U {5, a} 5 (ху? - ху + y”) = 42) — 5xy — y”) 30x y* 6x9 y* 7. V6erV3r (Assume that r = 0.) 8.4/50- 3/18 о, —— 5 + V2 10. Yi6x* 11 . ~6 12. 5X10 20 x 108 (Express the answer in scientific notation.) x? + 2x — 3 " х? = 3x +2 Summary, Review, and Test 91 103. x° — x 104. x* + 5x? — 2x — 10 105. x* + 18x + 81 — у? In Exercises 106-108, factor and simplify each algebraic expression. 3 1 106. 16x 4 + 32x* 1 3 107. (x? — 40x? +3) — (x? — “x? +3)? 1 3 108. 12x 2 + 6x 2 P.6 In Exercises 109-111, simplify each rational expression. Also, list all numbers that must be excluded from the domain. x* + 2х? x? + 3x — 18 x? + 2x 109. — — — 110. e х + 2 xXx — 36 Xx + 4х + 4 In Exercises 112-114, multiply or divide as indicated. ? + 6x + + + 112. X 6x 9 x 3 113. bx +2 3% X х° = 4 x-2 x? = 1 x —1 2 _ 2 п © 5x 24 x 10x + 16 x“ —= x — 12 xr +x—6 In Exercises 115-118, add or subtract as indicated. 2x — 7 x —10 3x X 115 — 116 + x* — 9 x” — 9 x+2 x-2 x — 1 117. Xx —9 x—5x+6 Ax — 118. x — 1 x+3 27 +5x-3 6x2 +x —2 In Exercises 119-121, simplify each complex rational expression. 1 1 a 1 х 2 3+ > 7773 119. 120. X 11. ——— Lx | - © 34 À 3 6 x? x + 3 13. (2x — 5)(х? — 4x + 3) 14. (5x + Зу)? 2х + 8х? + 5х + 4 ох 5 15. + 16. + x — 3 xs x+3 x-3 11 2x +3 2 x 3 17. — 18. x*—"7x+12 x-3 1 X In Exercises 19-24, factor completely, or state that the polynomial is prime. 20. x + 2x? +3x +6 22. 36x” — 84x + 49 24. x? + 10x + 25 — 9y* 19. x? — 9x + 18 21. 25x* — 9 23. y? — 125 25. Factor and simplify: 3 2 x(x + 3) 5 + (x + 3%. 26. List all the rational numbers in this set: (-7,-%,0,0.25,V3,V4,%,7). P Section P.1 Check Point Exercises | 1608 299209 33,7) 403456789 5aV9 b0,V9 e-9,0,V9 d-9,-13,0,03,V9 e z Y -9,-13,0,03,7,V9, V10 8aV-1 bdar-3 el 79 8387+23x 9.42 — 4x Concept and Vocabulary Check 1. expression 2. b to the nth power; base; exponent 3. formula; modeling; models 4. intersection; A NB 5. union; AUB 6. natural 7. whole 8. integers 9. rational 10. irrational 11. rational; irrational 12. absolute value; x; —x 13. b + a; ba 14. a + (b + 6); (ab) 15. ab + ac 16. 0; inverse; 0; identity 17. inverse; 1; identity 18. simplified 19. a Exercise Set P.1 15 310 58 710 944 11.46 13.10 15. -8 17. 10°C 19.60ft 21. {2,4} 23 sen 25. 2 27. 3 29. {1,2,3,4,5} 31. [1,2,3,4,5,6,7,8,10) 33 (а, е,1,0,и} 3B a Vid 6. 0,V100 e —9,0,vi100 4 — 4 — d. -9,-=,0,0.25, 92, V100 e Vi 1-9,-7,0,025, V3,92, V100 37.a V64 ».0,V64 e —11,0, V6 5 5 | d. e 0, 0.75,V64 e. V5, т f. de 0, 0.75, V5, т, V64 39. 0 41. Answers may vary; an example is 2. 43, true 45. true — 47. true 49. true 51.300 53. 12 — п || 55. 5 — V2 57. —1 50.4 8.3 63.7 65-1 67.17- 2.15 69. 15 — (—2);7 71. 4 — (-19),15 73. |+1.4 — (-3.6)1;2.2 75. commutative property of addition 77. associative property of addition 79. commutative property of addition 81. distributive property of multiplication over addition 83. inverse property of multiplication 85. 15x + 16 87. 27x — 10 89. 29y — 29 91. 8у — 12 93, 16y — 25 95, 12x? + 11 97. 14x 99, —2x + 3y + 6 101. x 1 8 1 103. > 105. = 107. < 100. = 111.45 118.757 118. 14H Meo 12127 (x + 4); —4 123. 6(—5x); — 30x 125. 5х — 2x; 3x 127. 8x — (Bx + 6);5x — 6 129. a. 140 beats per minute b. 160 beats per minute 131. a. $26,317 b. overestimates by $44 c. $30,568 133. a. 1200 — 0.07х b. $780 145. does not make sense 147. does not make sense + 149. false + 151. false + 153. false 155. true 157. > 159. a. bb. bY °c Add the exponents. 160. a. b* b. bé c. Subtract the exponents. 161. It moves the decimal point 3 places to the right. Section 2 Check Point Exercises 1 1 3y° 1. a 3°0r243 b 40x%y 2 a (-3°0r-27 b9x"y® Baz b= 26.16 @ 2 4 a 301729 bb. + 25 27 x6 y! ; 32 x? 12,24 3,8 y y | ce. b2 5. -64х° 6. а. === bir 7a lx y” b-18y се. =~ @ 8. 8 —2,600,000,000 b. 0.000003017 y° 27 x 25х? 9. а. 5.21 х 10” hb. —6.893 x 10% 10. 4.1 х 10° 11. a. 3,55 x 10! b. 4 x 10° 12. $8300 Concept and Vocabulary Check 1 1, bp" add 2, b""" subtract 3.1 4. pa 5. false 6. b” 7. true 8. a number greater than or equal to 1 and less than 10; integer 9. true 10. false Exercise Set P.2 1 1 1 y 1.50 3.64 85-64 7.1 9. —1 11. — 13. 32 15. 64 17. 16 19.— . + ‚у? xXx 6 я 5 27% 23. > 25. y 27. x 64 29. х° . 31. x 33 — 95. x7 | 37. 39,64% 41.—— 43.9 45.61! 47. 18% - 49.4x% 51.—5ab xX X 2 1 3 14 2 2765 . | 5.5 8— 8. a. 59. — 61.2 6a 1 65380 67. 0.0006 69 —7,160,000 71.079 73. —0.00415 b 16x 4x 25x a 75. —60,000,100,000 77. 3.2 х 10* 79. 6.38 x 107 81. —5.716 x 10° 88. 2.7 x 1073 85. -504 x 107 87. 6,3 х 10’ 89. 64 x 10% 91. 122 x 1071 93. 2.67 x 10** 95. 2.1 x 10° 97. 4 X 10° 90. 2 x 10° 101.5x 10? 103. 4 x 105 АА? Answers to Selected Exercises y 1 x'8y5 105. 9 x 107 107.1 108. loss Maz 18 115. а. 2.17 X 10° b.3.09x10% с. $7023 хо Xx “y zZ 117. $1.0586 x 10'% 119. 1.06 x 10g 121. 4.064 x 10% — 193. makessense — 195. makessense — 137. true — 199. false 141. false | 143. true 145. A=C+D 148. а. 8 b8 ce V16-V4= V16-4 149. a 1732 b 1732 с. У/З00 = 10\/3 В 150. a. 31x b. 31V2 a Section P.3 м Check Point Exercises - С 1 5 1 a9 b-3 € 5 d 10 е 14 2a5V3 b5V7 3 a 7 + SxV3 4 a 17V13 b —19V17X 5 a 175 ов. 5V3 8(4 — V5) 32 — 8% 5 | в b. 10V2x 6. a. b. V3 7. ( = ) or a v5 8. a. 2V5 - b 25 e. 3 9 5V3 10a5 b2 6-3 » | 04 1 1 | | | d-2 ед Mas b8 67 Mal b. 4x” 18 Vx Concept and Vocabulary Check 1. principal 2. 8 3h 4va vb 5 6. 18V3 7.56V3 87-Vi 9 VIO + V2 10. index: radicand 11. (-2)° 12 ala) 18 Va 14 2:8 SIS Exercise Set P.3 1.6 8-6 5 notarealnumber 7.3 9.1 11.13 1352 153xV5 17.273 19. xVE 22V/ 2 5 27. 4х 29.5xV2x 31.275 8313Vi 835. -2VI7x% 37, SVT 39. 3V0 437 14320V7-5vi 7 4 A V7 V10 133 - VI 45. 47, > 49. BE = vib), 51. 7(V5 + 2) 53. 3(V5 — V3) 55. 5 57. —2 59. notarealnumber 61.3 1 63.3 65. — 67, 274 ex n37% ax BIO 711.1393 70. -yV2x 81. V2+2 86 85.2 — 9 87. 25 89. wo 14x"? 93, 4x4 95. x? 97, 5x|y|3 09.27 101. V5 408. x2 105 М? 107. Vx*y 109, 3 à . . 5+1 1 11, — 113. = 115. Paige Fox is bad at math. 117. v5 ; 1.62101 119. Р = 18 \/5 ft; A = 100 ft? 129. does not make sense 77” 1 | 131. does not make sense 133. false 135. false 187. Let J = 3. 139. 4 141. a. 8 b. 142. 10x7y* 143. 16x° + 6x 144. 2x? + 11x2 + 22x + 15 4 Section P.A Check Point Exercises 1. а. —х” + х* — &х — 20 b. 20x? — 11x? -2x-8 2. 15x? — 3x? + 30x = 8 3 28¢% — 41x + 15 4 a 49% ~64 b 4y6—25 5. a. x“ + 20x + 100 b. 25x? + 40x +16 6. а. х? - 18x +81 b. 49% -42x+9 7 2x’y + 5ху? — 2у° 8. a 21x? — 25xy + 6y“ b. 4x* + 16xy + 16y° | | Concept and Vocabulary Check 1. whole 2. standard 3. monomial 4. binomial 5. trinomial 6.n 7. like 8. distributive: 4x5 — 8x2 + 6; 7x 9. 5x; 3; like of the terms; plus 14. n +m | и ‘ Exercise Set P.A 1. уев; 3х” + 2х = 5 300 852 74 9. 1х3 + 7х? - 12х = 4;3 11. 12x? + 4x? + 12x — 14:3 18. 6x2 — 6x + 2;2 15. х° + 117. 2х3 — 9x* + 19x — 15 19. х? + 10х + 21 M 2 ~2x—15 28. 6х? + 13х + 5 25 10 - 9x — 9 27. 15х" — 47х° + 28 29. 8х° — 40х® + 3х? — 15° 31. х?2 = 9 33 9x2 —4 85 25 — 49x2 397. 16x* — 25х39, 1 — у“ 41. x“ +4x +4 48. 4х? + 12x +9 45. х? - 6х + 9 47. 16x* — 8x2 +1 49. 4x? — 28x +49 51. x} + 3x2 + 3x + 1 53, 8x) + 36x? + 54x + 27 55, x? — 9x% + 27x — 27 57. 27x* — 108x? + 144x — 64 59. 7x"y — 4xy is of degree 3 61. 2x*y + 13xy + 13isof degree 3 63. —5x* + 8xy — 9y"isofdegree3 65, х“у? + 8х?у + у — 6xisofdegree6 67. Tx? + 38xy + 19% 69. 2x“ + xy — 21y* 71, 15x74? + xy —2 78. 49х? + 70xy + 25y? 75 4-6 +9 77. xP — y? 79. 9х? — 25)? 81. 49x°y* — 100y? 83. 48xy 85 —Ox2 + 3x +9 87. 16x*- 625 89. 4х? — 28х + 49 oi a $54,647; overestimates by $556 b. M — W = 1.8х° — 82х? + 2644х — 11,449 e. $14,434 d. $15,136; underestimates by $702 93. 4x? — 36x2 + 80x 95. 6x + 22 103. makes sense 105. makes sense 107. false 109. false 111. 49x% + 70x + 25 — 16y” 113. 6x" — 13 115.4 116.2 117.3 Mid-Chapter P Check Point x2 1. 12х = х = 35 2-x+12 8.10V6 43V3 5. x+45 6 64x2 -48x+9 7.5 в y 3 9. —х? + 5х — 6 10. 2х — 11x + 17x = 5 11. —x% + 2х3 12. 18a? — 11ab — 1062 13. (a,c,d,e,f,h} 14 {c,d} 15. 5) + 2wy - Y Ajo Answers to Selected Exercises AA3 12 15 6 3 + 8 4177 18 112x107 20.27 5-4 2x +4l+4 2.1073 2 7+1v3 ; т ‘( 46 25. "va 26. 1 0,045 V25 27. V13-2 28. -x? 20. 1.38 x 10°pounds — 30. 4times 31. a. model2 b. overestimates by $5 Section P.5 Check Point Exercises 1. а. 2х*(5х — 2) b. (x — Dx + 3) 2. (х + 5)(х° — 2) 3 (x + 8)(x + 5) or (x + 5)(х + 8) 4, (x — Dx +2)or (x + Xx —7) 5 (3x — 1)(2х + 7) от (2х + 7)(3х — 1) 6. (3x — y)(x — 4y) or (x — 4y)(3x — y) 7. a. (x + 9)(x — 9) b. (6x + 5)(6x — 5) 8. (9х? + 4)(3х + 2)(3х - 2) 9 a (x +77? в. (4х — 7)? 10. а. (х + )(х — х + 1) b (5x — 2)(25x* + 10x + 4) 11. 3x(x — 5° 2x — 1 | 12. (x + 10 + 6a)(x + 10 — 6a) 13. a - DE Concept and Vocabulary Check 1d 2g 3b 4c 5e 6a 7.Ё 8 (x+ 1) Exercise Set P.§ | | 1. 9(2х + 3) 3 3x(x+2) |5. 9х%(х? — 2х + 3) T(x+SHx+3) %A-3IEÉ+1) 11.([email protected]?*+5) 134-167 +2) 15. (3x — 2)(х? — 2) 17. (x + 2)(x + 3) 19. (x — 5)(х + 3) 21. (x — 5)(x — 3) 23. (3x + 2Xx — 1) 25. (3х — 28)(х + 1) 27. (2х — 1)(3х — 4) 29. (2х + 3)(2х + 5) 231. 3x- 2)3x— 1) 133. (5x + 8)(4х — 1) 386 (2х + у)(х + у) 37. (3x + 2y)(2x — 3y) 39. (x + 10)(x — 10) 41. (6х + 7)(бх — 7) 48. (3x + 5у)(3х — 5у) 45. (x“ + (x + (x — 2) | 47. (4х? + 9)(2х + 3)(2х — 3) 49. (х +1? 51. (к — 7)? 58. (2х + 1)? 86 (3х = 1)? 57. (x + 3)(x* — 3x +9) 59 (x — 4)(х? + 4х + 16) 61. (2х — 14x? + 2х + 1) 68. (4x + 3)(16х? — 12х + 9) 66 3x(x + (x — 1) 67. 4(x + 2)(x — 3) 69. 2(x* + 9)(х + 3)(х — 3) 71. (x — 3)(x + 3)(x +2) 78 20x — 8)(х +7) | ТБ. х(х = 2)(х + 2) 77 prime 78 (x — 2)(х + 2)? 81. yO” + 9 + 3 — 3) 83. 5y(2y + 3)(2y — 3) 85. (x —6 + 7у)(х — 6 — 7y) 87. (x + y)(3b + 4)(3b — 4) — 89. (y — 2)(x + 4)(x — 4) 05 4(1 + 2x) x+4 4(4x — D(x — 1) TB (x + 5% vi 3 103. (x + 1)(5x — 6)(2x +1) 105, (x? + 6)(6x? — 1) 107. y(Y? + 19 — y? +1) 109. (1 + 2» — ду)(х + y) — y) 111. (x — y)*(x —y +2)x — y —2) 113. (2x - y) — 37) 115. а. (x — 0.4х)(1 — 0.4) = (0.6x)(0.6) = 0.36x b: no;36% 117. а. 9х? — 16 b. (3x + 4)(3x — 4) 119, a. x(x + у) — у(х + y) b. (x + y)(x — y) 121. 40% — 4ab” = 4a(a + bY(a — b) 10 (x — (x + 592 91. 2x(x + 6 + 2a)(x + 6 — 2a) — 93. x*(x — 1) 97. —(x + 3Yx +2) 9. 131. makes sense 133. makes sense 135. true 137. false 139. —(x + 5)(x — 1) 141. — à ni (к + 5) + 1) _x+1 2 7 | . = — + > > . = 145, — Co 143. b = 0,3,4,—c(c + 4), where c > 0 is an integer 144 +5 -5) x—5 45 3 146 6 | section P.6 р Check Point Exercises ; 1. а. —5 b 6-6 —2,7 2 2x #3 b. rx q — 23 # 2,x% # —2,x # —3 | . à . 6, c. —2, . A. XX IT 3 &TDa+3p" x E | =D orn 8-2x2-1 в ЕЙ 41551 7 (em 3) — Be + Bore 3 + 3 | a+ ‚X E . —2, Xx «+ DE -D?* „X . (Xx — 3)(x — 3)(x + 3) or (x — 3)“(x + 3) | 2 — ; —x* + 11x — 20 2(2 — 3x) 4 1 an —— == A, XA р 2 — 5) ‚х * 5 9 1 + ar x # 0,x 7 3 Xe + 7) х É 0,x # —7 | Concept and Vocabulary Check : x 3 x" = х + 4 1. polynomials 2. domain;0 8. factoring; common factors 4. 15 5. 5 T3 7. х + 3andx — 2; x + 3andx +1; x — (x + 3) —3 1 _— + , + . . 1 10. = == (x + 3)(x — 2)(x + 1) 8 3x +4 9. complex; complex 30x + 3) 3x(x + 3) Xx + 3) Exercise Set P.6 13 85-5 5-1-10 ‚лез 0°" %:x6 2, yx12 10 F066 185x223 es x — 3 4 y —1 x — 6 3 A x — 3)(x +3 — 1 2.47 + 4 — 92 EME, 4 0,—4,3 19. x ,X * —2,-1,2,3 21. xo # —2,0,2 23. 1, #-i 25. ) ‚х # 0,—2,2 x(x + 4) x + 2 3x 9 2(x +3 — х + 2)(х + 4 — 27. +3). # 3,—3 29, x > e 1,—5 31. rx + 4) % —6,—3,—1,3,5 33. 2,x # _2 35. 2x Ly # 0, —3 ЗЕ 9х + 39 3 3х? + 4 à A Zak A A-5 48 ——— x #—-1,0 AX A, | (x + dx +35)" x(x + 1) Da 77 22 37. 3,x # 2 a0. — x # 3,4 mM 3° AA4 Answers to Selected Exercises 2x* + 50 13 4x + 16 Xx — x 4. ————————,X £ -5,5 AQ, — = Am ‚ ==, хо # =3 B3, ‚х * —5,2,-3 (x = 5)(x + 5)” 6(x +2)” (x + 372° (x + 5)(x — 2)(x + 3)" 2 Xx —2x +1 x—1 1 x +1 1 1 Le x 5 —1,1 87 x # —2, —1 > 61. ‚х + 0,= Lx #0, ‚х # — 5 Da 1? ух 59. 3x #3 я 0.3 63. Xx 7 0,y 7 0% % y — 14 — 2x + h 24+ 5х + 6. ——x 22,3 6x2 22 6 yw 2-13 NA AO UN pu x + 3 7 x +2 x*(x + hY | (x + 2)(x + 1) 1 2d ; 75. 2 77. — 79. 7% 1. a. 86. 2 586 i 40% i inst this strai O +5) ab = be 81. a. 86.67, 520, 1170; It costs $86,670,000 to inoculate 6 of the population against this strain of flu, $520,000,000 to inoculate 80% of the population, and $1,170,000,000 to inoculate 90% of the population. b. x = 100 e. The cost increases rapidly; it is impossible to inoculate 100% of the population. — 83. a. 2078; underestimates by 22 calories — b. 2662; underestimates by 38 calories —33х° + 263х + 515 4x” + 22x x—1 с. ‚ щие 99. does not make sense 101. makes sense 103. true 105. false 107. —60x” + 499x + 295 (x + 5)(x + 6) х + 3 109. It cubes x. 110. —3; 0; 3; 4; 3; 0; —5 111. —8; —3; 0; 1; 0; —3; —8 112. 3; 2; 1; 0; 1; 2; 3 Chapter P Review Exercises | 1.5 216 312 4 fac) 5. fa,b,c,d,e) 6 {a,b,c,d fg} 7 (a) 8a VEL b0v8 e —-17,0V8 d. —17, == 0,0.75,V81 e V2,m ff 7, 0,0.75,V2,7,V81 9.103 10. VZ-1 ня. VT7-3 12 |4- (-17)|; 21 13 13. commutative property of addition . 14. associative property of multiplication 15. distributive property of multiplication over addition 16. commutative property of multiplication 17. commutative property of multiplication 18. commutative property of addition 1 1 19. 17x — 15 20. 2x 21. 5y — 17 22 10x 23. 55; It’s the same. 24. —108 28. = 2.55 2.77 & —8x*y” 0 1 5 | 29. 10 30. —— 31. — 32. 37,400 33. 0.0000745 34. 3.59 x 10° 35. 7.25 x 107 36. 390,000 37. 0.023 x8 16x12 4x8 | ; 38. 135 x 10” 39. 3.2 X 107 40, 42,188 years 41. 10V3 42. 2|х| УЗ 4325 M VF 4, > 46. 4xV3 6 5(6 — V3 | | 47. 20\/5 48. 16V2 49.24V2-8V3 50.6V5 81. V6 52. 3) 53. 7(V7 + V5) 54.5 55. —2 33 1 56. not areal number 57,5 58. 33 | 59. у)? 60.245 61.137 62 xV2 63.4 Ga. = 65 6. 1 68. = 69. 20x12 70. 3x 71. 25x* 720 Vy 78 8x? + 10x? — 20x — 4;degree3 74. 8x* — 5x3 + 6; degree 4 75. 120° + x* — 21x +10 76. 6x* — Tx — 5 77. 16x32 ~ 25 78 4x? +20x +25 79. 9x2 — 24x +16 80. 8x3 + 12x2 + 6x +1 81. 125х° — 150х° + 60x — 8 82. —x* — 17xy — 3y*; degree 2 83. 24x)? + xP — 124% + 4; degree S 84. 3x? + 16xy — 35y* 85. 9x“ — 30xy + 25y* 86. 9x* + 12x’y + 4y? 87. 49x? ~ 16y> 88. a3-b? 89. 3х?(5х + 1) 90. (x — 4)(x — 7) 91. (3x + 1)(5x — 2) 92 (8 —x)(8 + x) 93. prime 94. 3x'(x — 5)(x +2) 95. 4x°(5x* - 9) 96. (x + 3)(x — 3% 97 (4x — 5 98. (х° + 4)(x + 2)(x —2) 99. (y—2)(y? +2y +4) 100. (x + 4)(x? — 4x +16) 101. 3xX(x — 2)(x + 2) 67. 16 о | | 16(1 + 2х) 102. (3x — 5)(9x* + 15х + 25) 103. x(x ~ D(x + Dx? + 1) 104. (x? — 2)(x + 5) 105, (x + 9 + y)(x + 9 — y) 06. er X | | 6(2x + 1 — 3 | 107. (x + 2)(x — 2)(x* + 3) 4{—x* + x? +13) 108. ex +1) 109. x2 x # —2 110. x 5-66 A111, —— x 5% ~2 | x ; x—6 x +2 x + 3 3 2 1 + ; 1 . . a 7 2,—2 18, ———— x #.0,1,~1, — 114, X =. # —3,4,2,8 118, ——, x # 3,—3 (x — 2) (x + 2) x(x + 1) 3 x — 4 x — 3”. 4х(х — 1) 2х? — 3 Mx ~ x — 11 1 2 3 | 116. —————mx € 2,-2 47. ‚х * 3, — 118. XxX E -,—-3,—= 419, — x # 0,2 | (x + 2)(x — 2) * (x — 3)(x + 3)(x — 2) 7 32 (2x — D(x + 3)(3x + 2) "72 3 X 3x 3x + 8 10 120. ZZ 7 = 0,4,—4 121. 3 + 10* = 7 Chapter P Test 5y* 36 — V2 1, 6x” — 27x 2. -6x +17 3 {5} 4 (1,2,5,4) 5. бх?у3 + 4ху + 2у? 6 2 7.3rV2 , 811V2 9. > X + 3 ` 2{x + 3 10. 2xW2x 11. ha £21 12.25 x 100 13 2x? — 13x% + 26x — 15 14 256% + 30xy + 992 45. etd) £ 3,-1,-4,73 x — 2 х +1 x“ + 2х + 15 11 3-х . ‚ —3 17. с ‚4 (x — 20. (x* + + 16 (x + 3)}x - 3)" 73 (x — 3)(x — 4) x #3 18 354 = 0 19. (х — 3) — 6) {x“ + 3)(х + 2) + 21. (5х — 3)(5х + 3) 22. (6x -7)* 28. (у - 5)()? + 5) + 25) 24 (х + 5 + Зу)(х + 5 — Зу) 25 с: 7 | X 4 22 . " en dinar ны A 26. —7,——, 0, 0.25, V4, 7 27. commutative property of addition 28. distributive property of multiplication over addition 29. 7.6 x 10 5 ; —0.28n + 4 30. 743 31, 1.32 x 101° 32. a. 43.08%; overestimates by 0.08% b. R = 0.28n 7 degrees for every two men.; It describes the projections exactly. 2 3 i helor's 0.281 + 53 e = Three women will receive bac A y dr

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