algebra & trigonometry graphs & models 3rd ed - marvin l. bittinger

R ina wants to establish a college fund for her newborn daughter that will have accumulated $120,000 at the end of 18 yr. If she can count on an interest rate of 6%, compounded monthly, how much should she deposit each month to accomplish this? This problem appears as Exercise 95 in Section R.2. G Basic Concepts of Algebra R.1 The Real-Number System R.2 Integer Exponents, Scientific Notation, and Order of Operations R.3 Addition, Subtraction, and Multiplication of Polynomials R.4 Factoring R.5 Rational Expressions R.6 Radical Notation and Rational Exponents R.7 The Basics of Equation Solving SUMMARY AND REVIEW TEST APPLICATION BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 1 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley 2.1 Polynomial Functions and Modeling 2 Chapter R • Basic Concepts of Algebra R.1 The Real-Number System Identify various kinds of real numbers. Use interval notation to write a set of numbers. Identify the properties of real numbers. Find the absolute value of a real number. Real Numbers In applications of algebraic concepts, we use real numbers to represent quantities such as distance, time, speed, area, profit, loss, and tempera- ture. Some frequently used sets of real numbers and the relationships among them are shown below. Real numbers Rational numbers Negative integers: −1, −2, −3, … Natural numbers (positive integers): 1, 2, 3, … Zero: 0 −, − −, −−, −−, 8.3, 0.56, … 2 3 4 5 19 −5 −7 8 − Whole numbers: 0, 1, 2, 3, … Rational numbers that are not integers: Integers: …, −3, −2, −1, 0, 1, 2, 3, … Irrational numbers: −4.030030003…, … √2, p, −√3, √27, 54 Numbers that can be expressed in the form , where p and q are in- tegers and , are rational numbers.Decimal notation for rational numbers either terminates (ends) or repeats.Each of the following is a rational number. a) 0 for any nonzero integer a b) Ϫ7 ,or c) Te r minating decimal d) Repeating decimalϪ 5 11 ෇ Ϫ0.45 1 4 ෇ 0.25 7 ؊1 ؊7 ؍ ؊7 1 0 ؍ 0 a q  0 p͞q BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 2 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley The real numbers that are not rational are irrational numbers.Decimal notation for irrational numbers neither terminates nor repeats. Each of the following is an irrational number. a) There is no repeating block of digits. and 3.14 are rational approximations of the irrational number b) There is no repeating block of digits. c) Although there is a pattern, there is no repeating block of digits. The set of all rational numbers combined with the set of all irrational numbers gives us the set of real numbers.The real numbers are modeled using a number line, as shown below. Each point on the line represents a real number, and every real number is represented by a point on the line. The order of the real numbers can be determined from the number line. If a number a is to the left of a number b, then a is less than b .Similarly, a is greater than b if a is to the right of b on the number line. For example, we see from the number line above that ,because Ϫ2.9 is to the left of . Also, , because is to the right of . The statement , read “a is less than or equal to b,” is true if either is true or is true. The symbol ʦ is used to indicate that a member, or element,belongs to a set. Thus if we let represent the set of rational numbers, we can see from the diagram on page 2 that . We can also write to indi- cate that is not an element of the set of rational numbers. When all the elements of one set are elements of a second set, we say that the first set is a subset of the second set. The symbol ʕ is used to denote this. For instance, if we let represent the set of real numbers, we can see from the diagram that (read “ is a subset of ”). Interval Notation Sets of real numbers can be expressed using interval notation.For example, for real numbers a and b such that , the open interval is the set of real numbers between, but not including, a and b.That is, . The points a and b are endpoints of the interval. The parentheses indicate that the endpoints are not included in the interval. Some intervals extend without bound in one or both directions. The interval , for example, begins at a and extends to the right without bound. That is, . The bracket indicates that a is included in the interval. ͓a, ϱ͒ ෇ ͕x ͉x Ն a͖ ͓a, ϱ͒ ͑a, b͒ ෇ ͕x ͉a Ͻ x Ͻ b͖ ͑a, b͒a Ͻ b ޒޑޑ ʕ ޒ ޒ ͙ 2 ͙ 2  ޑ0.56 ʦ ޑ ޑ a ෇ ba Ͻ b a Յ b ͙ 3 17 4 17 4 Ͼ ͙ 3Ϫ 3 5 Ϫ2.9 ϽϪ 3 5 ͑a Ͼ b͒͑a Ͻ b͒ Ϫ2.9 ϪE ͙3 p * Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 123450 Ϫ6.12122122212222 ͙ 2 ෇ 1.414213562 ␲ . ͒͑ 22 7 ␲ ෇ 3.1415926535 Section R.1 • The Real-Number System 3 ( ) ab (a, b) [ a [a, ∞) BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 3 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley The various types of intervals are listed below. 4 Chapter R • Basic Concepts of Algebra The interval , graphed below, names the set of all real num- bers, . EXAMPLE 1 Write interval notation for each set and graph the set. a) b) c) d) Solution a) ; b) ; c) ; 01Ϫ4Ϫ5 Ϫ2 Ϫ1Ϫ3 2345 ͕x͉Ϫ5 Ͻ x ՅϪ2͖ ෇ ͑Ϫ5,Ϫ2͔ 01Ϫ4Ϫ5 Ϫ2 Ϫ1Ϫ3 2345 ͕x͉x Ն 1.7͖ ෇ ͓1.7,ϱ͒ 01Ϫ4Ϫ5 Ϫ2 Ϫ1Ϫ3 2345 ͕x͉Ϫ4 Ͻ x Ͻ 5͖ ෇ ͑Ϫ4, 5͒ ͕ x ͉ x Ͻ ͙ 5 ͖ ͕x͉Ϫ5 Ͻ x ՅϪ2͖ ͕x͉x Ն 1.7͖͕x͉Ϫ4 Ͻ x Ͻ 5͖ ޒ ͑Ϫ ϱ, ϱ͒ Intervals: Types, Notation, and Graphs INTERVAL SET TYPE NOTATION NOTATION GRAPH Open Closed Half-open Half-open Open Half-open Open Half-open ] b ͕x ͉ x Յ b͖͑Ϫϱ, b͔ ) b ͕x ͉ x Ͻ b͖͑Ϫϱ, b͒ [ a ͕x ͉ x Ն a͖͓a, ϱ͒ ( a ͕x ͉ x Ͼ a͖͑a, ϱ͒ ( ] ab ͕x ͉ a Ͻ x Յ b͖͑a, b͔ [ ) ab ͕x ͉ a Յ x Ͻ b͖͓a, b͒ [ ] ab ͕x ͉ a Յ x Յ b͖͓a, b͔ ( ) ab ͕x ͉ a Ͻ x Ͻ b͖͑a, b͒ BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 4 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Section R.1 • The Real-Number System 5 d) ; Properties of the Real Numbers The following properties can be used to manipulate algebraic expressions as well as real numbers. Properties of the Real Numbers For any real numbers a, b, and c: and Commutative properties of addition and multiplication and Associative properties of addition and multiplication Additive identity property Additive inverse property Multiplicative identity property Multiplicative inverse property Distributive property Note that the distributive property is also true for subtraction since . EXAMPLE 2 State the property being illustrated in each sentence. a) b) c) d) e) Solution SENTENCE PROPERTY a) Commutative property of multiplication: b) Associative property of addition: c) Additive inverse property: d) Multiplicative identity property: e) Distributive property: a͑b ϩ c͒ ෇ ab ϩ ac 2͑a Ϫ b͒ ෇ 2a Ϫ 2b a и 1 ෇ 1 и a ෇ a 6 и 1 ෇ 1 и 6 ෇ 6 a ϩ ͑Ϫa͒ ෇ 014 ϩ ͑Ϫ14͒ ෇ 0 a ϩ ͑b ϩ c͒ ෇ ͑a ϩ b͒ ϩ c 5 ϩ ͑m ϩ n͒ ෇ ͑5 ϩ m͒ ϩ n ab ෇ ba 8 и 5 ෇ 5 и 8 2͑a Ϫ b͒ ෇ 2a Ϫ 2b 6 и 1 ෇ 1 и 6 ෇ 614 ϩ ͑Ϫ14͒ ෇ 0 5 ϩ ͑m ϩ n͒ ෇ ͑5 ϩ m͒ ϩ n8 и 5 ෇ 5 и 8 a͑b Ϫ c͒ ෇ a͓b ϩ ͑Ϫc͔͒ ෇ ab ϩ a͑Ϫc͒ ෇ ab Ϫ ac a͑b ϩ c͒ ෇ ab ϩ ac ͑a  0͒a и 1 a ෇ 1 a и a ෇ 1 a и 1 ෇ 1 и a ෇ a Ϫa ϩ a ෇ a ϩ ͑Ϫa͒ ෇ 0 a ϩ 0 ෇ 0 ϩ a ෇ a a͑bc͒ ෇ ͑ab͒c a ϩ ͑b ϩ c͒ ෇ ͑a ϩ b͒ ϩ c ab ෇ ba a ϩ b ෇ b ϩ a 01Ϫ4Ϫ5 Ϫ2 Ϫ1Ϫ32345 ͕ x ͉ x Ͻ ͙ 5 ͖ ෇ ͑ Ϫϱ, ͙ 5 ͒ BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 5 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley 6 Chapter R • Basic Concepts of Algebra ab ͉a Ϫ b͉ ϭ ͉b Ϫ a͉ Absolute Value The number line can be used to provide a geometric interpretation of absolute value.The absolute value of a number a,denoted , is its dis- tance from 0 on the number line. For example, , because the distance of Ϫ5 from 0 is 5. Similarly, , because the distance of from 0 is . Absolute Value For any real number a, When a is nonnegative, the absolute value of a is a.When a is negative, the absolute value of a is the opposite, or additive inverse, of a.Thus, is never negative; that is, for any real number a,. Absolute value can be used to find the distance between two points on the number line. Distance Between Two Points on the Number Line For any real numbers a and b, the distance between a and b is ,or equivalently, . EXAMPLE 3 Find the distance between Ϫ2 and 3. Solution The distance is ,or equivalently, . We can also use the absolute-value operation on a graphing calculator to find the distance between two points. On many graphing calculators, ab- solute value is denoted “abs” and is found in the MATH NUM menu and also in the CATALOG. 5 abs (3Ϫ( Ϫ 2)) 5 abs ( Ϫ 2Ϫ3) ͉3 Ϫ ͑Ϫ2͉͒ ෇ ͉3 ϩ 2͉ ෇ ͉5͉ ෇ 5 ͉Ϫ2 Ϫ 3͉ ෇ ͉Ϫ5͉ ෇ 5 ͉b Ϫ a͉͉a Ϫ b͉ ͉a͉ Ն 0͉a͉ ͉a͉ ෇ ͭ a, Ϫa, if a Ն 0, if a Ͻ 0. 3 4 3 4 Խ 3 4 Խ ෇ 3 4 ͉Ϫ5͉ ෇ 5 ͉a͉ GCM BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 6 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Section R.1 • The Real-Number System 7 In Exercises 1– 10, consider the numbers Ϫ12,,, ,,0,,,,Ϫ1.96, 9, ,,,. 1. Which are whole numbers? ,0,9, 2. Which are integers? Ϫ12, , 0,9, 3. Which are irrational numbers? 4. Which are natural numbers? ,9, 5. Which are rational numbers? 6. Which are real numbers? All of them 7. Which are rational numbers but not integers? 8. Which are integers but not whole numbers? Ϫ12 9. Which are integers but not natural numbers? Ϫ12, 0 10. Which are real numbers but not integers? Ճ Write interval notation. Then graph the interval. 11. Ճ 12. Ճ 13. Ճ 14. Ճ 15. Ճ 16. Ճ 17. Ճ 18. Ճ 19. Ճ 20. Ճ Write interval notation for the graph. 21. 22. 23. 24. 25. 26. 27. 28. In Exercises 29 –46, the following notation is used: the set of natural numbers, the set of whole numbers, the set of integers, the set of rational numbers, the set of irrational numbers, and the set of real numbers. Classify the statement as true or false. 29. Tr ue 30. Tr ue 31. False 32. Tr ue 33. Tr ue 34. False 35. False 36. False 37. False 38. Tr ue 39. True 40. Tr ue 41. Tr ue 42. False 43. True 44. Tr u e 45. False 46. False Name the property illustrated by the sentence. 47. Commutative property of multiplication 48. Associative property of addition 49. 50. Ճ Multiplicative identity property 51. Ճ 52. Distributive property 4͑y Ϫ z͒ ෇ 4y Ϫ 4z5͑ab͒ ෇ ͑5a͒b x ϩ 4 ෇ 4 ϩ xϪ3 и 1 ෇ Ϫ3 3 ϩ ͑x ϩ y͒ ෇ ͑3 ϩ x͒ ϩ y 6 и x ෇ x и 6 ޑ ʕ މޒ ʕ ޚ ޚ ʕ ޑޑ ʕ ޒ ޚ ʕ ގޗ ʕ ޚ ގ ʕ ޗ1.089  މ 1 ʦ ޚ24  ޗ Ϫ1 ʦ ޗ ͙ 11  ޒ Ϫ ͙ 6 ʦ ޑϪ 11 5 ʦ ޑ Ϫ10.1 ʦ ޒ3.2 ʦ ޚ 0  ގ6 ʦ ގ ޒ ෇ މ ෇ ޑ ෇ޚ ෇ ޗ ෇ގ ෇ q ] ͑Ϫϱ, q͔ p ( ͑ p, ϱ͒ ( ] xx ϩ h ͑x, x ϩ h͔ [ ] xx ϩ h ͓x, x ϩ h͔ Ϫ10 Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ3 Ϫ2 Ϫ1 012Ϫ4 ( ] ͑Ϫ9, Ϫ5͔ Ϫ10 Ϫ9 Ϫ8 Ϫ7 Ϫ6 Ϫ5 Ϫ3 Ϫ2 Ϫ1 012Ϫ4 [ ) ͓Ϫ9, Ϫ4͒ Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 1234560 [ ] ͓Ϫ1, 2͔ Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 1234560 ( ) ͑0, 5͒ ͕x ͉Ϫ3 Ͼ x͖͕x ͉7 Ͻ x͖ ͕ x ͉x Ն ͙ 3 ͖ ͕x ͉x Ͼ 3.8͖ ͕x ͉x ϾϪ5͖͕x͉ x ՅϪ2͖ ͕x ͉1 Ͻ x Յ 6͖͕x ͉Ϫ4 Յ x ϽϪ1͖ ͕x ͉Ϫ4 Ͻ x Ͻ 4͖͕x ͉Ϫ3 Յ x Յ 3͖ ͙ 25 ͙ 3 8 ͙ 25 ͙ 3 8 ͙ 25 ͙ 3 8 5 7 ͙ 3 4 ͙ 254 2 3 ͙ 5 5Ϫ ͙ 145.242242224 . . . ͙ 3 8Ϫ 7 3 5.3 ͙ 7 Exercise Set R.1 ,, ,,͙ 3 4 ͙ 5 5Ϫ ͙ 14 5.242242224 . . . ͙ 7 ,,Ϫ1.96, , 5 7 4 2 3 Ϫ 7 3 5.3 Ϫ12, , , , 0, Ϫ1.96, 9, , , 5 7 ͙ 254 2 3 ͙ 3 8Ϫ 7 3 5.3 Ճ Answers to Exercises 10–20, 50, and 51 can be found on p. IA-1. BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 7 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley 53. 54. Commutative property of multiplication 55. Commutative property of addition 56. Additive identity property 57. Multiplicative inverse property 58. Distributive property Simplify. 59. 7.1 60. 86.2 61. 347 62. 54 63. 64. 65. 0 66. 15 67. 68. Find the distance between the given pair of points on the number line. 69. Ϫ5, 6 11 70. Ϫ2.5, 0 2.5 71. Ϫ8, Ϫ2 6 72. , 73. 6.7, 12.1 5.4 74. Ϫ14, Ϫ3 11 75. , 76. Ϫ3.4, 10.2 13.6 77. Ϫ7, 0 7 78. 3, 19 16 Collaborative Discussion and Writing To the student and the instructor: The Collaborative Discussion and Writing exercises are meant to be answered with one or more sentences. These exercises can also be discussed and answered collaboratively by the entire class or by small groups. Because of their open-ended nature, the answers to these exercises do 21 8 15 8 Ϫ 3 4 1 24 23 12 15 8 ͙ 3 Խ Ϫ ͙ 3 Խ 5 4 ͉ 5 4 ͉ ͉15͉͉0͉ 12 19 ͉ 12 19 ͉ ͙ 97 Խ Ϫ ͙ 97 Խ ͉Ϫ54͉͉347͉ ͉Ϫ86.2͉͉Ϫ7.1͉ 9x ϩ 9y ෇ 9͑x ϩ y͒ 8 и 1 8 ෇ 1 t ϩ 0 ෇ t Ϫ6͑m ϩ n͒ ෇ Ϫ6͑n ϩ m͒ Ϫ7 ϩ 7 ෇ 02͑a ϩ b͒ ෇ ͑a ϩ b͒2 8 Chapter R • Basic Concepts of Algebra Ճ Answer to Exercise 85 can be found on p. IA-1. not appear at the back of the book. They are denoted by the words “Discussion and Writing.” 79. How would you convince a classmate that division is not associative? 80. Under what circumstances is a rational number? Synthesis To the student and the instructor: The Synthesis exercises found at the end of every exercise set challenge students to combine concepts or skills studied in that section or in preceding parts of the text. Between any two (different) real numbers there are many other real numbers. Find each of the following. Answers may vary. 81. An irrational number between 0.124 and 0.125 Answers may vary; 82. A rational number between and Answers may vary; Ϫ1.415 83. A rational number between and Answers may vary; Ϫ0.00999 84. An irrational number between and Answers may vary; 85. The hypotenuse of an isosceles right triangle with legs of length 1 unit can be used to “measure” a value for by using the Pythagorean theorem, as shown. c 1 1 ͙ 2 ͙ 5.995 ͙ 6 ͙ 5.99 Ϫ 1 100 Ϫ 1 101 Ϫ ͙ 2Ϫ ͙ 2.01 0.124124412444 . . . ͙ a c ෇ ͙ 2 c 2 ෇ 2 c 2 ෇ 1 2 ϩ 1 2 Draw a right triangle that could be used to “measure” units. Ճ ͙ 10 Additive inverse property BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 8 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley Section R.2 • Integer Exponents, Scientific Notation, and Order of Operations 9 R.2 Integer Exponents, Scientific Notation, and Order of Operations Simplify expressions with integer exponents. Solve problems using scientific notation. Use the rules for order of operations. Integers as Exponents When a positive integer is used as an exponent, it indicates the number of times a factor appears in a product. For example, means and means 5. For any positive integer n, , n factors where a is the base and n is the exponent. Zero and negative-integer exponents are defined as follows. For any nonzero real number a and any integer m, and . EXAMPLE 1 Simplify each of the following. a) b) Solution a) b) EXAMPLE 2 Write each of the following with positive exponents. a) b) c) Solution a) b) c) x Ϫ3 y Ϫ8 ෇ x Ϫ3 и 1 y Ϫ8 ෇ 1 x 3 и y 8 ෇ y 8 x 3 1 ͑0.82͒ Ϫ7 ෇ ͑0.82͒ Ϫ͑Ϫ7͒ ෇ ͑0.82͒ 7 4 Ϫ5 ෇ 1 4 5 x Ϫ3 y Ϫ8 1 ͑0.82͒ Ϫ7 4 Ϫ5 ͑Ϫ3.4͒ 0 ෇ 16 0 ෇ 1 ͑Ϫ3.4͒ 0 6 0 a Ϫm ෇ 1 a m a 0 ෇ 1 a n ෇ a и a и a иииa 5 1 7 и 7 и 77 3        BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 9 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley 10 Chapter R • Basic Concepts of Algebra The results in Example 2 can be generalized as follows. For any nonzero numbers a and b and any integers m and n, . (A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed.) EXAMPLE 3 Write an equivalent expression without negative exponents: . Solution Since each exponent is negative, we move each factor to the other side of the fraction bar and change the sign of each exponent: . The following properties of exponents can be used to simplify expressions. Properties of Exponents For any real numbers a and b and any integers m and n, assuming 0 is not raised to a nonpositive power: Product rule Quotient rule Power rule Raising a product to a power Raising a quotient to a power EXAMPLE 4 Simplify each of the following. a) b) c) d) e) ͩ 45x Ϫ4 y 2 9z Ϫ8 ͪ Ϫ3 ͑2s Ϫ2 ͒ 5 ͑t Ϫ3 ͒ 5 48x 12 16x 4 y Ϫ5 и y 3 ͑b  0͒ ͩ a b ͪ m ෇ a m b m ͑ab͒ m ෇ a m b m ͑a m ͒ n ෇ a mn ͑a  0͒ a m a n ෇ a mϪn a m и a n ෇ a mϩn x Ϫ3 y Ϫ8 z Ϫ10 ෇ z 10 x 3 y 8 x Ϫ3 y Ϫ8 z Ϫ10 a Ϫm b Ϫn ෇ b n a m BBEPMC0R_0312279093.QXP 12/2/04 2:42 PM Page 10 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley [...]... used for numbers greater than or equal to 10 and negative exponents for numbers between 0 and 1 EXAMPLE 5 Undergraduate Enrollment In a recent year, there were 16,539,000 undergraduate students enrolled in post-secondary institutions in the United States (Source: U.S National Center for Education Statistics) Convert this number to scientific notation Solution We want the decimal point to be positioned... computation in Example 9(b) is entered in a calculator, enter the computation without using these parentheses What is the result? Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP 14 Chapter R • 12/2/04 2:42 PM Page 14 Basic Concepts of Algebra EXAMPLE 10 Compound Interest If a principal P is invested at an interest rate r, compounded n times per year, in t... 13-week period? $3.4749 ϫ 10 7 77 Chesapeake Bay Bridge-Tunnel The 17.6-mile-long Chesapeake Bay Bridge-Tunnel was completed in 1964 Construction costs were $210 million Find the average cost per mile $1.19 ϫ 10 7 64 The mass of a proton is about 1.67 ϫ 10 Ϫ24 g 78 Personal Space in Hong Kong The area of Hong Kong is 412 square miles It is estimated that the population of Hong Kong will be 9,600,000... that $2125 is invested at 6.2%, compounded semiannually How much is in the account at the end of 5 yr? $2883.67 88 Suppose that $9550 is invested at 5.4%, compounded semiannually How much is in the account at the end of 7 yr? $13,867.23 gives the amount S accumulated in a savings plan when a deposit of P dollars is made each month for t years in an account with interest rate r, compounded monthly Use this... daughter that will have accumulated $120,000 at the end of 18 yr If she can count on an interest rate of 6%, compounded monthly, how much should she deposit each month to accomplish this? $309.79 89 Suppose that $6700 is invested at 4.5%, compounded quarterly How much is in the account at the end of 6 yr? $8763.54 90 Suppose that $4875 is invested at 5.8%, compounded quarterly How much is in the account... checked by multiplying Some trials show that the desired factorization is ͑3x ϩ 2͒ ͑x Ϫ 4͒ The Grouping Method The second method for factoring trinomials of the type ax 2 ϩ bx ϩ c , a 1, is known as the grouping method, or the ac-method Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP 26 Chapter R 12/2/04 2:42 PM Page 26 • Basic Concepts of Algebra. .. FACTORS Ϫ1, Ϫ12 Ϫ13 Ϫ2, Ϫ6 Ϫ8 Ϫ3, Ϫ4 Ϫ7 The numbers we need are ؊3 and ؊4 Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP 24 Chapter R 12/2/04 2:42 PM Page 24 • Basic Concepts of Algebra The factorization of y 2 Ϫ 7y ϩ 12 is ͑ y Ϫ 3͒ ͑ y Ϫ 4͒ We must also include the common factor that we factored out earlier Thus we have 2y 2 Ϫ 14y ϩ 24 ෇ 2͑ y Ϫ 3͒... Ϫ1, SUMS OF FACTORS 8 7 1, Ϫ8 Ϫ7 Ϫ2, 4 2 2, Ϫ4 Ϫ2 The numbers we need are 2 and ؊4 We might have observed at the outset that since the sum of the factors is Ϫ2, a negative number, we need consider only pairs of factors for which the negative factor has the greater absolute value Thus only the pairs 1, Ϫ8 and 2, Ϫ4 need have been considered Using the pair of factors 2 and Ϫ4, we see that the factorization... Copyright © 2006 Pearson Education, Inc., publishing as Pearson Addison-Wesley BBEPMC0R_0312279093.QXP 20 Chapter R • 12/2/04 2:42 PM Page 20 Basic Concepts of Algebra We can find the product of two binomials by multiplying the First terms, then the Outer terms, then the Inner terms, then the Last terms Then we combine like terms, if possible This procedure is sometimes called FOIL EXAMPLE 6 Solution... month beginning at age 40 If the investment earns 5% interest, compounded monthly, how much will have accumulated in the account when she retires 27 yr later? $170,797.30 94 Gordon deposits $100 in a retirement account each month beginning at age 25 If the investment earns 4% interest, compounded monthly, how much will have accumulated in the account when Gordon retires at age 65? $118,196.13 95 Gina . polynomial. •Add, subtract, and multiply polynomials. Polynomials Polynomials are a type of algebraic expression that you will often encounter in your study of algebra. Some examples of polynomials. various types of intervals are listed below. 4 Chapter R • Basic Concepts of Algebra The interval , graphed below, names the set of all real num- bers, . EXAMPLE 1 Write interval notation for each. Chesapeake Bay Bridge-Tunnel. The 17.6-mile-long Chesapeake Bay Bridge-Tunnel was completed in 1964. Construction costs were $210 million. Find the average cost per mile. 78. Personal Space in Hong