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Applied Mathematics For The Managerial, Life, And Social Sciences, 6th Edition Solution Manual

Applied Mathematics For The Managerial, Life, And Social Sciences, 6th Edition Solution Manual makes tackling textbook exercises a breeze, with clear and concise answers to every problem.

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    CONTENTS CHAPTER 1 Fundamentals of Algebra 1 1.1 Real Numbers 1 1.2 Polynomials 3 1.3 Factoring Polynomials 6 1.4 Rational Expressions 8 1.5 Integral Exponents 12 1.6 Solving Equations 14 1.7 Rational Exponents and Radicals 21 1.8 Quadratic Equations 25 1.9 Inequalities and Absolute Value 35 Chapter 1 Review 41 Chapter 1 Before Moving On 47 CHAPTER 2 Functions and Their Graphs 49 2.1 The Cartesian Coordinate System and Straight Lines 49 2.2 Equations of Lines 51 2.3 Functions and Their Graphs 61 2.4 The Algebra of Functions 71 2.5 Linear Functions 78 2.6 Quadratic Functions 84 2.7 Functions and Mathematical Models 93 Chapter 2 Review 101 Chapter 2 Before Moving On 107 CHAPTER 3 Exponential and Logarithmic Functions 111 3.1 Exponential Functions 111 3.2 Logarithmic Functions 117 3.3 Exponential Functions as Mathematical Models 122 Chapter 3 Review 127 Chapter 3 Before Moving On 130 v

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    vi CONTENTS CHAPTER 4 Mathematics of Finance 133 4.1 Compound Interest 133 4.2 Annuities 139 4.3 Amortization and Sinking Funds 144 4.4 Arithmetic and Geometric Progressions 155 Chapter 4 Review 158 Chapter 4 Before Moving On 162 CHAPTER 5 Systems of Linear Equations and Matrices 167 5.1 Systems of Linear Equations: An Introduction 167 5.2 Systems of Linear Equations: Unique Solutions 172 5.3 Systems of Linear Equations: Underdetermined and Overdetermined Systems 186 5.4 Matrices 196 5.5 Multiplication of Matrices 202 5.6 The Inverse of a Square Matrix 213 Chapter 5 Review 231 Chapter 5 Before Moving On 240 CHAPTER 6 Linear Programming 245 6.1 Graphing Systems of Linear Inequalities in Two Variables 245 6.2 Linear Programming Problems 251 6.3 Graphical Solutions of Linear Programming Problems 260 6.4 The Simplex Method: Standard Maximization Problems 280 6.5 The Simplex Method: Standard Minimization Problems 309 Chapter 6 Review 330 Chapter 6 Before Moving On 343 CHAPTER 7 Sets and Counting 349 7.1 Sets and Set Operations 349 7.2 The Number of Elements in a Finite Set 354 7.3 The Multiplication Principle 360 7.4 Permutations and Combinations 363 7.5 Experiments, Sample Spaces, and Events 369

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    CONTENTS vii 7.6 Probability 372 7.7 Rules of Probability 376 Chapter 7 Review 382 Chapter 7 Before Moving On 386 CHAPTER 8 Additional Topics in Probability 389 8.1 Use of Counting Techniques in Probability 389 8.2 Conditional Probability and Independent Events 393 8.3 Bayes ’ Theorem 401 8.4 Distributions of Random Variables 410 8.5 Expected Value 413 8.6 Variance and Standard Deviation 418 Chapter 8 Review 425 Chapter 8 Before Moving On 428 CHAPTER 9 The Derivative 431 9.1 Limits 431 9.2 One-Sided Limits and Continuity 439 9.3 The Derivative 448 9.4 Basic Rules of Differentiation 461 9.5 The Product and Quotient Rules; Higher-Order Derivatives 469 9.6 The Chain Rule 480 9.7 Differentiation of Exponential and Logarithmic Functions 492 9.8 Marginal Functions in Economics 499 Chapter 9 Review 503 Chapter 9 Before Moving On 514 CHAPTER 10 Applications of the Derivative 523 10.1 Applications of the First Derivative 523 10.2 Applications of the Second Derivative 537 10.3 Curve Sketching 550 10.4 Optimization I 571 10.5 Optimization II 586 Chapter 10 Review 594 Chapter 10 Before Moving On 609

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    viii CONTENTS CHAPTER 11 Integration 615 11.1 Antiderivatives and the Rules of Integration 615 11.2 Integration by Substitution 623 11.3 Area and the Definite Integral 630 11.4 The Fundamental Theorem of Calculus 633 11.5 Evaluating Definite Integrals 638 11.6 Area Between Two Curves 645 11.7 Applications of the Definite Integral to Business and Economics 655 Chapter 11 Review 659 Chapter 11 Before Moving On 667 CHAPTER 12 Calculus of Several Variables 673 12.1 Functions of Several Variables 673 12.2 Partial Derivatives 678 12.3 Maxima and Minima of Functions of Several Variables 684 Chapter 12 Review 693 Chapter 12 Before Moving On 698

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    1 FUNDAMENTALS OF ALGEBRA 1.1 Real Numbers Concept Questions page 6 1. The set of natural numbers is N = { 1 , 2 , 3 , . . . } ; the set of whole numbers is W = { 0 , 1 , 2 , 3 , . . . } ; the set of integers is I = { . . ., − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , . . . } ; the set of rational numbers is Q = { a / b | a and b are integers and b / = 0 } (example: 1 / 2), and the set of irrational numbers contains all real numbers that cannot be expressed in the form a / b , where a and b are integers and b / = 0 (example: π ). The set of real numbers contains all irrational and rational numbers. 2. a. The associative law of addition states that a + ( b + c ) = ( a + b ) + c . b. The distributive law states that ab + ac = a ( b + c ) . 3. If ab / = 0, then neither a nor b is equal to zero. If abc / = 0, then none of a , b , and c is equal to zero. Exercises page 6 1. The number − 3 is an integer, a rational number, and a real number. 2. The number − 420 is an integer, a rational number, and a real number. 3. The number 3 8 is a rational real number. 4. The number − 4 125 is a rational real number. 5. The number √ 11 is an irrational real number. 6. The number − √ 5 is an irrational real number. 7. The number π 2 is an irrational real number. 8. The number 2 π is an irrational real number. 9. The number 2 . 421 is a rational real number. 10. The number 2 . 71828 . . . is an irrational real number. 11. False. − 2 is not a whole number. 12. True. 13. True. 14. True. 15. False. No natural number is irrational. 16. True. 17. ( 2 x + y ) + z = z + ( 2 x + y ) : The Commutative Law of Addition. 18. 3 x + ( 2 y + z ) = ( 3 x + 2 y ) + z : The Associative Law of Addition. 19. u ( 3 v + w) = ( 3 v + w) u : The Commutative Law of Multiplication. 20. a 2 b b 2 c c = b a 2 b 2 c c : The Associative Law of Multiplication. 1

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    2 1 FUNDAMENTALS OF ALGEBRA 21. u ( 2 v + w) = 2 u v + u w : The Distributive Law. 22. ( 2 u + v) w = 2 u w + vw : The Distributive Law. 23. ( 2 x + 3 y ) + ( x + 4 y ) = 2 x + d 3 y + ( x + 4 y ) e : The Associative Law of Addition. 24. ( a + 2 b ) ( a − 3 b ) = a ( a − 3 b ) + 2 b ( a − 3 b ) : The Distributive Law. 25. a − [ − ( c + d ) ] = a + ( c + d ) : Property 1 of negatives. 26. − ( 2 x + y ) d − ( 3 x + 2 y ) e = ( 2 x + y ) ( 3 x + 2 y ) : Property 3 of negatives. 27. 0 ( 2 a + 3 b ) = 0: Property 1 involving zero. 28. If ( x − y ) ( x + y ) = 0, then x = y or x = − y . Property 2 involving zero. 29. If ( x − 2 ) ( 2 x + 5 ) = 0, then x = 2, or x = − 5 2 . Property 2 involving zero. 30. If x ( 2 x − 9 ) = 0, then x = 0 or x = 9 2 . Property 2 involving zero. 31. ( x + 1 ) ( x − 3 ) ( 2 x + 1 ) ( x − 3 ) = x + 1 2 x + 1 . Property 2 of quotients. 32. ( 2 x + 1 ) ( x + 3 ) ( 2 x − 1 ) ( x + 3 ) = 2 x + 1 2 x − 1 . Property 2 of quotients. 33. a + b b ÷ a − b ab = a ( a + b ) a − b . Properties 2 and 5 of quotients. 34. x + 2 y 3 x + y ÷ x 6 x + 2 y = x + 2 y 3 x + y · 2 ( 3 x + y ) x = 2 ( x + 2 y ) x . Properties 2 and 5 of quotients and the Distributive Law. 35. a b + c + c b = ab + bc + c 2 b ( b + c ) . Property 6 of quotients and the Distributive Law. 36. x + y x + 1 − y x = x 2 − y x ( x + 1 ) . Property 7 of quotients and the Distributive Law. 37. False. Consider a = 2 and b = 1 2 . Then ab = 1, but a / = 1 and b / = 1. 38. True. Multiplying both sides of the equation by 1 a (which exists because a / = 0), we have 1 a ( ab ) = 1 a ( 0 ) , or b = 0. 39. False. Consider a = 3 and b = 2. Then a − b = 3 − 2 / = b − a = 2 − 3 = − 1. 40. False. Consider a = 3 and b = 2. Then a b = 3 2 / = b a = 2 3 . 41. False. Consider a = 1, b = 2, and c = 3. Then ( a − b ) − c = ( 1 − 2 ) − 3 = − 4 / = a − ( b − c ) = 1 − ( 2 − 3 ) = 2. 42. False. Consider a = 1, b = 2, and c = 3. Then a b / c = 1 2 / 3 = 3 2 / = a / b c = 1 / 2 3 = 1 6 .

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    1.2 POLYNOMIALS 3 1.2 Polynomials Concept Questions page 13 1. A polynomial of degree n in x is an expression of the form a n x n + a n − 1 x n − 1 + · · · + a 1 x + a 0 , where n is a nonnegative integer and a 0 , a 1 , . . . , a n are real numbers with a n / = 0. One polynomial of degree 4 in x is x 4 + 2 x 3 − 2 x 2 − 5 x − 7. 2. (a) 1 + 2 b + b 2 b. a 2 − 2 ab + b 2 c. a 2 − b 2 Exercises page 13 1. 3 4 = 3 · 3 · 3 · 3 = 81. 2. ( − 2 ) 5 = ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 ) ( − 2 ) = − 32. 3. r 2 3 s 3 = r 2 3 s r 2 3 s r 2 3 s = 8 27 4. r − 3 4 s 2 = r − 3 4 s r − 3 4 s = 9 16 . 5. − 3 4 = − 3 · 3 · 3 · 3 = − 81. 6. − r − 4 5 s 3 = − r − 4 5 s r − 4 5 s r − 4 5 s = 64 125 . 7. − 3 r 3 5 s 3 = ( − 3 ) r 3 5 s r 3 5 s r 3 5 s = − 81 125 . 8. r − 2 3 s 2 r − 3 4 s 3 = r 4 9 s r − 27 64 s = − 3 16 . 9. 2 3 · 2 5 = 2 8 = 256. 10. ( − 3 ) 2 · ( − 3 ) 3 = ( − 3 ) 5 = − 243. 11. ( 3 y ) 2 ( 3 y ) 3 = ( 3 y ) 5 = 243 y 5 . 12. ( − 2 x ) 3 ( − 2 x ) 2 = ( − 2 x ) 5 = − 32 x 5 . 13. ( 2 x + 3 ) + ( 4 x − 6 ) = 2 x + 3 + 4 x − 6 = 6 x − 3. 14. ( − 3 x + 2 ) − ( 4 x − 3 ) = − 3 x + 2 − 4 x + 3 = − 7 x + 5. 15. b 7 x 2 − 2 x + 5 c + b 2 x 2 + 5 x − 4 c = 7 x 2 − 2 x + 5 + 2 x 2 + 5 x − 4 = 7 x 2 + 2 x 2 − 2 x + 5 x + 5 − 4 = 9 x 2 + 3 x + 1. 16. b 3 x 2 + 5 x y + 2 y c + b 4 − 3 x y − 2 x 2 c = x 2 + 2 x y + 2 y + 4. 17. b 5 y 2 − 2 y + 1 c − b y 2 − 4 y − 8 c = 5 y 2 − 2 y + 1 − y 2 + 4 y + 8 = 5 y 2 − y 2 − 2 y + 4 y + 1 + 8 = 4 y 2 + 2 y + 9. 18. b 2 x 2 − 3 x + 4 c − b − x 2 + 2 x − 6 c = 2 x 2 − 3 x + 4 + x 2 − 2 x + 6 = 3 x 2 − 5 x + 10. 19. b 2 . 4 x 3 − 3 x 2 + 1 . 7 x − 6 . 2 c − b 1 . 2 x 3 + 1 . 2 x 2 − 0 . 8 x + 2 c = 2 . 4 x 3 − 3 x 2 + 1 . 7 x − 6 . 2 − 1 . 2 x 3 − 1 . 2 x 2 + 0 . 8 x − 2 = 1 . 2 x 3 − 4 . 2 x 2 + 2 . 5 x − 8 . 2. 20. b 1 . 4 x 3 − 1 . 2 x 2 + 3 . 2 c − b − 0 . 8 x 3 − 2 . 1 x − 1 . 8 c = 1 . 4 x 3 − 1 . 2 x 2 + 3 . 2 + 0 . 8 x 3 + 2 . 1 x + 1 . 8 = 2 . 2 x 3 − 1 . 2 x 2 + 2 . 1 x + 5. 21. b 3 x 2 c b 2 x 3 c = 6 x 5 . 22. b − 2 rs 2 c b 4 r 2 s 2 c ( 2 s ) = − 16 r 3 s 5 . 23. − 2 x b x 2 − 2 c + 4 x 3 = − 2 x 3 + 4 x + 4 x 3 = 2 x 3 + 4 x .

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    4 1 FUNDAMENTALS OF ALGEBRA 24. x y ( 2 y − 3 x ) = 2 x y 2 − 3 x 2 y . 25. 2 m ( 3 m − 4 ) + m ( m − 1 ) = 6 m 2 − 8 m + m 2 − m = 7 m 2 − 9 m . 26. − 3 x b 2 x 2 + 3 x − 5 c + 2 x b x 2 − 3 c = − 6 x 3 − 9 x 2 + 15 x + 2 x 3 − 6 x = − 4 x 3 − 9 x 2 + 9 x . 27. 3 ( 2 a − b ) − 4 ( b − 2 a ) = 6 a − 3 b − 4 b + 8 a = 6 a + 8 a − 3 b − 4 b = 14 a − 7 b . 28. 2 ( 3 m − 1 ) − 3 ( − 4 m + 2 n ) = 6 m − 2 + 12 m − 6 n = 18 m − 6 n − 2. 29. ( 2 x + 3 ) ( 3 x − 2 ) = 2 x ( 3 x − 2 ) + 3 ( 3 x − 2 ) = 6 x 2 − 4 x + 9 x − 6 = 6 x 2 + 5 x − 6. 30. ( 3 r − 1 ) ( 2 r + 5 ) = 3 r ( 2 r + 5 ) − ( 2 r + 5 ) = 6 r 2 + 15 r − 2 r − 5 = 6 r 2 + 13 r − 5. 31. ( 2 x − 3 y ) ( 3 x + 2 y ) = 2 x ( 3 x + 2 y ) − 3 y ( 3 x + 2 y ) = 6 x 2 + 4 x y − 9 x y − 6 y 2 = 6 x 2 − 5 x y − 6 y 2 . 32. ( 5 m − 2 n ) ( 5 m + 3 n ) = 5 m ( 5 m + 3 n ) − 2 n ( 5 m + 3 n ) = 25 m 2 + 15 mn − 10 mn − 6 n 2 = 25 m 2 + 5 mn − 6 n 2 . 33. ( 3 r + 2 s ) ( 4 r − 3 s ) = 3 r ( 4 r − 3 s ) + 2 s ( 4 r − 3 s ) = 12 r 2 − 9 rs + 8 rs − 6 s 2 = 12 r 2 − rs − 6 s 2 . 34. ( 2 m + 3 n ) ( 3 m − 2 n ) = 2 m ( 3 m − 2 n ) + 3 n ( 3 m − 2 n ) = 6 m 2 − 4 mn + 9 mn − 6 n 2 = 6 m 2 + 5 mn − 6 n 2 . 35. ( 0 . 2 x + 1 . 2 y ) ( 0 . 3 x − 2 . 1 y ) = 0 . 2 x ( 0 . 3 x − 2 . 1 y ) + 1 . 2 y ( 0 . 3 x − 2 . 1 y ) = 0 . 06 x 2 − 0 . 42 x y + 0 . 36 x y − 2 . 52 y 2 = 0 . 06 x 2 − 0 . 06 x y − 2 . 52 y 2 . 36. ( 3 . 2 m − 1 . 7 n ) ( 4 . 2 m + 1 . 3 n ) = 3 . 2 m ( 4 . 2 m + 1 . 3 n ) − 1 . 7 n ( 4 . 2 m + 1 . 3 n ) = 13 . 44 m 2 + 4 . 16 mn − 7 . 14 mn − 2 . 21 n 2 = 13 . 44 m 2 − 2 . 98 mn − 2 . 21 n 2 . 37. ( 2 x − y ) b 3 x 2 + 2 y c = 2 x b 3 x 2 + 2 y c − y b 3 x 2 + 2 y c = 6 x 3 − 3 x 2 y + 4 x y − 2 y 2 . 38. b 3 m − 2 n 2 c b 2 m 2 + 3 n c = 3 m b 2 m 2 + 3 n c − 2 n 2 b 2 m 2 + 3 n c = 6 m 3 + 9 mn − 4 m 2 n 2 − 6 n 3 . 39. ( 2 x + 3 y ) 2 = ( 2 x ) 2 + 2 ( 2 x ) ( 3 y ) + ( 3 y ) 2 = 4 x 2 + 12 x y + 9 y 2 . 40. ( 3 m − 2 n ) 2 = ( 3 m ) 2 − 2 ( 3 m ) ( 2 n ) + ( 2 n ) 2 = 9 m 2 − 12 mn + 4 n 2 . 41. ( 2 u − v) ( 2 u + v) = ( 2 u ) 2 − v 2 = 4 u 2 − v 2 . 42. ( 3 r + 4 s ) ( 3 r − 4 s ) = ( 3 r ) 2 − ( 4 s ) 2 = 9 r 2 − 16 s 2 . 43. ( 2 x − 1 ) 2 + 3 x − 2 b x 2 + 1 c + 3 = 4 x 2 − 4 x + 1 + 3 x − 2 x 2 − 2 + 3 = 2 x 2 − x + 2. 44. ( 3 m + 2 ) 2 − 2 m ( 1 − m ) − 4 = 9 m 2 + 12 m + 4 − 2 m + 2 m 2 − 4 = 11 m 2 + 10 m . 45. ( 2 x + 3 y ) 2 − ( 2 y + 1 ) ( 3 x − 2 ) + 2 ( x − y ) = 4 x 2 + 12 x y + 9 y 2 − 6 x y − 3 x + 4 y + 2 + 2 x − 2 y = 4 x 2 + 6 x y + 9 y 2 − x + 2 y + 2. 46. ( x − 2 y ) ( y + 3 x ) − 2 x y + 3 ( x + y − 1 ) = x y + 3 x 2 − 2 y 2 − 6 x y − 2 x y + 3 x + 3 y − 3 = 3 x 2 − 7 x y − 2 y 2 + 3 x + 3 y − 3.

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    1.2 POLYNOMIALS 5 47. b t 2 − 2 t + 4 c b 2 t 2 + 1 c = b t 2 − 2 t + 4 c b 2 t 2 c + b t 2 − 2 t + 4 c ( 1 ) = 2 t 4 − 4 t 3 + 8 t 2 + t 2 − 2 t + 4 = 2 t 4 − 4 t 3 + 9 t 2 − 2 t + 4. 48. b 3 m 2 − 1 c b 2 m 2 + 3 m − 4 c = 3 m 2 b 2 m 2 + 3 m − 4 c − b 2 m 2 + 3 m − 4 c = 6 m 4 + 9 m 3 − 12 m 2 − 2 m 2 − 3 m + 4 = 6 m 4 + 9 m 3 − 14 m 2 − 3 m + 4. 49. 2 x − { 3 x − [ x − ( 2 x − 1 ) ] } = 2 x − { 3 x − [ x − 2 x + 1] } = 2 x − [3 x − ( − x + 1 ) ] = 2 x − ( 3 x + x − 1 ) = 2 x − ( 4 x − 1 ) = 2 x − 4 x + 1 = − 2 x + 1. 50. 3 m − 2 { m − 3 [2 m − ( m − 5 ) ] + 4 } = 3 m − 2 [ m − 3 ( 2 m − m + 5 ) + 4] = 3 m − 2 [ m − 3 ( m + 5 ) + 4] = 3 m − 2 ( m − 3 m − 15 + 4 ) = 3 m − 2 ( − 2 m − 11 ) = 3 m + 4 m + 22 = 7 m + 22. 51. x − { 2 x − [ − x − ( 1 + x ) ] } = x − [2 x − ( − x − 1 − x ) ] = x − [2 x − ( − 2 x − 1 ) ] = x − ( 2 x + 2 x + 1 ) = x − 4 x − 1 = − 3 x − 1. 52. 3 x 2 − j x 2 + 1 − x [ x − ( 2 x − 1 ) ] k + 2 = 3 x 2 − d x 2 + 1 − x ( x − 2 x + 1 ) e + 2 = 3 x 2 − d x 2 + 1 − x ( − x + 1 ) e + 2 = 3 x 2 − b x 2 + 1 + x 2 − x c + 2 = 3 x 2 − 2 x 2 − 1 + x + 2 = x 2 + x + 1. 53. ( 2 x − 3 ) 2 − 3 ( x + 4 ) ( x − 4 ) + 2 ( x − 4 ) + 1 = ( 2 x ) 2 − 2 ( 2 x ) ( 3 ) + 3 2 − 3 b x 2 − 16 c + 2 x − 8 + 1 = 4 x 2 − 12 x + 9 − 3 x 2 + 48 + 2 x − 7 = x 2 − 10 x + 50. 54. ( x − 2 y ) 2 + 2 ( x + y ) ( x − 3 y ) + x ( 2 x + 3 y + 2 ) = x 2 − 2 x ( 2 y ) + ( 2 y ) 2 + 2 b x 2 − 3 x y + x y − 3 y 2 c + 2 x 2 + 3 x y + 2 x = x 2 − 4 x y + 4 y 2 + 2 x 2 − 4 x y − 6 y 2 + 2 x 2 + 3 x y + 2 x = 5 x 2 − 5 x y − 2 y 2 + 2 x . 55. 2 x { 3 x [2 x − ( 3 − x ) ] + ( x + 1 ) ( 2 x − 3 ) } = 2 x d 3 x ( 2 x − 3 + x ) + 2 x 2 − 3 x + 2 x − 3 e = 2 x d 3 x ( 3 x − 3 ) + 2 x 2 − x − 3 e = 2 x b 9 x 2 − 9 x + 2 x 2 − x − 3 c = 2 x b 11 x 2 − 10 x − 3 c = 22 x 3 − 20 x 2 − 6 x . 56. − 3 d ( x + 2 y ) 2 − ( 3 x − 2 y ) 2 + ( 2 x − y ) ( 2 x + y ) e = − 3 d x 2 + 4 x y + 4 y 2 − b 9 x 2 − 12 x y + 4 y 2 c + b 4 x 2 − y 2 ce = − 3 b x 2 + 4 x y + 4 y 2 − 9 x 2 + 12 x y − 4 y 2 + 4 x 2 − y 2 c = − 3 b − 4 x 2 + 16 x y − y 2 c = 12 x 2 − 48 x y + 3 y 2 . 57. The total weekly pro fi t is given by the revenue minus the cost: b − 0 . 04 x 2 + 2000 x c − b 0 . 000002 x 3 − 0 . 02 x 2 + 1000 x + 120,000 c = − 0 . 04 x 2 + 2000 x − 0 . 000002 x 3 + 0 . 02 x 2 − 1000 x − 120,000 = − 0 . 000002 x 3 − 0 . 02 x 2 + 1000 x − 120,000. 58. The total revenue is given by x p = x ( − 0 . 0004 x + 10 ) = − 0 . 0004 x 2 + 10 x . Therefore, the total pro fi t is given by the revenue minus the cost: − 0 . 0004 x 2 + 10 x − b 0 . 0001 x 2 + 4 x + 400 c = − 0 . 0005 x 2 + 6 x − 400. 59. The total revenue is given by b 0 . 2 t 2 + 150 t c + b 0 . 5 t 2 + 200 t c = 0 . 7 t 2 + 350 t thousand dollars t months from now, where 0 ≤ t ≤ 12.

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    6 1 FUNDAMENTALS OF ALGEBRA 60. In month t , the revenue of the second gas station will exceed that of the fi rst gas station by b 0 . 5 t 2 + 200 t c − b 0 . 2 t 2 + 150 t c = 0 . 3 t 2 + 50 t thousand dollars, where 0 < t ≤ 12. 61. The gap is given by b 3 . 5 t 2 + 26 . 7 t + 436 . 2 c − ( 24 . 3 t + 365 ) = 3 . 5 t 2 + 2 . 4 t + 71 . 2. 62. The difference is given by b 2 . 5 t 2 + 18 . 5 t + 509 c − b − 1 . 1 t 2 + 29 . 1 t + 429 c = 3 . 6 t 2 − 10 . 6 t + 80 dollars. The difference at the beginning of 1998 is obtained by replacing t with 4, giving 3 . 6 ( 4 ) 2 − 10 . 6 ( 4 ) + 80 = 95 . 2, or $95 . 20. The difference at the beginning of 2000 is given by 3 . 6 ( 6 ) 2 − 10 . 6 ( 6 ) + 80 = 146, or $146. 63. False. Let a = 2, b = 3, m = 3, and n = 2. Then 2 3 · 3 2 = 8 · 9 = 72 / = ( 2 · 3 ) 3 + 2 = 6 5 . 64. True. 65. False. For example, x 2 + 1 is a polynomial of degree 2 and x is a polynomial of degree 1, but b x 2 + 1 c x = x 3 + x is a polynomial of degree 3, not 2. 66. False. For example, p = x 3 + x + 1 is a polynomial of degree 3 and q = − x 3 + 2 is a polynomial of degree 3, but p + q = x 3 + x + 1 + b − x 3 + 2 c = x + 3 is a polynomial of degree 1. 67. The degree of p − q is m . To see this, suppose that p = a m x m + · · · + a n x n + · · · + a 0 and q = b n x n + · · · + b 0 . Because m > n , p − q = a m x m + · · · + ( a n − b n ) x n + · · · + ( a 0 − b 0 ) has degree m . 1.3 Factoring Polynomials Concept Questions page 19 1. A polynomial is completely factored over the set of integers if it is expressed as a product of prime polynomials with integral coef fi cients. An example is 4 x 2 − 9 y 2 = ( 2 x − 3 y ) ( 2 x + 3 y ) . 2. a. ( a + b ) b a 2 − ab + b 2 c b. ( a − b ) b a 2 + ab + b 2 c Exercises page 19 1. 6 m 2 − 4 m = 2 m ( 3 m − 2 ) . 2. 4 t 4 − 12 t 3 = 4 t 3 ( t − 3 ) . 3. 9 ab 2 − 6 a 2 b = 3 ab ( 3 b − 2 a ) . 4. 12 x 3 y 5 + 16 x 2 y 3 = 4 x 2 y 3 b 3 x y 2 + 4 c . 5. 10 m 2 n − 15 mn 2 + 20 mn = 5 mn ( 2 m − 3 n + 4 ) . 6. 6 x 4 y − 4 x 2 y 2 + 2 x 2 y 3 = 2 x 2 y b 3 x 2 − 2 y + y 2 c . 7. 3 x ( 2 x + 1 ) − 5 ( 2 x + 1 ) = ( 2 x + 1 ) ( 3 x − 5 ) . 8. 2 u b 3 v 2 + w c + 5 v b 3 v 2 + w c = b 3 v 2 + w c ( 2 u + 5 v) . 9. ( 3 a + b ) ( 2 c − d ) + 2 a ( 2 c − d ) 2 = ( 2 c − d ) [3 a + b + 2 a ( 2 c − d ) ] = ( 2 c − d ) ( 3 a + b + 4 ac − 2 ad ) . 10. 4 u v 2 ( 2 u − v) + 6 u 2 v (v − 2 u ) = b 4 u v 2 − 6 u 2 v c ( 2 u − v) = 2 u v ( 2 u − v) ( 2 v − 3 u ) . 11. 2 m 2 − 11 m − 6 = ( 2 m + 1 ) ( m − 6 ) . 12. 6 x 2 − x − 1 = ( 3 x + 1 ) ( 2 x − 1 ) . 13. x 2 − x y − 6 y 2 = ( x − 3 y ) ( x + 2 y ) . 14. 2 u 2 + 5 u v − 12 v 2 = ( 2 u − 3 v) ( u + 4 v) .

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    1.3 FACTORING POLYNOMIALS 7 15. x 2 − 3 x − 1 is prime. 16. m 2 + 2 m + 3 is prime. 17. 4 a 2 − b 2 = ( 2 a − b ) ( 2 a + b ) . 18. 12 x 2 − 3 y 2 = 3 b 4 x 2 − y 2 c = 3 ( 2 x − y ) ( 2 x + y ) . 19. u 2 v 2 − w 2 = ( u v) 2 − w 2 = ( u v − w) ( u v + w) . 20. 4 a 2 b 2 − 25 c 2 = ( 2 ab ) 2 − ( 5 c ) 2 = ( 2 ab − 5 c ) ( 2 ab + 5 c ) . 21. z 2 + 4 is prime. 22. u 2 + 25 v 2 is prime. 23. x 2 + 6 x y + y 2 is prime. 24. 4 u 2 − 12 u v + 9 v 2 = ( 2 u − 3 v) 2 . 25. x 2 + 3 x − 4 = ( x + 4 ) ( x − 1 ) . 26. 3 m 3 + 3 m 2 − 18 m = 3 m b m 2 + m − 6 c = 3 m ( m + 3 ) ( m − 2 ) . 27. 12 x 2 y − 10 x y − 12 y = 2 y b 6 x 2 − 5 x − 6 c = 2 y ( 3 x + 2 ) ( 2 x − 3 ) . 28. 12 x 2 y − 2 x y − 24 y = 2 y b 6 x 2 − x − 12 c = 2 y ( 3 x + 4 ) ( 2 x − 3 ) . 29. 35 r 2 + r − 12 = ( 7 r − 4 ) ( 5 r + 3 ) . 30. 6 u v 2 + 9 u v − 6 v = 3 v ( 2 u v + 3 u − 2 ) . 31. 9 x 3 y − 4 x y 3 = x y b 9 x 2 − 4 y 2 c = x y d ( 3 x ) 2 − ( 2 y ) 2 e = x y ( 3 x − 2 y ) ( 3 x + 2 y ) . 32. 4 u 4 v − 9 u 2 v 3 = u 2 v b 4 u 2 − 9 v 2 c = u 2 v d ( 2 u ) 2 − ( 3 v) 2 e = u 2 v ( 2 u − 3 v) ( 2 u + 3 v) . 33. x 4 − 16 y 2 = b x 2 c 2 − ( 4 y ) 2 = b x 2 − 4 y c b x 2 + 4 y c . 34. 16 u 4 v − 9 v 3 = v b 16 u 4 − 9 v 2 c = v K b 4 u 2 c 2 − ( 3 v) 2 L = v b 4 u 2 − 3 v c b 4 u 2 + 3 v c . 35. ( a − 2 b ) 2 − ( a + 2 b ) 2 = [ ( a − 2 b ) − ( a + 2 b ) ] [ ( a − 2 b ) + ( a + 2 b ) ] = ( − 4 b ) ( 2 a ) = − 8 ab . 36. 2 x ( x + y ) 2 − 8 x b x + y 2 c 2 = 2 x K ( x + y ) 2 − 4 b x + y 2 c 2 L = 2 x d ( x + y ) − 2 b x + y 2 ce d ( x + y ) + 2 b x + y 2 ce = 2 x b y − x − 2 y 2 c b 3 x + y + 2 y 2 c . 37. 8 m 3 + 1 = ( 2 m ) 3 + 1 = ( 2 m + 1 ) b 4 m 2 − 2 m + 1 c . 38. 27 m 3 − 8 = ( 3 m ) 3 − 2 3 = ( 3 m − 2 ) b 9 m 2 + 6 m + 4 c . 39. 8 r 3 − 27 s 3 = ( 2 r ) 3 − ( 3 s ) 3 = ( 2 r − 3 s ) b 4 r 2 + 6 rs + 9 s 2 c . 40. x 3 + 64 y 3 = x 3 + ( 4 y ) 3 = ( x + 4 y ) b x 2 − 4 x y + 16 y 2 c . 41. u 2 v 6 − 8 u 2 = u 2 b v 6 − 8 c = u 2 b v 2 − 2 c b v 4 + 2 v 2 + 4 c .

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    8 1 FUNDAMENTALS OF ALGEBRA 42. r 6 s 6 + 8 s 3 = s 3 b r 6 s 3 + 8 c = s 3 K b r 2 s c 3 + 2 3 L = s 3 b r 2 s + 2 c b r 4 s 2 − 2 r 2 s + 4 c . 43. 2 x 3 + 6 x + x 2 + 3 = 2 x b x 2 + 3 c + b x 2 + 3 c = b x 2 + 3 c ( 2 x + 1 ) . 44. 2 u 4 − 4 u 2 + 2 u 2 − 4 = 2 u 4 − 2 u 2 − 4 = 2 b u 4 − u 2 − 2 c = 2 b u 2 + 1 c b u 2 − 2 c . 45. 3 ax + 6 ay + bx + 2 by = 3 a ( x + 2 y ) + b ( x + 2 y ) = ( x + 2 y ) ( 3 a + b ) . 46. 6 ux − 4 uy + 3 v x − 2 v y = 2 u ( 3 x − 2 y ) + v ( 3 x − 2 y ) = ( 3 x − 2 y ) ( 2 u + v) . 47. u 4 − v 4 = b u 2 c 2 − b v 2 c 2 = b u 2 − v 2 c b u 2 + v 2 c = ( u − v) ( u + v) b u 2 + v 2 c . 48. u 4 − u 2 v 2 − 6 v 4 = b u 2 − 3 v 2 c b u 2 + 2 v 2 c . 49. 4 x 3 − 9 x y 2 + 4 x 2 y − 9 y 3 = x b 4 x 2 − 9 y 2 c + y b 4 x 2 − 9 y 2 c = d ( 2 x ) 2 − ( 3 y ) 2 e ( x + y ) = ( 2 x − 3 y ) ( 2 x + 3 y ) ( x + y ) . 50. 4 u 4 + 11 u 2 v 2 − 3 v 4 = b 4 u 2 − v 2 c b u 2 + 3 v 2 c = ( 2 u − v) ( 2 u + v) b u 2 + 3 v 2 c . 51. x 4 + 3 x 3 − 2 x − 6 = x 3 ( x + 3 ) − 2 ( x + 3 ) = ( x + 3 ) b x 3 − 2 c . 52. a 2 − b 2 + a + b = ( a − b ) ( a + b ) + ( a + b ) = ( a + b ) ( a − b + 1 ) . 53. au 2 + ( a + c ) u + c = au 2 + au + cu + c = au ( u + 1 ) + c ( u + 1 ) = ( u + 1 ) ( au + c ) . 54. ax 2 − ( 1 + ab ) x y + by 2 = ax 2 − x y − abx y + by 2 = ax ( x − by ) − y ( x − by ) = ( x − by ) ( ax − y ) . 55. P + Prt = P ( 1 + rt ) . 56. − t 3 + 6 t 2 + 15 t = − t b t 2 − 6 t − 15 c . 57. 8000 x − 100 x 2 = 100 x ( 80 − x ) . 58. R = k Qx − kx 2 = kx ( Q − x ) . 59. k M x − kx 2 = kx ( M − x ) . 60. − 0 . 1 x 2 + 500 x = − 0 . 1 x ( x − 5000 ) . 61. V = V 0 + V 0 273 T = V 0 273 ( 273 + T ) . 62. k D 2 2 − D 3 3 = D 2 t k 2 − D 3 u . 1.4 Rational Expressions Concept Questions page 25 1. a. Quotients of polynomials are rational expressions; 2 x 2 + 1 3 x 2 − 3 x + 4 . b. Any polynomial P can be written in the form P 1 , but not all rational expressions can be written as a polynomial. 2. a. P R Q S ; P S R Q . b. P + Q R ; P − Q R .

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    1.4 RATIONAL EXPRESSIONS 9 Exercises page 25 1. 28 x 2 7 x 3 = 4 x . 2. 3 y 4 18 y 2 = 1 6 y 2 . 3. 4 x + 12 5 x + 15 = 4 ( x + 3 ) 5 ( x + 3 ) = 4 5 . 4. 12 m − 6 18 m − 9 = 6 ( 2 m − 1 ) 9 ( 2 m − 1 ) = 2 3 . 5. 6 x 2 − 3 x 6 x 2 = 3 x ( 2 x − 1 ) 6 x 2 = 2 x − 1 2 x . 6. 8 y 2 4 y 3 − 4 y 2 + 8 y = 8 y 2 4 y b y 2 − y + 2 c = 2 y y 2 − y + 2 . 7. x 2 + x − 2 x 2 + 3 x + 2 = ( x + 2 ) ( x − 1 ) ( x + 2 ) ( x + 1 ) = x − 1 x + 1 . 8. 2 y 2 − y − 3 2 y 2 + y − 1 = ( 2 y − 3 ) ( y + 1 ) ( 2 y − 1 ) ( y + 1 ) = 2 y − 3 2 y − 1 . 9. x 2 − 9 2 x 2 − 5 x − 3 = ( x − 3 ) ( x + 3 ) ( 2 x + 1 ) ( x − 3 ) = x + 3 2 x + 1 . 10. 6 y 2 + 11 y + 3 4 y 2 − 9 = ( 3 y + 1 ) ( 2 y + 3 ) ( 2 y − 3 ) ( 2 y + 3 ) = 3 y + 1 2 y − 3 . 11. x 3 + y 3 x 2 − x y + y 2 = ( x + y ) b x 2 − x y + y 2 c x 2 − x y + y 2 = x + y . 12. 8 r 3 − s 3 2 r 2 + rs − s 2 = ( 2 r − s ) b 4 r 2 + 2 rs + s 2 c ( 2 r − s ) ( r + s ) = 4 r 2 + 2 rs + s 2 r + s . 13. 6 x 3 32 · 8 3 x 2 = 1 2 x . 14. 25 y 4 12 y · 3 y 2 5 y 3 = 5 4 y 2 . 15. 3 x 3 8 x 2 ÷ 15 x 4 16 x 5 = 3 x 3 8 x 2 · 16 x 5 15 x 4 = 2 x 8 5 x 6 = 2 5 x 2 16. 6 x 5 21 x 2 ÷ 4 x 7 x 3 = 6 x 5 21 x 2 · 7 x 3 4 x = 1 2 x 5 . 17. 3 x x + 2 y · 5 x + 10 y 6 = ( 3 x ) 5 ( x + 2 y ) 6 ( x + 2 y ) = 5 x 2 . 18. 4 y + 12 y + 2 · 3 y + 6 2 y − 1 = 4 ( y + 3 ) 3 ( y + 2 ) ( y + 2 ) ( 2 y − 1 ) = 12 ( y + 3 ) 2 y − 1 . 19. 2 m + 6 3 ÷ 3 m + 9 6 = 2 ( m + 3 ) 3 · 6 3 ( m + 3 ) = 4 3 . 20. 3 y − 6 4 y + 6 ÷ 6 y + 24 8 y + 12 = 3 ( y − 2 ) 2 ( 2 y + 3 ) · 4 ( 2 y + 3 ) 6 ( y + 4 ) = y − 2 y + 4 . 21. 6 r 2 − r − 2 2 r + 4 · 6 r + 12 4 r + 2 = ( 3 r − 2 ) ( 2 r + 1 ) 6 ( r + 2 ) 2 ( r + 2 ) 2 ( 2 r + 1 ) = 3 ( 3 r − 2 ) 2 . 22. x 2 − x − 6 2 x 2 + 7 x + 6 · 2 x 2 − x − 6 x 2 + x − 6 = ( x − 3 ) ( x + 2 ) ( 2 x + 3 ) ( x − 2 ) ( 2 x + 3 ) ( x + 2 ) ( x + 3 ) ( x − 2 ) = x − 3 x + 3 . 23. k 2 − 2 k − 3 k 2 − k − 6 ÷ k 2 − 6 k + 8 k 2 − 2 k − 8 = ( k − 3 ) ( k + 1 ) ( k − 3 ) ( k + 2 ) · ( k − 4 ) ( k + 2 ) ( k − 4 ) ( k − 2 ) = k + 1 k − 2 . 24. 6 y 2 − 5 y − 6 6 y 2 + 13 y + 6 ÷ 6 y 2 − 13 y + 6 9 y 2 − 12 y + 4 = ( 3 y + 2 ) ( 2 y − 3 ) ( 3 y + 2 ) ( 2 y + 3 ) · ( 3 y − 2 ) ( 3 y − 2 ) ( 3 y − 2 ) ( 2 y − 3 ) = 3 y − 2 2 y + 3 . 25. 2 2 x + 3 + 3 2 x − 1 = 2 ( 2 x − 1 ) + 3 ( 2 x + 3 ) ( 2 x + 3 ) ( 2 x − 1 ) = 4 x − 2 + 6 x + 9 ( 2 x + 3 ) ( 2 x − 1 ) = 10 x + 7 ( 2 x + 3 ) ( 2 x − 1 ) . 26. 2 x − 1 x + 2 − x + 3 x − 1 = ( 2 x − 1 ) ( x − 1 ) − ( x + 3 ) ( x + 2 ) ( x + 2 ) ( x − 1 ) = 2 x 2 − 3 x + 1 − x 2 − 5 x − 6 ( x + 2 ) ( x − 1 ) = x 2 − 8 x − 5 ( x + 2 ) ( x − 1 ) .

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    10 1 FUNDAMENTALS OF ALGEBRA 27. 3 x 2 − x − 6 + 2 x 2 + x − 2 = 3 ( x − 3 ) ( x + 2 ) + 2 ( x + 2 ) ( x − 1 ) = 3 ( x − 1 ) + 2 ( x − 3 ) ( x − 3 ) ( x + 2 ) ( x − 1 ) = 3 x − 3 + 2 x − 6 ( x − 3 ) ( x + 2 ) ( x − 1 ) = 5 x − 9 ( x − 3 ) ( x + 2 ) ( x − 1 ) . 28. 4 x 2 − 9 − 5 x 2 − 6 x + 9 = 4 ( x − 3 ) ( x + 3 ) − 5 ( x − 3 ) 2 = 4 ( x − 3 ) − 5 ( x + 3 ) ( x + 3 ) ( x − 3 ) 2 = 4 x − 12 − 5 x − 15 ( x + 3 ) ( x − 3 ) 2 = − x + 27 ( x + 3 ) ( x − 3 ) 2 . 29. 2 m 2 m 2 − 2 m − 1 + 3 2 m 2 − 3 m + 3 = 2 m b 2 m 2 − 3 m + 3 c + 3 b 2 m 2 − 2 m − 1 c b 2 m 2 − 2 m − 1 c b 2 m 2 − 3 m + 3 c = 4 m 3 − 6 m 2 + 6 m + 6 m 2 − 6 m − 3 b 2 m 2 − 2 m − 1 c b 2 m 2 − 3 m + 3 c = 4 m 3 − 3 b 2 m 2 − 2 m − 1 c b 2 m 2 − 3 m + 3 c . 30. t t 2 + t − 2 − 2 t − 1 2 t 2 + 3 t − 2 = t ( t + 2 ) ( t − 1 ) − 2 t − 1 ( t + 2 ) ( 2 t − 1 ) = t ( t + 2 ) ( t − 1 ) − 1 t + 2 = t − 1 ( t − 1 ) ( t + 2 ) ( t − 1 ) = t − t + 1 ( t + 2 ) ( t − 1 ) = 1 ( t + 2 ) ( t − 1 ) . 31. x 1 − x + 2 x + 3 x 2 − 1 = − x x − 1 + 2 x + 3 ( x + 1 ) ( x − 1 ) = − x ( x + 1 ) + 2 x + 3 ( x + 1 ) ( x − 1 ) = − x 2 − x + 2 x + 3 ( x + 1 ) ( x − 1 ) = − x 2 − x − 3 ( x + 1 ) ( x − 1 ) . 32. 2 + 1 a + 2 − 2 a a − 2 = 2 ( a + 2 ) ( a − 2 ) + a − 2 − 2 a ( a + 2 ) ( a + 2 ) ( a − 2 ) = 2 a 2 − 8 + a − 2 − 2 a 2 − 4 a ( a + 2 ) ( a − 2 ) = − 3 a + 10 ( a + 2 ) ( a − 2 ) . 33. x − x 2 x + 2 + 2 x − 2 = x ( x + 2 ) ( x − 2 ) − x 2 ( x − 2 ) + 2 ( x + 2 ) ( x + 2 ) ( x − 2 ) = x 3 − 4 x − x 3 + 2 x 2 + 2 x + 4 ( x + 2 ) ( x − 2 ) = 2 x 2 − 2 x + 4 ( x + 2 ) ( x − 2 ) = 2 b x 2 − x + 2 c ( x + 2 ) ( x − 2 ) . 34. y y 2 − 1 + y − 1 y + 1 − 2 y 1 − y = y ( y + 1 ) ( y − 1 ) + y − 1 y + 1 + 2 y y − 1 = y + ( y − 1 ) ( y − 1 ) + 2 y ( y + 1 ) ( y + 1 ) ( y − 1 ) = y + y 2 − 2 y + 1 + 2 y 2 + 2 y ( y + 1 ) ( y − 1 ) = 3 y 2 + y + 1 ( y + 1 ) ( y − 1 ) . 35. x x 2 + 5 x + 6 + 2 x 2 − 4 − 3 x 2 + 3 x + 2 = x ( x + 3 ) ( x + 2 ) + 2 ( x − 2 ) ( x + 2 ) − 3 ( x + 1 ) ( x + 2 ) = x ( x − 2 ) ( x + 1 ) + 2 ( x + 3 ) ( x + 1 ) − 3 ( x + 3 ) ( x − 2 ) ( x + 3 ) ( x + 2 ) ( x − 2 ) ( x + 1 ) = x 3 − x 2 − 2 x + 2 x 2 + 8 x + 6 − 3 x 2 − 3 x + 18 ( x + 3 ) ( x + 2 ) ( x − 2 ) ( x + 1 ) = x 3 − 2 x 2 + 3 x + 24 ( x + 3 ) ( x + 2 ) ( x − 2 ) ( x + 1 ) . 36. 2 x + 1 2 x 2 − x − 1 − x + 1 2 x 2 + 3 x + 1 + 4 x 2 + 2 x − 3 = 2 x + 1 ( 2 x + 1 ) ( x − 1 ) − x + 1 ( 2 x + 1 ) ( x + 1 ) + 4 ( x + 3 ) ( x − 1 ) = 1 x − 1 − 1 2 x + 1 + 4 ( x + 3 ) ( x − 1 ) = ( 2 x + 1 ) ( x + 3 ) − ( x − 1 ) ( x + 3 ) + 4 ( 2 x + 1 ) ( x − 1 ) ( 2 x + 1 ) ( x + 3 ) = 2 x 2 + 7 x + 3 − x 2 − 2 x + 3 + 8 x + 4 ( x − 1 ) ( 2 x + 1 ) ( x + 3 ) = x 2 + 13 x + 10 ( x − 1 ) ( 2 x + 1 ) ( x + 3 )

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    1.4 RATIONAL EXPRESSIONS 11 37. x ax − ay + y by − bx = x a ( x − y ) − y b ( x − y ) = bx − ay ab ( x − y ) . 38. ax + by ax − bx + ay − bx by − ay = ax + by x ( a − b ) + ay − bx − y ( a − b ) = − y ( ax + by ) + x ( ay − bx ) − ( a − b ) x y = − by 2 − bx 2 − ( a − b ) x y = − b b x 2 + y 2 c − x y ( a − b ) = b b x 2 + y 2 c ( a − b ) x y . 39. 1 + 1 x 1 − 1 x = x + 1 x x − 1 x = x + 1 x · x x − 1 = x + 1 x − 1 . 40. 2 + 2 x x − 2 x = 2 x + 2 x x 2 − 2 x = 2 ( x + 1 ) x · x x 2 − 2 = 2 ( x + 1 ) x 2 − 2 . 41. 1 x + 1 y 1 − 1 x y = y + x x y x y − 1 x y = y + x x y · x y x y − 1 = y + x x y − 1 . 42. 1 + x y 1 − x 2 y 2 = y + x y y 2 − x 2 y 2 = x + y y · y 2 ( y − x ) ( y + x ) = y y − x . 43. 1 x 2 − 1 y 2 x + y = y 2 − x 2 x 2 y 2 x + y = ( y + x ) ( y − x ) x 2 y 2 · 1 x + y = y − x x 2 y 2 . 44. 1 x 3 − 1 y 3 1 x − 1 y = y 3 − x 3 x 3 y 3 y − x x y = y 3 − x 3 x 3 y 3 · x y y − x = ( y − x ) b y 2 + x y + x 2 c x 2 y 2 ( y − x ) = y 2 + x y + x 2 x 2 y 2 . 45. 1 2 ( x + h ) − 1 2 x h = x − ( x + h ) 2 x ( x + h ) h = − h 2 x ( x + h ) · 1 h = − 1 2 x ( x + h ) . 46. 1 ( x + h ) 2 − 1 x 2 h = x 2 − ( x + h ) 2 x 2 ( x + h ) 2 h = x 2 − x 2 − 2 xh − h 2 x 2 ( x + h ) 2 · 1 h = − 2 x + h x 2 ( x + h ) 2 . 47. a. 2 . 2 + 2500 x = 2 . 2 x + 2500 x . b. The total cost is x t 2 . 2 x + 2500 x u = 2 . 2 x + 2500. 48. A = km q + cm + hq 2 = 2 km + 2 cmq + hq 2 2 q . 49. P = R i − R i ( 1 + i ) n = R ( 1 + i ) n − R i ( 1 + i ) n = R d ( 1 + i ) n − 1 e i ( 1 + i ) n . 50. P = kT V − b + ab V 2 ( V − b ) − a V ( V − b ) = kT V 2 + ab − aV V 2 ( V − b ) .

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