Appendix A.4 Rational Expressions A39 A.4 RATIONAL EXPRESSIONS What you should learn • Find domains of algebraic expressions. • Simplify rational expressions. • Add, subtract, multiply, and divide rational expressions. • Simplify complex fractions and rewrite difference quotients. Domain of an Algebraic Expression The set of real numbers for which an algebraic expression is defined is the domain of the expression. Two algebraic expressions are equivalent if they have the same domain and yield the same values for all numbers in their domain. For instance, 共x ⫹ 1兲 ⫹ 共x ⫹ 2兲 and 2x ⫹ 3 are equivalent because 共x ⫹ 1兲 ⫹ 共x ⫹ 2兲 ⫽ x ⫹ 1 ⫹ x ⫹ 2 ⫽x⫹x⫹1⫹2 Why you should learn it Rational expressions can be used to solve real-life problems. For instance, in Exercise 102 on page A48, a rational expression is used to model the projected numbers of U.S. households banking and paying bills online from 2002 through 2007. ⫽ 2x ⫹ 3. Example 1 Finding the Domain of an Algebraic Expression a. The domain of the polynomial 2x 3 ⫹ 3x ⫹ 4 is the set of all real numbers. In fact, the domain of any polynomial is the set of all real numbers, unless the domain is specifically restricted. b. The domain of the radical expression 冪x ⫺ 2 is the set of real numbers greater than or equal to 2, because the square root of a negative number is not a real number. c. The domain of the expression x⫹2 x⫺3 is the set of all real numbers except x ⫽ 3, which would result in division by zero, which is undefined. Now try Exercise 7. The quotient of two algebraic expressions is a fractional expression. Moreover, the quotient of two polynomials such as 1 , x 2x ⫺ 1 , x⫹1 or x2 ⫺ 1 x2 ⫹ 1 is a rational expression. Simplifying Rational Expressions Recall that a fraction is in simplest form if its numerator and denominator have no factors in common aside from ± 1. To write a fraction in simplest form, divide out common factors. a b ⭈ c ⫽ a, ⭈c b c⫽0 A40 Appendix A Review of Fundamental Concepts of Algebra The key to success in simplifying rational expressions lies in your ability to factor polynomials. When simplifying rational expressions, be sure to factor each polynomial completely before concluding that the numerator and denominator have no factors in common. Example 2 WARNING / CAUTION In Example 2, do not make the mistake of trying to simplify further by dividing out terms. x⫹6 x⫹6 ⫽ ⫽x⫹2 3 3 Remember that to simplify fractions, divide out common factors, not terms. Write Simplifying a Rational Expression x 2 ⫹ 4x ⫺ 12 in simplest form. 3x ⫺ 6 Solution x2 ⫹ 4x ⫺ 12 共x ⫹ 6兲共x ⫺ 2兲 ⫽ 3x ⫺ 6 3共x ⫺ 2兲 ⫽ x⫹6 , 3 x⫽2 Factor completely. Divide out common factors. Note that the original expression is undefined when x ⫽ 2 (because division by zero is undefined). To make sure that the simplified expression is equivalent to the original expression, you must restrict the domain of the simplified expression by excluding the value x ⫽ 2. Now try Exercise 33. Sometimes it may be necessary to change the sign of a factor by factoring out 共⫺1兲 to simplify a rational expression, as shown in Example 3. Example 3 Write Simplifying Rational Expressions 12 ⫹ x ⫺ x2 in simplest form. 2x2 ⫺ 9x ⫹ 4 Solution 12 ⫹ x ⫺ x2 共4 ⫺ x兲共3 ⫹ x兲 ⫽ 2x2 ⫺ 9x ⫹ 4 共2x ⫺ 1兲共x ⫺ 4兲 ⫽ ⫺ 共x ⫺ 4兲共3 ⫹ x兲共2x ⫺ 1兲共x ⫺ 4兲 ⫽⫺ 3⫹x , x⫽4 2x ⫺ 1 Factor completely. 共4 ⫺ x兲 ⫽ ⫺ 共x ⫺ 4兲 Divide out common factors. Now try Exercise 39. In this text, when a rational expression is written, the domain is usually not listed with the expression. It is implied that the real numbers that make the denominator zero are excluded from the expression. Also, when performing operations with rational expressions, this text follows the convention of listing by the simplified expression all values of x that must be specifically excluded from the domain in order to make the domains of the simplified and original expressions agree. In Example 3, for instance, the restriction x ⫽ 4 is listed with the simplified expression 1 to make the two domains agree. Note that the value x ⫽ 2 is excluded from both domains, so it is not necessary to list this value. Appendix A.4 Rational Expressions A41 Operations with Rational Expressions To multiply or divide rational expressions, use the properties of fractions discussed in Appendix A.1. Recall that to divide fractions, you invert the divisor and multiply. Example 4 Multiplying Rational Expressions 2x2 ⫹ x ⫺ 6 x2 ⫹ 4x ⫺ 5 ⭈ x3 ⫺ 3x2 ⫹ 2x 共2x ⫺ 3兲共x ⫹ 2兲 ⫽ 4x2 ⫺ 6x 共x ⫹ 5兲共x ⫺ 1兲 ⫽ ⭈ x共x ⫺ 2兲共x ⫺ 1兲 2x共2x ⫺ 3兲共x ⫹ 2兲共x ⫺ 2兲 , x ⫽ 0, x ⫽ 1, x ⫽ 32 2共x ⫹ 5兲 Now try Exercise 53. In Example 4, the restrictions x ⫽ 0, x ⫽ 1, and x ⫽ 32 are listed with the simplified expression in order to make the two domains agree. Note that the value x ⫽ ⫺5 is excluded from both domains, so it is not necessary to list this value. Example 5 Dividing Rational Expressions x 3 ⫺ 8 x 2 ⫹ 2x ⫹ 4 x 3 ⫺ 8 ⫼ ⫽ 2 x2 ⫺ 4 x3 ⫹ 8 x ⫺4 ⫽ x3 ⫹ 8 ⭈ x 2 ⫹ 2x ⫹ 4 Invert and multiply. 共x ⫺ 2兲共x2 ⫹ 2x ⫹ 4兲共x ⫹ 2兲共x2 ⫺ 2x ⫹ 4兲 ⭈ 共x2 ⫹ 2x ⫹ 4兲共x ⫹ 2兲共x ⫺ 2兲 ⫽ x 2 ⫺ 2x ⫹ 4, x ⫽ ± 2 Divide out common factors. Now try Exercise 55. To add or subtract rational expressions, you can use the LCD (least common denominator) method or the basic definition a c ad ± bc ± ⫽ , b d bd b ⫽ 0, d ⫽ 0. Basic definition This definition provides an efficient way of adding or subtracting two fractions that have no common factors in their denominators. Example 6 WARNING / CAUTION When subtracting rational expressions, remember to distribute the negative sign to all the terms in the quantity that is being subtracted. Subtracting Rational Expressions x 2 x共3x ⫹ 4兲 ⫺ 2共x ⫺ 3兲 ⫺ ⫽ x ⫺ 3 3x ⫹ 4 共x ⫺ 3兲共3x ⫹ 4兲 Basic definition ⫽ 3x 2 ⫹ 4x ⫺ 2x ⫹ 6 共x ⫺ 3兲共3x ⫹ 4兲 Distributive Property ⫽ 3x 2 ⫹ 2x ⫹ 6 共x ⫺ 3兲共3x ⫹ 4兲 Combine like terms. Now try Exercise 65. A42 Appendix A Review of Fundamental Concepts of Algebra For three or more fractions, or for fractions with a repeated factor in the denominators, the LCD method works well. Recall that the least common denominator of several fractions consists of the product of all prime factors in the denominators, with each factor given the highest power of its occurrence in any denominator. Here is a numerical example. 1 3 2 1⭈2 3⭈3 2⭈4 ⫹ ⫺ ⫽ ⫹ ⫺ 6 4 3 6⭈2 4⭈3 3⭈4 ⫽ 2 9 8 ⫹ ⫺ 12 12 12 ⫽ 3 12 ⫽ 1 4 The LCD is 12. Sometimes the numerator of the answer has a factor in common with the denominator. In such cases the answer should be simplified. For instance, in the 3 example above, 12 was simplified to 14. Example 7 Combining Rational Expressions: The LCD Method Perform the operations and simplify. 2 x⫹3 3 ⫺ ⫹ 2 x⫺1 x x ⫺1 Solution Using the factored denominators 共x ⫺ 1兲, x, and 共x ⫹ 1兲共x ⫺ 1兲, you can see that the LCD is x共x ⫹ 1兲共x ⫺ 1兲. 3 2 x⫹3 ⫺ ⫹ x⫺1 x 共x ⫹ 1兲共x ⫺ 1兲 ⫽ 共x ⫹ 3兲共x兲 3共x兲共x ⫹ 1兲 2共x ⫹ 1兲共x ⫺ 1兲 ⫺ ⫹ x共x ⫹ 1兲共x ⫺ 1兲 x共x ⫹ 1兲共x ⫺ 1兲 x共x ⫹ 1兲共x ⫺ 1兲 ⫽ 3共x兲共x ⫹ 1兲 ⫺ 2共x ⫹ 1兲共x ⫺ 1兲 ⫹ 共x ⫹ 3兲共x兲 x共x ⫹ 1兲共x ⫺ 1兲 ⫽ 3x 2 ⫹ 3x ⫺ 2x 2 ⫹ 2 ⫹ x 2 ⫹ 3x x共x ⫹ 1兲共x ⫺ 1兲 Distributive Property ⫽ 3x 2 ⫺ 2x 2 ⫹ x 2 ⫹ 3x ⫹ 3x ⫹ 2 x共x ⫹ 1兲共x ⫺ 1兲 Group like terms. ⫽ 2x2 ⫹ 6x ⫹ 2 x共x ⫹ 1兲共x ⫺ 1兲 Combine like terms. ⫽ 2共x 2 ⫹ 3x ⫹ 1兲 x共x ⫹ 1兲共x ⫺ 1兲 Factor. Now try Exercise 67. Appendix A.4 Rational Expressions A43 Complex Fractions and the Difference Quotient Fractional expressions with separate fractions in the numerator, denominator, or both are called complex fractions. Here are two examples. 冢x冣冢x冣 1 x2 ⫹ 1 1 and 冢x 2 1 ⫹1 冣 To simplify a complex fraction, combine the fractions in the numerator into a single fraction and then combine the fractions in the denominator into a single fraction. Then invert the denominator and multiply. Example 8 Simplifying a Complex Fraction 2 ⫺ 3共x兲 x ⫽ 1 1共x ⫺ 1兲 ⫺ 1 1⫺ x⫺1 x⫺1 冢 x ⫺ 3冣冤 2 冢冥冣冤冥 Combine fractions. 2 ⫺ 3x 冢 x 冣 ⫽ x⫺2 冢x ⫺ 1冣 Simplify. x⫺1 ⫽ 2 ⫺ 3x x ⫽ 共2 ⫺ 3x兲共x ⫺ 1兲 , x⫽1 x共x ⫺ 2兲 ⭈x⫺2 Invert and multiply. Now try Exercise 73. Another way to simplify a complex fraction is to multiply its numerator and denominator by the LCD of all fractions in its numerator and denominator. This method is applied to the fraction in Example 8 as follows. 冢 x ⫺ 3冣冢 x ⫺ 3冣 2 冢 1 1⫺ x⫺1 2 ⫽ 冣冢 1 1⫺ x⫺1 x共x ⫺ 1兲冣 ⭈ x共x ⫺ 1兲冢2 ⫺x 3x冣 ⭈ x共x ⫺ 1兲 ⫽ 冢xx ⫺⫺ 21冣 ⭈ x共x ⫺ 1兲 ⫽ 共2 ⫺ 3x兲共x ⫺ 1兲 , x⫽1 x共x ⫺ 2兲 LCD is x共x ⫺ 1兲. A44 Appendix A Review of Fundamental Concepts of Algebra The next three examples illustrate some methods for simplifying rational expressions involving negative exponents and radicals. These types of expressions occur frequently in calculus. To simplify an expression with negative exponents, one method is to begin by factoring out the common factor with the smaller exponent. Remember that when factoring, you subtract exponents. For instance, in 3x⫺5兾2 ⫹ 2x⫺3兾2 the smaller exponent is ⫺ 52 and the common factor is x⫺5兾2. 3x⫺5兾2 ⫹ 2x⫺3兾2 ⫽ x⫺5兾2关3共1兲 ⫹ 2x⫺3兾2⫺ 共⫺5兾2兲兴 ⫽ x⫺5兾2共3 ⫹ 2x1兲 ⫽ Example 9 3 ⫹ 2x x 5兾2 Simplifying an Expression Simplify the following expression containing negative exponents. x共1 ⫺ 2x兲⫺3兾2 ⫹ 共1 ⫺ 2x兲⫺1兾2 Solution Begin by factoring out the common factor with the smaller exponent. x共1 ⫺ 2x兲⫺3兾2 ⫹ 共1 ⫺ 2x兲⫺1兾2 ⫽ 共1 ⫺ 2x兲⫺3兾2关 x ⫹ 共1 ⫺ 2x兲(⫺1兾2)⫺(⫺3兾2)兴 ⫽ 共1 ⫺ 2x兲⫺3兾2关x ⫹ 共1 ⫺ 2x兲1兴 ⫽ 1⫺x 共1 ⫺ 2x兲 3兾2 Now try Exercise 81. A second method for simplifying an expression with negative exponents is shown in the next example. Example 10 Simplifying an Expression with Negative Exponents 共4 ⫺ x 2兲1兾2 ⫹ x 2共4 ⫺ x 2兲⫺1兾2 4 ⫺ x2 ⫽ 共4 ⫺ x 2兲1兾2 ⫹ x 2共4 ⫺ x 2兲⫺1兾2 共4 ⫺ x 2兲1兾2 ⭈ 共4 ⫺ x 2兲1兾2 4 ⫺ x2 ⫽ 共4 ⫺ x 2兲1 ⫹ x 2共4 ⫺ x 2兲 0 共4 ⫺ x 2兲 3兾2 ⫽ 4 ⫺ x2 ⫹ x2 共4 ⫺ x 2兲 3兾2 ⫽ 4 共4 ⫺ x 2兲 3兾2 Now try Exercise 83. Appendix A.4 Example 11 Rational Expressions A45 Rewriting a Difference Quotient The following expression from calculus is an example of a difference quotient. 冪x ⫹ h ⫺ 冪x h Rewrite this expression by rationalizing its numerator. Solution 冪x ⫹ h ⫺ 冪x h You can review the techniques for rationalizing a numerator in Appendix A.2. ⫽ 冪x ⫹ h ⫺ 冪x h 冪x ⫹ h ⫹ 冪x ⭈ 冪x ⫹ h ⫹ 冪x ⫽ 共冪x ⫹ h 兲2 ⫺ 共冪x 兲2 h共冪x ⫹ h ⫹ 冪x 兲 ⫽ h h共冪x ⫹ h ⫹ 冪x 兲 ⫽ 1 , 冪x ⫹ h ⫹ 冪x h⫽0 Notice that the original expression is undefined when h ⫽ 0. So, you must exclude h ⫽ 0 from the domain of the simplified expression so that the expressions are equivalent. Now try Exercise 89. Difference quotients, such as that in Example 11, occur frequently in calculus. Often, they need to be rewritten in an equivalent form that can be evaluated when h ⫽ 0. Note that the equivalent form is not simpler than the original form, but it has the advantage that it is defined when h ⫽ 0. A.4 EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises. VOCABULARY: Fill in the blanks. 1. The set of real numbers for which an algebraic expression is defined is the ________ of the expression. 2. The quotient of two algebraic expressions is a fractional expression and the quotient of two polynomials is a ________ ________. 3. Fractional expressions with separate fractions in the numerator, denominator, or both are called ________ fractions. 4. To simplify an expression with negative exponents, it is possible to begin by factoring out the common factor with the ________ exponent. 5. Two algebraic expressions that have the same domain and yield the same values for all numbers in their domains are called ________. 6. An important rational expression, such as a ________ ________. 共x ⫹ h兲2 ⫺ x2 , that occurs in calculus is called h A46 Appendix A Review of Fundamental Concepts of Algebra SKILLS AND APPLICATIONS 44. Error Analysis In Exercises 7–22, find the domain of the expression. 7. 3x 2 ⫺ 4x ⫹ 7 9. 4x 3 ⫹ 3, x ⱖ 0 11. 1 3⫺x 13. ⫺1 x2 ⫺ 2x ⫹ 1 8. 2x 2 ⫹ 5x ⫺ 2 10. 6x 2 ⫺ 9, x > 0 12. x2 14. x⫹6 3x ⫹ 2 x2 ⫺ 5x ⫹ 6 x2 ⫺ 4 x2 ⫺ 2x ⫺ 3 15. 2 x ⫺ 6x ⫹ 9 x2 ⫺ x ⫺ 12 16. 2 x ⫺ 8x ⫹ 16 17. 冪x ⫹ 7 19. 冪2x ⫺ 5 18. 冪4 ⫺ x 20. 冪4x ⫹ 5 21. 1 22. 冪x ⫺ 3 x2 1 冪x ⫹ 2 25. 15x 2 10x 26. 18y 2 60y 5 45. 3xy xy ⫹ x 28. 2x xy ⫺ y 29. 4y ⫺ 8y 2 10y ⫺ 5 30. 9x 2 ⫹ 9x 2x ⫹ 2 x⫺5 10 ⫺ 2x 32. x 3 ⫹ 5x 2 ⫹ 6x 35. x2 ⫺ 4 y 2 ⫺ 7y ⫹ 12 37. 2 y ⫹ 3y ⫺ 18 x 2 ⫺ 7x ⫹ 6 38. 2 x ⫹ 11x ⫹ 10 2 ⫺ x ⫹ 2x 2 ⫺ x 3 x2 ⫺ 4 z3 ⫺ 8 41. 2 z ⫹ 2z ⫹ 4 39. 43. Error Analysis 5x3 2x3 ⫹ 4 ⫽ x2 ⫺ 9 ⫹ x 2 ⫺ 9x ⫺ 9 y 3 ⫺ 2y 2 ⫺ 3y 42. y3 ⫹ 1 40. 1 2 3 4 x3 Describe the error. 5x3 5 5 ⫽ ⫽ 3 2x ⫹ 4 2 ⫹ 4 6 5 6 x2 ⫺ 2x ⫺ 3 x⫺3 x⫹1 46. 0 x 1 2 3 4 5 6 x⫺3 x2 ⫺ x ⫺ 6 1 x⫹2 GEOMETRY In Exercises 47 and 48, find the ratio of the area of the shaded portion of the figure to the total area of the figure. 47. 48. x+5 2 r 2x + 3 x+5 2 12 ⫺ 4x x⫺3 x 2 ⫺ 25 34. 5⫺x x 2 ⫹ 8x ⫺ 20 36. 2 x ⫹ 11x ⫹ 10 y 2 ⫺ 16 33. y⫹4 0 x 2y 27. 31. x共x ⫹ 5兲共x ⫺ 5兲 x共x ⫹ 5兲 ⫽ 共x ⫺ 5兲共x ⫹ 3兲 x⫹3 In Exercises 45 and 46, complete the table. What can you conclude? 3共䊏兲 3 24. ⫽ 4 4共x ⫹ 1兲 In Exercises 25–42, write the rational expression in simplest form. x共x 2 ⫹ 25兲 x 3 ⫹ 25x ⫽ ⫺ 2x ⫺ 15 共x ⫺ 5兲共x ⫹ 3兲 ⫽ In Exercises 23 and 24, find the missing factor in the numerator such that the two fractions are equivalent. 5 5共䊏兲 23. ⫽ 2x 6x 2 Describe the error. x+5 In Exercises 49–56, perform the multiplication or division and simplify. 5 x⫺1 ⭈ x ⫺ 1 25共x ⫺ 2兲 r r2 51. ⫼ 2 r⫺1 r ⫺1 49. x共x ⫺ 3兲 5 4⫺y 4y ⫺ 16 52. ⫼ 5y ⫹ 15 2y ⫹ 6 50. x ⫹ 13 x 3共3 ⫺ x兲 t2 ⫺ t ⫺ 6 t⫹3 ⭈ t 2 ⫹ 6t ⫹ 9 t 2 ⫺ 4 x x 2 ⫹ xy ⫺ 2y 2 54. ⭈ 3 2 2 x ⫹x y x ⫹ 3xy ⫹ 2y 2 53. 55. x 2 ⫺ 36 x 3 ⫺ 6x 2 ⫼ 2 x x ⫹x 56. x 2 ⫺ 14x ⫹ 49 3x ⫺ 21 ⫼ x 2 ⫺ 49 x⫹7 ⭈ Appendix A.4 In Exercises 57–68, perform the addition or subtraction and simplify. 57. 6 ⫺ 5 x⫹3 5 x ⫹ x⫺1 x⫺1 3 5 61. ⫹ x⫺2 2⫺x 59. 63. 4 x ⫺ 2x ⫹ 1 x ⫹ 2 58. 2x ⫺ 1 1 ⫺ x ⫹ x⫹3 x⫹3 5 2x 62. ⫺ x⫺5 5⫺x 60. 64. 3x1兾3 ⫺ x⫺2兾3 3x⫺2兾3 ⫺x 3共1 ⫺ x 2兲⫺1兾2 ⫺ 2x共1 ⫺ x 2兲1兾2 84. x4 1 2 1 ⫹ 2 ⫹ 3 x x ⫹1 x ⫹x 2 1 2 68. ⫹ ⫹ x ⫹ 1 x ⫺ 1 x2 ⫺ 1 67. ⫺ x ⫹ 4 3x ⫺ 8 x ⫹ 4 ⫺ 3x ⫺ 8 ⫺ ⫽ x⫹2 x⫹2 x⫹2 ⫺2x ⫺ 4 ⫺2共x ⫹ 2兲 ⫽ ⫽ ⫽ ⫺2 x⫹2 x⫹2 6⫺x x⫹2 8 70. ⫹ ⫹ 2 2 x共x ⫹ 2兲 x x 共x ⫹ 2兲 x共6 ⫺ x兲 ⫹ 共x ⫹ 2兲 2 ⫹ 8 ⫽ x 2共x ⫹ 2兲 6x ⫺ x 2 ⫹ x 2 ⫹ 4 ⫹ 8 ⫽ x 2共x ⫹ 2兲 6共x ⫹ 2兲 6 ⫽ 2 ⫽ x 共x ⫹ 2兲 x 2 69. In Exercises 71–76, simplify the complex fraction. x 73. 共x ⫺ 2兲 x2 共x ⫹ 1兲2 冤冥 x 冤共x ⫹ 1兲冥 3 冢冪x ⫺ 2冪x冣 1 75. 冪x In Exercises 85– 88, simplify the difference quotient. 1 1 1 1 ⫺ 2 ⫺ 2 共 x ⫹ h 兲 x x⫹h x 85. 86. h h x ⫹ h x 1 1 ⫺ ⫺ x⫹h⫹1 x⫹1 x⫹h⫺4 x⫺4 87. 88. h h 冢 ERROR ANALYSIS In Exercises 69 and 70, describe the error. 共x ⫺ 4兲 x 4 ⫺ 4 x x2 ⫺ 1 x 74. 共x ⫺ 1兲2 x t2 ⫺ 冪t 2 ⫹ 1 冪t 2 ⫹ 1 76. t2 72. 2x共x ⫺ 5兲⫺3 ⫺ 4x 2共x ⫺ 5兲⫺4 2x 2共x ⫺ 1兲1兾2 ⫺ 5共x ⫺ 1兲⫺1兾2 4x 3共2x ⫺ 1兲3兾2 ⫺ 2x共2x ⫺ 1兲⫺1兾2 83. 1 x ⫺ x 2 ⫺ x ⫺ 2 x 2 ⫺ 5x ⫹ 6 10 2 66. 2 ⫹ 2 x ⫺ x ⫺ 2 x ⫹ 2x ⫺ 8 71. 78. x5 ⫺ 5x⫺3 x 5 ⫺ 2x⫺2 x 2共x 2 ⫹ 1兲⫺5 ⫺ 共x 2 ⫹ 1兲⫺4 In Exercises 83 and 84, simplify the expression. 5x 2 ⫹ x ⫺ 3 3x ⫹ 4 65. 冢 2 ⫺ 1冣 In Exercises 77–82, factor the expression by removing the common factor with the smaller exponent. 77. 79. 80. 81. 82. 3 ⫺5 x⫺1 冢冢冤冢 A47 Rational Expressions 冣冣冥冣冤冣冢冥冢冣冣 In Exercises 89–94, simplify the difference quotient by rationalizing the numerator. 89. 91. 93. 94. 冪x ⫹ 2 ⫺ 冪x 90. 2 冪t ⫹ 3 ⫺ 冪3 92. t 冪z ⫺ 3 ⫺ 冪z 3 冪x ⫹ 5 ⫺ 冪5 x 冪x ⫹ h ⫹ 1 ⫺ 冪x ⫹ 1 h 冪x ⫹ h ⫺ 2 ⫺ 冪x ⫺ 2 h PROBABILITY In Exercises 95 and 96, consider an experiment in which a marble is tossed into a box whose base is shown in the figure. The probability that the marble will come to rest in the shaded portion of the box is equal to the ratio of the shaded area to the total area of the figure. Find the probability. 95. 96. x 2 x 2x + 1 x+4 x x x+2 4 x (x + 2) 97. RATE A digital copier copies in color at a rate of 50 pages per minute. (a) Find the time required to copy one page. A48 Appendix A Review of Fundamental Concepts of Algebra (b) Find the time required to copy x pages. (c) Find the time required to copy 120 pages. 98. RATE After working together for t hours on a common task, two workers have done fractional parts of the job equal to t兾3 and t兾5, respectively. What fractional part of the task has been completed? 102. INTERACTIVE MONEY MANAGEMENT The table shows the projected numbers of U.S. households (in millions) banking online and paying bills online from 2002 through 2007. (Source: eMarketer; Forrester Research) FINANCE In Exercises 99 and 100, the formula that approximates the annual interest rate r of a monthly installment loan is given by 24共NM ⴚ P兲 [ ] N rⴝ 冸P ⴙ NM 12 冹 where N is the total number of payments, M is the monthly payment, and P is the amount financed. 4t 2 ⫹ 16t ⫹ 75 2 ⫹ 4t ⫹ 10 冢t 冣 0 2 4 6 8 10 14 16 18 2002 2003 2004 2005 2006 2007 21.9 26.8 31.5 35.0 40.0 45.0 13.7 17.4 20.9 23.9 26.7 29.1 Number banking online ⫽ ⫺0.728t2 ⫹ 23.81t ⫺ 0.3 ⫺0.049t2 ⫹ 0.61t ⫹ 1.0 and Number paying bills online ⫽ 4.39t ⫹ 5.5 0.002t2 ⫹ 0.01t ⫹ 1.0 where t represents the year, with t ⫽ 2 corresponding to 2002. (a) Using the models, create a table to estimate the projected numbers of households banking online and the projected numbers of households paying bills online for the given years. (b) Compare the values given by the models with the actual data. (c) Determine a model for the ratio of the projected number of households paying bills online to the projected number of households banking online. (d) Use the model from part (c) to find the ratios for the given years. Interpret your results. 20 TRUE OR FALSE? In Exercises 103 and 104, determine whether the statement is true or false. Justify your answer. 103. x 2n ⫺ 12n ⫽ x n ⫹ 1n x n ⫺ 1n 104. x 2 ⫺ 3x ⫹ 2 ⫽ x ⫺ 2, for all values of x x⫺1 12 T t Paying Bills EXPLORATION where T is the temperature (in degrees Fahrenheit) and t is the time (in hours). (a) Complete the table. t Banking Mathematical models for these data are 99. (a) Approximate the annual interest rate for a four-year car loan of $20,000 that has monthly payments of $475. (b) Simplify the expression for the annual interest rate r, and then rework part (a). 100. (a) Approximate the annual interest rate for a fiveyear car loan of $28,000 that has monthly payments of $525. (b) Simplify the expression for the annual interest rate r, and then rework part (a). 101. REFRIGERATION When food (at room temperature) is placed in a refrigerator, the time required for the food to cool depends on the amount of food, the air circulation in the refrigerator, the original temperature of the food, and the temperature of the refrigerator. The model that gives the temperature of food that has an original temperature of 75⬚F and is placed in a 40⬚F refrigerator is T ⫽ 10 Year 22 T (b) What value of T does the mathematical model appear to be approaching? 105. THINK ABOUT IT How do you determine whether a rational expression is in simplest form? 106. CAPSTONE In your own words, explain how to divide rational expressions.