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bel84775_ch04_c.qxd 10/13/04 20:26 Page 336 336 Chapter 4 4-46 Exponents and Polynomials 4.5 ADDITION AND SUBTRACTION OF POLYNOMIALS To Succeed, Review How To . . . 1. Add and subtract like terms (pp. 98–101). Objectives A Add polynomials. B Subtract polynomials. 2. Remove parentheses in expressions preceded by a minus sign (p. 101). C Find areas by adding polynomials. D Solve applications. GETTING STARTED The annual amount of waste (in millions of tons) generated in the United States is approximated by G(t) 0.001t3 0.06t2 2.6t 88.6, where t is the number of years after 1960. How much of this waste is recovered? That amount can be Waste Generated approximated by R(t) 0.06t2 0.59t 6.4. From these two approximations, we can estimate that the amount of waste actually “wasted” (not recovered) is G(t) R(t). To find this difference, we simply subtract like terms. To make the procedure more familiar, we write it in columns: 300 Tons (in millions) Wasted Waste 200 G(t) 0.001t3 0.06t2 2.6t 88.6 () R(t) () 100 0.001t3 3.19t 82.2 Thus the amount of waste generated and not recovered is 0 0 10 20 30 35 40 Year (1960 0) G(t) R(t) 0.001t3 3.19t 82.2. Let’s see what this means in millions of tons. Since t is the number of years after 1960, t 0 in 1960, and the amount of waste generated, the amount of waste recycled, and the amount of waste not recovered are as follows: Waste Recovered 80 Tons (in millions) 0.06t2 0.59t 6.4 G(0) 0.001(0)3 0.06(0)2 2.6(0) 88.6 88.6 (million tons) 60 R(0) 0.06(0)2 0.59(0) 6.4 6.4 G(0) R(0) 88.6 6.4 82.2 40 20 0 0 10 20 30 35 Year (1960 0) 40 (million tons) (million tons) As you can see, there is much more material not recovered than material recovered. How can we find out whether the situation is changing? One way is to predict how much waste will be produced and how much recovered, say, in the year 2010. The amount can be approximated by G(50) R(50). Then we find out if, percentagewise, the situation is getting better. In 1960, the percent of materials recovered was 6.488.6, or about 7.2%. What percent would it be in the year 2010? In this section we will learn how to add and subtract polynomials and use these ideas to solve applications. bel84775_ch04_c.qxd 10/13/04 20:26 Page 337 4-47 4.5 A Addition and Subtraction of Polynomials 337 Adding Polynomials The addition of monomials and polynomials is a matter of combining like terms. For example, suppose we wish to add 3x 2 7x 3 and 5x 2 2x 9; that is, we wish to find (3x 2 7x 3) (5x 2 2x 9) Web It Using the commutative, associative, and distributive properties, we write To practice combining like terms, go to link 4-5-1 on the Bello Website at mhhe.com/ bello. (3x 2 7x 3) (5x 2 2x 9) (3x 2 5x 2) (7x 2x) (3 9) (3 5)x 2 (7 2)x (3 9) 8x 2 5x 6 3 1 Similarly, the sum of 4x 3 7 x 2 2x 3 and 6x 3 7 x 2 9 is written as 4x 3 3 1 x 2 2x 3 6x 3 x 2 9 7 7 3 1 (4x 3 6x 3) x 2 x 2 (2x) (3 9) 7 7 2 10x 3 x 2 2x 12 7 In both examples, the polynomials have been written in descending order for convenience in combining like terms. EXAMPLE 1 PROBLEM 1 Adding polynomials Add: 3x 7x 2 7 and 4x 2 9 3x Add: 5x 8x 2 3 and 3x 2 8 5x SOLUTION We first write both polynomials in descending order and then combine like terms to obtain (7x 2 3x 7) (4x 2 3x 9) (7x 2 4x 2) (3x 3x) (7 9) 3x 2 0 2 3x 2 2 As in arithmetic, the addition of polynomials can be done by writing the polynomials in descending order and then placing like terms in columns. In arithmetic, you add 345 and 678 by writing the numbers in a column: 345 678 Units Tens Hundreds Thus to add 4x 3 3x 7 and 7x 3x 3 x 2 9, we first write both polynomials in descending order with like terms in the same column, leaving space for any missing terms. We then add the terms in each of the columns: The x 2 term is missing in 4x 3 3x 7. Answer 1. 5x 2 5 4x 3 3x 7 3x 3 x 2 7x 9 x 3 x 2 10x 2 bel84775_ch04_c.qxd 10/13/04 20:26 Page 338 338 Chapter 4 4-48 Exponents and Polynomials EXAMPLE 2 PROBLEM 2 More practice adding polynomials Add: 3x 7x 2 2 and 4x 2 3 5x Add: 5y 8y 2 3 and 5y 2 4 6y SOLUTION We first write both polynomials in descending order, place like terms in a column, and then add as shown: 7x 2 3x 2 4x 2 5x 3 3x 2 2x 5 Horizontally, we write: (7x 2 3x 2) (4x 2 5x 3) 14243 (7x 2 4x 2) 14243 (3x 5x) 1 (2 3) 42 43 3x 2 3x 2 B 2x 2x (5) 5 Subtracting Polynomials To subtract polynomials, we first recall that a (b c) a b c Web It To practice with the subtraction of polynomials, go to link 4-5-2 on the Bello Website at mhhe.com/bello and enter your problem. Be careful with parentheses! For a more conventional approach and different strategies, try link 4-5-3. To remove the parentheses from an expression preceded by a minus sign, we must change the sign of each term inside the parentheses. This is the same as multiplying each term inside the parentheses by 1. Thus (3x 2 2x 1) (4x 2 5x 2) 3x 2 2x 1 4x 2 5x 2 (3x 2 4x 2) (2x 5x) (1 2) x 2 7x (1) x 2 7x 1 Here’s how we do it using columns: 3x 2 2x 1 3x 2 2x 1 is written () 4x 2 5x 2 () 4x 2 5x 2 x 2 7x 1 Note that we changed the sign of every term in 4x 2 5x 2 and wrote 4x 2 5x 2. NOTE “Subtract b from a” means to find a b. EXAMPLE 3 PROBLEM 3 Subtracting polynomials Subtract 4x 3 7x 2 from 5x 2 3x. SOLUTION and add: Subtract 5y 4 8y 2 from 6y 2 4y. We first write the problem in columns, then change the signs 5x 2 3x is written ()7x 2 4x 3 5x 2 3x ()7x 2 4x 3 2x 2 7x 3 Thus the answer is 2x 2 7x 3. Answers 2. 3y 2 y 7 3. 2y 2 9y 4 bel84775_ch04_c.qxd 10/13/04 20:26 Page 339 4-49 4.5 Addition and Subtraction of Polynomials 339 To do it horizontally, we write (5x 2 3x) (7x 2 4x 3) 5x 2 3x 7x 2 4x 3 Change the sign of every term in 7x 2 4x 3. (5x 2 7x 2) (3x 4x) 3 Use the commutative and associative properties. 2x 2 7x 3 Just as in arithmetic, we can add or subtract more than two polynomials. For example, to add the polynomials 7x x 2 3, 6x 2 8 2x, and 3x x 2 5, we simply write each of the polynomials in descending order with like terms in the same column and add: x 2 7x 3 6x 2 2x 8 x 2 3x 5 6x 2 2x 6 Or, horizontally, we write (x 2 7x 3) (6x 2 2x 8) (x 2 3x 5) (x 2 6x 2 x 2) (7x 2x 3x) (3 8 5) 6x 2 (2x) (6) 6x 2 2x 6 EXAMPLE 4 PROBLEM 4 More practice adding polynomials Add: x 2x 3x 5, 8 2x 5x , and 7x 4x 9 3 2 2 Add: y 3 3y 4y 2 6, 9 3y 6y 2, and 6y 3 5y 8 3 SOLUTION We first write all the polynomials in descending order with like terms in the same column and then add: x 3 3x 2 2x 5 5x 2 2x 8 3 7x 4x 9 8x 3 8x 2 4 Horizontally, we have (x 3 3x 2 2x 5) (5x 2 2x 8) (7x 3 4x 9) (x 3 7x 3) (3x 2 5x 2) (2x 2x 4x) (5 8 9) 8x 3 (8x 2) 0x (4) 8x 3 8x 2 4 C Finding Areas Addition of polynomials can be used to find the sum of the areas of several rectangles. To find the total area of the shaded rectangles, add the individual areas. 5 2 Answer 4. 7y 10y y 7 3 2 A 5 3 B 2 2 D C 3 5 bel84775_ch04_c.qxd 10/13/04 20:26 Page 340 340 Chapter 4 4-50 Exponents and Polynomials Web It Since the area of a rectangle is the product of its length and its width, we have To practice adding polynomials using sums of areas, go back to link 4-5-3 on the Bello Website at mhhe.com/ bello. Area of A 14243 Area of B Area of C Area of D 14243 14243 14243 52 23 32 55 10 6 6 25 Thus 10 6 6 25 47 (square units) This same procedure can be used when some of the lengths are represented by variables, as shown in the next example. EXAMPLE 5 PROBLEM 5 Finding sums of areas Find the sum of the areas of the shaded rectangles: 3 B x SOLUTION 3x x A 5 Find the sum of the areas of the shaded rectangles: x C 3 D 3 B y A 3x y 6 The total area in square units is Area of A 14243 Area of B 14243 Area of C 14243 Area of D 14243 5x 3x 3x 1444442444443 11x (3x)2 9x 2 4y y D C 3 4y or 9x 2 11x D In descending order Solving Applications The concepts we’ve just studied can be used to examine important issues in American life. Let’s see how we can use these concepts to study health care. EXAMPLE 6 PROBLEM 6 Paying to get well Based on Social Security Administration statistics, the annual amount of money spent on hospital care by the average American can be approximated by H(t) 6.6t2 30t 682 How much was spent on hospital care and doctors’ services in 2000? (dollars) while the annual amount spent for doctors’ services is approximated by D(t) 41t 293 where t is the number of years after 1985. a. Find the annual amount of money spent for doctors’ services and hospital care. b. How much was spent for doctors’ services in 1985? c. How much was spent on hospital care in 1985? d. How much would you predict will be spent on hospital care and doctors’ services in the year 2005? Answers 5. 12y 16y 2 6. $3525 bel84775_ch04_c.qxd 10/13/04 20:26 Page 341 4-51 4.5 Addition and Subtraction of Polynomials 341 SOLUTION a. To find the annual amount spent for doctors’ services and hospital care, we find D(t) H(t): D(t) 41t 293 H(t) 6.6t 2 30t 682 6.6t 2 71t 975 Thus the annual amount spent for doctors’ services and hospital care is 6.6t2 71t 975. D(t) H(t) b. To find how much was spent for doctors’ services in 1985 (t 0), find D(0) 41(0) 293. Thus $293 was spent for doctors’ services in 1985. c. The amount spent on hospital care in 1985 (t 0) was H(0) 6.6(0)2 30(0) 682 682. Thus $682 was spent for hospital care in 1985. d. The year 2005 corresponds to t 2005 1985 20. To predict the amount to be spent on hospital care and doctors’ services in the year 2005, we find 6.6(20)2 71(20) 975 $5035. Note that it’s easier to evaluate D(t) H(t) 6.6t2 71t 975 for t 20 than it is to evaluate D(20) and H(20) and then add. EXAMPLE 7 More drinking In Example 7 of Section 4.4, we introduced the polynomial equations y 0.0226x 0.1509 and y 0.0257x 0.1663 approximating the blood alcohol level (BAL) for a 150-pound male or female, respectively. In these equations x represents the time since consuming 3 ounces of alcohol. It is known that the burn-off rate of alcohol is 0.015 per hour (that is, the BAL is reduced by 0.015 per hour if no additional alcohol is consumed). Find a polynomial that would approximate the BAL for a male x hours after consuming the 3 ounces of alcohol. SOLUTION The initial BAL is 0.0226x 0.1509, but this level is decreased by 0.015 each hour. Thus, the actual BAL after x hours is 0.0226x 0.1509 0.015x, or 0.0376x 0.1509. (Check with the calculator at link 4-4-5 on the Bello Website at mhhe.com/bello.) Answer 7. 0.0407x 0.1663 PROBLEM 7 Find a polynomial that would approximate the BAL for a female x hours after consuming 3 ounces of alcohol. bel84775_ch04_c.qxd 10/13/04 20:26 Page 342 342 Chapter 4 4-52 Exponents and Polynomials Exercises 4.5 Boost your GRADE at mathzone.com! In Problems 1–30, add as indicated. 1. (5x 2 2x 5) (7x 2 3x 1) • • • • • 2. (3x 2 5x 5) (9x 2 2x 1) Practice Problems Self-Tests Videos NetTutor e-Professors 3. (3x 5x 2 1) (7 2x 7x 2 ) 4. (3 2x 2 7x) (6 2x 2 5x) 5. (2x 5x 2 2) (3 5x 8x 2 ) 6. 3x 2 3x 2 and 4 5x 6x 2 7. 2 5x and 3 x 2 5x 8. 4x 2 6x 2 and 2 5x 9. x 3 2x 3 and 2x 2 x 5 10. x 4 3 2x 3x 3 and 3x 4 2x 2 5 x 11. 6x 3 2x 4 x and 2x 2 5 2x 2x 3 1 1 3 1 12. x3 x2 x and x x3 3x2 2 5 5 2 1 2 3 1 1 2 13. x2 x and x x2 3 5 4 4 5 3 14. 0.3x 0.1 0.4x 2 and 0.1x 2 0.1x 0.6 1 1 1 15. 0.2x 0.3 0.5x2 and x x 2 10 10 10 16. x 2 5x 2 () 3x 2 7x 2 2x 4 18. () 3x 4 20. () 22. 3x 3 17. 2x 1 x 3x 5 19. 2x 2 5x 5 2x 5x 7 21. x1 x 2x 5 x 23. 3 3x 2 2x 4 () x 2 4x 7 () 3 5x 4 () 5x 2 3 5x 3x 2 5 5x 3 3x 2 2 () 5x 3 3x 4 x 3 2x 5 3x 4 () 3x 2 3 3 5x 9 7 bel84775_ch04_c.qxd 10/13/04 20:26 Page 343 4-53 24. 4.5 1 x 3 3 1 x 5 2 1 2 1 x x 1 5 2 2 () x 3 3 26. 28. 30. 25. x2 1 1 1 x 2 x 8 3 5 3 2 x 3 x 2 8 5 2 4 () 3x 3 x 3 5 2x 4 5x 3 2x 2 3x 5 8x 3 2x 5 4 x 3x 2 x 2 () 6x 3 2x 5 27. Addition and Subtraction of Polynomials 2 1 x 3 x 2 2 7 6 1 x 3 5x 3 7 5 () x 2 1 6 1 x 3 7 2 1 x 2 x 3 9 2 3 2 2 () x x 2x 5 7 9 29. 6x 3 2x 2 1 x 3x 3 5x 2 3x x 3 7x 2 4 () 3x 3x 1 4 3x 4 2x 2 5 4 3 x x 2x 7x 2 5x 2x 4 2x 2 7 5 3 () 7x 2x 2x 5 In Problems 31–50, subtract as indicated. 31. (7x 2 2) (3x 2 5) 32. (8x 2 x) (7x 2 3x) 33. (3x 2 2x 1) (4x 2 2x 5) 34. (3x x 2 1) (5x 1 3x 2) 35. (1 7x 2 2x) (5x 3x 2 7) 36. (7x 3 x 2 x 1) (2x 2 3x 6) 37. (5x 2 2x 5) (3x 3 x 2 5) 38. (3x 2 x 7) (5x 3 5 x 2 2x) 39. (6x 3 2x 2 3x 1) (x 3 x 2 5x 7) 40. (x 3x 2 x 3 9) (8 7x x 2 x 3 ) 41. 6x 2 3x 5 () 3x 2 4x 2 42. 7x 2 4x 5 () 9x 2 2x 5 343 bel84775_ch04_c.qxd 10/13/04 20:26 Page 344 344 43. Chapter 4 3x 2 2x 1 () 3x 2 2x 1 4x 3 45. 44. 5x 2 1 () 3x 2 2x 1 2x 5 3x 5x 1 46. 3x 2 5x 2 () x 2x 2 5 2 2x x 6 48. 2 () 3x 3 47. 3 2 () 5x 3 49. 4-54 Exponents and Polynomials x2 5x 3x 7 6x 3 50. 2 () x 2 2x 1 () 3x x 2 5x 2 3 () 2x 5 3x 2 x In Problems 51–55, find the sum of the areas of the shaded rectangles. 51. 52. x x A x x B 2 x C D 4 3 A x 53. 2 x B x x x A B x C x x x x D 4 A 4 x B 4 2x C 2x D 2x 3x 54. x 3 C x D 4 x 55. 3x A 2x 3x B 5 3x C 3x 3x D 3x APPLICATIONS 56. Annual tuna consumption Based on Census Bureau estimates, the annual consumption of canned tuna (in pounds) per person is C(t) 3.87 0.13t, while the consumption of turkey (in pounds) is T(t) 0.15t 2 0.8t 13, where t is the number of years after 1989. a. How many pounds of canned tuna and how many pounds of turkey were consumed per person in 1989? b. Find the difference between the amount of turkey consumed and the amount of canned tuna consumed. c. Use your answer to part b to find the difference in the amount of turkey and the amount of canned tuna you would expect to be consumed in the year 2000. 57. Annual poultry consumption Poultry products consist of turkey and chicken. Annual chicken consumption per person (in pounds) is C(t) 0.05t 3 0.7t 2 t 36 and annual turkey consumption (in pounds) is T(t) 0.15t 2 1.6t 9, where t is the number of years after 1985. a. Find the total annual poultry consumption per person. b. How much poultry was consumed per person in 1995? How much poultry do you predict would be consumed in 2005? bel84775_ch04_c.qxd 10/13/04 20:26 Page 345 4-55 4.5 Addition and Subtraction of Polynomials 345 58. College costs How much are you paying for tuition and fees? In a four-year public institution, the amount T(t) you pay for tuition and fees (in dollars) can be approximated by T(t) 45t2 110t 3356, where t is the number of years after 2000 (2000 0). a. What would you predict tuition and fees to be in 2005? b. The cost of books t years after 2000 can be approximated by B(t) 27.5t 680. What would be the cost of books in 2005? c. What polynomial would represent the cost of tuition and fees and books t years after 2000? d. What would you predict the cost of tuition and fees and books would be in 2005? 59. College expenses The three major college expenses are: tuition and fees, books, and room and board. They can be approximated, respectively, by: 60. Student loans If you are an undergraduate dependent student you can apply for a Stafford Loan. The amount of these loans can be approximated by S(t) 562.5t2 312.5t 2625, where t is between 0 and 2 inclusive. If t is between 3 and 5 inclusive, then S(t) $5500. a. How much money can you get from a Stafford Loan the first year? b. What about the second year? c. What about the fifth year? 61. College financial aid Assume that you have to pay tuition and fees, books, and room and board in 2003 but have your Stafford Loan to decrease expenses. (See Problems 59 and 60.) Write a polynomial that would approximate how much you would have to pay t years after 2000 (t between 0 and 2). T(t) 45t 2 110t 3356 B(t) 27.5t 680 R(t) 32t 2 200t 4730 where t is the number of years after 2000 (2000 0). a. Write a polynomial representing the total cost of tuition and fees, books, and room and board t years after 2000. b. What was the cost of tuition and fees, books, and room and board in 2000? c. What would you predict the cost of tuition and fees, books, and room and board would be in 2005? SKILL CHECKER Try the Skill Checker Exercises so you’ll be ready for the next section. Simplify: 62. (5x 3 ) (2x 4 ) 63. (2x 4 ) (3x 5 ) 65. 6(y 4) 66. 3(2y 3) 64. 5(x 3) USING YOUR KNOWLEDGE Business Polynomials Polynomials are also used in business and economics. For example, the revenue R may be obtained by subtracting the cost C of the merchandise from its selling price S. In symbols, this is RSC Now the cost C of the merchandise is made up of two parts: the variable cost per item and the fixed cost. For example, if you decide to manufacture Frisbees™, you might spend $2 per Frisbee in materials, labor, and so forth. In addition, you might have $100 of fixed expenses. Then the cost for manufacturing x Frisbees is cost per fixed Cost C of merchandise is Frisbee and expenses. 14444244443 { 123 123 14243 C 2x 100 bel84775_ch04_c.qxd 10/13/04 20:26 Page 346 346 Chapter 4 4-56 Exponents and Polynomials If x Frisbees are then sold for $3 each, the total selling price S is 3x, and the revenue R would be RSC 3x (2x 100) 3x 2x 100 x 100 Thus if the selling price S is $3 per Frisbee, the variable costs are $2 per Frisbee, and the fixed expenses are $100, the revenue after manufacturing x Frisbees is given by R x 100 In Problems 67–69, find the revenue R for the given cost C and selling price S. 67. C 3x 50; S 4x 68. C 6x 100; S 8x 70. In Problem 68, how many items were manufactured if the revenue was zero? 69. C 7x; S 9x 71. If the merchant of Problem 68 suffered a $40 loss ($40 revenue), how many items were produced? WRITE ON 72. Write the procedure you use to add polynomials. 73. Write the procedure you use to subtract polynomials. 74. Explain the difference between “subtract x 2 3x 5 from 7x 2 2x 9” and “subtract 7x 2 2x 9 from x 2 3x 5.” What is the answer in each case? 75. List the advantages and disadvantages of adding (or subtracting) polynomials horizontally or in columns. MASTERY TEST If you know how to do these problems, you have learned your lesson! Add: 76. 3x 3x 2 6 and 5x 2 10 x 77. 5x 8x 2 3 and 3x 2 4 8x Subtract: 78. 3 4x 2 5x from 9x 2 2x 79. 9 x 3 3x 2 from 10 7x 2 5x 3 80. Add 2x 3 3x 5x 2 2, 6 5x 2x 2, and 6x 3 2x 8. 81. Find the sum of the areas of the shaded rectangles: 3 C x A 3 2x B 2x x x D 2 82. The number of robberies (per 100,000 population) can be approximated by R(t) 1.85t 2 19.14t 262, while the number of aggravated assaults is approximated by A(t) 0.2t 3 4.7t 2 15t 300, where t is the number of years after 1960. a. Were there more aggravated assaults or more robberies per 100,000 in 1960? b. Find the difference between the number of aggravated assaults and the number of robberies per 100,000. c. What would this difference be in the year 2000? In 2010? bel84775_ch04_c.qxd 10/13/04 20:26 Page 347 4-57 4.6 4.6 Multiplication of Polynomials 347 MULTIPLICATION OF POLYNOMIALS To Succeed, Review How To . . . 1. Multiply expressions (pp. 92, 98, 101, 294, 301). 2. Use the distributive property to remove parentheses in an expression (p. 101). Objectives A Multiply two monomials. B Multiply a monomial and a binomial. C Multiply two binomials using the FOIL method. D Solve an application. GETTING STARTED Deflections on a Bridge How much does the beam bend (deflect) when a car or truck goes over the bridge? There’s a formula that can tell us. For a certain beam of length L, the deflection at a distance x from one end is given by (x L)(x 2L) To multiply these two binomials, we must first learn how to do several related types of multiplication. A Multiplying Two Monomials We’ve already multiplied two monomials in Section 4.1. The idea is to use the associative and commutative properties and the rules of exponents, as shown in the next example. Web It For practice problems and a tutorial on multiplying monomials, go to link 4-6-1 on the Bello Website at mhhe.com/bello. EXAMPLE 1 Multiplying two monomials Multiply: (3x 2) by (2x 3) Multiply: (4y 3) by (5y 4) SOLUTION (3x 2)(2x 3) (3 2)(x 2 x 3) 6x 23 6x5 Answer 1. 20y7 PROBLEM 1 Use the associative and commutative properties. Use the rules of exponents. bel84775_ch04_c.qxd 10/13/04 20:26 Page 348 348 Chapter 4 4-58 Exponents and Polynomials B Multiplying a Monomial and a Binomial In Sections 1.6 and 1.7, we also multiplied a monomial and a binomial. The procedure was based on the distributive property, as shown next. EXAMPLE 2 PROBLEM 2 Multiplying a monomial by a binomial Remove parentheses (simplify): a. 5(x 2y) Simplify: b. (x 2 2x)3x 4 a. 4(a 3b) b. (a2 3a)4a5 SOLUTION a. 5(x 2y) 5x 5 2y 5x 10y b. (x 2 2x)3x 4 x 2 3x 4 2x 3x 4 3 x2 x4 2 3 x x4 Since (a b)c ac bc 3x 6 6x 5 NOTE You can use the commutative property first and write (x 2 2x)3x 4 3x 4(x 2 2x) 3x 6 6x 5 C Web It (Same answer!) Multiplying Two Binomials Using the FOIL Method Another way to multiply (x 2)(x 3) is to use the distributive property a(b c) ab ac. Think of x 2 as a, which makes x like b and 3 like c. Here’s how it’s done. a (b c) a b a c 678 } } 678} 678} (x 2) (x 3) (x 2)x (x 2)3 To learn about four different methods of multiplying binomials, go to link 4-6-2 on the Bello Website at mhhe.com/ bello. 123 x x 123 2 x 123 x 3 123 23 x 2 2x 3x 6 14243 x2 5x 6 Similarly, (x 3)(x 5) (x 3)x (x 3)5 123 x x (3) x 123 x 5 (3) 5 14243 14243 x2 x 2 3x 5x 1442443 2x Can you see a pattern developing? Look at the answers: (x 2)(x 3) x 2 5x 6 Answers 2. a. 4a 12b b. 4a7 12a6 (x 3)(x 5) x 2 2x 15 15 15 bel84775_ch04_c.qxd 10/13/04 20:26 Page 349 4-59 4.6 Multiplication of Polynomials 349 It seems that the first term in each answer (x2) is obtained by multiplying the first terms in the factors (x and x). Similarly, the last terms (6 and 15) are obtained by multiplying the last terms (2 3 and 3 5). Here’s how it works so far: Need the middle term xx (x 2)(x 3) x 2 _____ 6 Web It 23 Need the middle term To practice FOIL with your very own polynomials, go to link 4-6-3 on the Bello Website at mhhe.com/bello. xx (x 3)(x 5) x 2 _____ 15 3 5 But what about the middle terms? In (x 2)(x 3), the middle term is obtained by adding 3x and 2x, which is the same as the result we got when we multiplied the outer terms (x and 3) and added the product of the inner terms (2 and x). Here’s a diagram that shows how the middle term is obtained: Outer terms x3 (x 2)(x 3) x 2 3x 2x 6 2x Inner terms Outer terms x5 (x 3)(x 5) x 2 5x 3x 15 3 x Inner terms Do you see how it works now? Here is a summary of this method. PROCEDURE FOIL Method for Multiplying Binomials First terms are multiplied first. Outer terms are multiplied second. Inner terms are multiplied third. Last terms are multiplied last. Of course, we call this method the FOIL method. We shall do one more example, step by step, to give you additional practice. F O I L (x 7)(x 4) → x 2 (x 7)(x 4) → x 2 4x (x 7)(x 4) → x 2 4x 7x (x 7)(x 4) x 2 4x 7x 28 x 2 3x 28 First: x x Outer: 4 x Inner: 7 x Last: 7 (4) bel84775_ch04_c.qxd 10/13/04 20:26 Page 350 350 Chapter 4 EXAMPLE 3 4-60 Exponents and Polynomials PROBLEM 3 Using FOIL to multiply two binomials Find: Find: a. (x 5)(x 2) a. (a 4)(a 3) b. (x 4)(x 3) b. (a 5)(a 4) SOLUTION (First) F a. (x 5)(x 2) x x 123 x2 b. (x 4)(x 3) x x 123 x2 (Outer) (Inner) O I (Last) L 2x 5x 5 2 1442443 123 3x 10 3x 4x 4 3 1442443 123 x 12 As in the case of arithmetic, we can use the ideas we’ve just discussed to do more complicated problems. Thus we can use the FOIL method to multiply expressions such as (2x 5) and (3x 4). We proceed as before; just remember your laws of exponents and the FOIL sequence. EXAMPLE 4 PROBLEM 4 Using FOIL to multiply two binomials Find: Find: a. (2x 5)(3x 4) a. (3a 5)(2a 3) b. (3x 2)(5x 1) b. (2a 3)(4a 1) SOLUTION (First) F (Outer) O (Inner) I (Last) L a. (2x 5)(3x 4) (2x)(3x) (2x)(4) 5(3x) (5)(4) 123 14243 123 123 6x 2 8x 15x 20 144424443 6x 2 7x 20 F O b. (3x 2)(5x 1) (3x)(5x) 123 15x 2 15x 2 I 3x(1) 2(5x) 14243 123 3x 10x 1442443 13x L 2(1) 123 2 2 Does FOIL work when the binomials to be multiplied contain more than one variable? Fortunately, yes. Again, just remember the sequence and the laws of exponents. For example, to multiply (3x 2y) by (2x 5y), we proceed as follows: F Answers 3. a. a2 a 12 b. a2 a 20 4. a. 6a 2 a 15 b. 8a2 14a 3 O I L (2x 5y)(3x 2y) (2x)(3x) (2x)(2y) (5y)(3x) (5y)(2y) 123 123 123 123 6x 2 4xy 15xy 10y 2 1442443 6x 2 19xy 10y 2 bel84775_ch04_c.qxd 10/13/04 20:26 Page 351 4-61 4.6 EXAMPLE 5 Multiplying binomials involving two variables Find: Multiplication of Polynomials 351 PROBLEM 5 Find: a. (5x 2y)(2x 3y) a. (4a 3b)(3a 5b) b. (3x y)(4x 3y) b. (2a b)(3a 4b) SOLUTION F O I L a. (5x 2y)(2x 3y) (5x)(2x) (5x)(3y) (2y)(2x) (2y)(3y) 123 123 123 123 10x 2 15xy 4xy 6y 2 1442443 10x 2 19xy 6y 2 F O I L b. (3x y)(4x 3y) (3x)(4x) (3x)(3y) (y)(4x) (y)(3y) 123 14243 14243 14243 12x 2 9xy 4xy 3y2 1442443 12x 2 13xy 3y2 Now one more thing. How do we multiply the expression in the Getting Started? (x L)(x 2L) We do it in the next example. EXAMPLE 6 Multiplying binomials involving two variables Perform the indicated operation: PROBLEM 6 Perform the indicated operation: (x L)(x 2L) (y 2L)(y 3L) SOLUTION F O I L (x L)(x 2L) x x (x)(2L) (L)(x) (L)(2L) 123 14243 14243 14243 x2 2xL xL 2L2 1442443 x2 3xL 2L2 D Answers 5. a. 12a2 29ab 15b2 b. 6a2 11ab 4b2 6. y 2 5Ly 6L2 Solving an Application Suppose we wish to find out how much is spent annually on hospital care. According to the American Hospital Association, average daily room charges can be approximated by C(t) 160 14t (dollars), where t is the number of years after 1990. On the other hand, the U.S. National Health Center for Health Statistics indicated that the average stay (in days) in the hospital can be approximated by D(t) 7 0.2t, where t is the number of years after 1990. The amount spent annually on hospital care is given by Cost per day Number of days C(t) D(t) We find this product next. bel84775_ch04_c.qxd 10/13/04 20:26 Page 352 352 Chapter 4 EXAMPLE 7 4-62 Exponents and Polynomials PROBLEM 7 Getting well in U.S. hospitals Find: C(t) D(t) (160 14t)(7 0.2t) SOLUTION Suppose that at a certain hospital, the cost per day is (200 15t) and the average stay (in days) is (10 0.1t). What is the amount spent annually at this hospital? We use the FOIL method: F O I L (160 14t)(7 0.2t) 160 7 160 (0.2t) 14t 7 14t (0.2t) 123 1442443 123 1442443 1120 32t 98t 2.8t 2 1442443 1120 66t 2.8t 2 Thus the total amount spent annually on hospital care is 1120 66t 2.8t2. Can you calculate what this amount was for 2000? What will it be for the year 2010? Some students prefer a grid method to multiply polynomials. Thus, to do Example 4(a), (2x 5) (3x 4), create a grid separated into four compartments. Place the term (2x 5) at the top and the term (3x 4) on the side of the grid. Multiply the rows and columns of the grid as shown. (After you get some practice, you can skip the initial step and write 6x 2, 15x, 8x, and 20 in the grid.) 2x 5 3x 3x 2x 6x 2 3x 5 15x 4 4 2x 8x 4 5 20 Finish by writing the results of each of the grid boxes: 6x 2 15x 8x 20 And combining like terms: 6x 2 7x 20 You can try using this technique in the margin problems or in the exercises! Calculate It Checking Equivalency In the Section 4.1 Calculate It, we agreed that two expressions are equivalent if their graphs are identical. Thus, to check Example 1 we have to check that (3x 2)(2x 3) 6x 5. Let Y1 (3x 2)(2x 3) and Y2 6x 5. Press and the graph shown here will appear. To confirm the result numerically, press and you get the result in the table. X 0 1 2 3 4 5 6 Y1 0 -6 -192 -1458 -6144 -18750 -46656 Y2 0 -6 -192 -1458 -6144 -18750 -46656 X=0 You can check the rest of the examples except Examples 5 and 6. Why? Answer 7. 1.5t 2 130t 2000 bel84775_ch04_c.qxd 10/13/04 20:26 Page 353 4-63 4.6 Multiplication of Polynomials Exercises 4.6 Boost your GRADE at mathzone.com! In Problems 1–6, find the product. 1. (5x 3)(9x 2) 2. (8x 4)(9x 3) 3. (2x)(5x 2) 4. (3y 2)(4y 3) 5. (2y 2)(3y) 6. (5z)(3z) • • • • • Practice Problems Self-Tests Videos NetTutor e-Professors In Problems 7–20, remove parentheses (simplify). 7. 3(x y) 8. 5(2x y) 9. 5(2x y) 10. 4(3x 4y) 11. 4x(2x 3) 12. 6x(5x 3) 13. (x 2 4x)x 3 14. (x 2 2x)x 2 15. (x x 2)4x 16. (x 3x 2)5x 17. (x y)3x 18. (x 2y)5x 2 19. (2x 3y)(4y 2) 20. (3x 2 4y)(5y3) In Problems 21–56, use the FOIL method to perform the indicated operation. 21. (x 1)(x 2) 22. (y 3)(y 8) 23. (y 4)(y 9) 24. (y 6)(y 5) 25. (x 7)(x 2) 26. (z 2)(z 9) 27. (x 3)(x 9) 28. (x 2)(x 11) 29. (y 3)(y 3) 30. (y 4)(y 4) 31. (2x 1)(3x 2) 32. (4x 3)(3x 5) 33. (3y 5)(2y 3) 34. (4y 1)(3y 4) 35. (5z 1)(2z 9) 36. (2z 7)(3z 1) 37. (2x 4)(3x 11) 38. (5x 1)(2x 1) 39. (4z 1)(4z 1) 40. (3z 2)(3z 2) 41. (3x y)(2x 3y) 42. (4x z)(3x 2z) 43. (2x 3y)(x y) 44. (3x 2y)(x 5y) 45. (5z y)(2z 3y) 46. (2z 5y)(3z 2y) 47. (3x 2z)(4x z) 48. (2x 3z)(5x z) 49. (2x 3y)(2x 3y) 50. (3x 5y)(3x 5y) 51. (3 4x)(2 3x) 52. (2 3x)(3 2x) 53. (2 3x)(3 x) 54. (3 2x)(2 x) 55. (2 5x)(4 2x) 56. (3 5x)(2 3x) 353 bel84775_ch04_c.qxd 10/13/04 20:26 Page 354 354 Chapter 4 4-64 Exponents and Polynomials APPLICATIONS 57. Area of a rectangle The area A of a rectangle is obtained by multiplying its length L by its width W; that is, A LW. Find the area of the rectangle shown in the figure. x2 x5 58. Area of a rectangle Use the formula in Problem 57 to find the area of a rectangle of width x 4 and length x 3. 59. Height of a thrown object The height reached by an 60. Resistance The resistance R of a resistor varies object t seconds after being thrown upward with a with the temperature T according to the equation velocity of 96 feet per second is given by 16t(6 t). R (T 100)(T 20). Use the distributive property to Use the distributive property to simplify this expression. simplify this expression. 61. Gas property expression In chemistry, when V is the volume and P is the pressure of a certain gas, we find the expression (V2 V1)(CP PR), where C and R are constants. Use the distributive property to simplify this expression. The garage shown is 40 feet by 20 feet. You want to convert it to a bigger garage with two storage areas, S1 and S2. 20 62. What is the area of the current garage? 5 Garage extension 64. Calculate the areas of S1, S2, and S3, and write your answers in the appropriate places in the diagram. S3 40 65. Determine the total area of the new garage by adding the area of the original garage to the areas of S1, S2, and S3; that is, add the answers you obtained in Problems 62 and 64. Garage 40 20 8 66. Is the area of the new garage (Problem 63) the same as the answer in Problem 65? S2 S1 63. If you extend the long side by 8 feet and the short side by 5 feet, what is the area of the new garage? Storage areas y 20 Garage extension 40 S3 S1 b. Find the area of S2. c. Find the area of S3. 68. The area of the new garage is (40 x)(20 y). Simplify this expression. Garage 40 20 x 67. If you are not sure how big you want the storage rooms, extend the long side of the garage by x feet and the short side by y feet. a. Find the area of S1. 69. Add the areas of S1, S2, S3, and the area of the original garage. Is the answer the same as the one you obtained in Problem 68? S2 Storage areas bel84775_ch04_c.qxd 10/13/04 20:26 Page 355 4-65 4.6 Multiplication of Polynomials 355 SKILL CHECKER Try the Skill Checker Exercises so you’ll be ready for the next section. Find: 70. (4y)2 71. (3x)2 73. (3A)2 74. A(A) 72. (A)2 USING YOUR KNOWLEDGE Profitable Polynomials Do you know how to find the profit made in a certain business? The profit P is the difference between the revenue R and the expense E of doing business. In symbols, PRE Of course, R depends on the number n of items sold and their price p (in dollars); that is, R np Thus P np E 75. If n 3p 60 and E 5p 100, find P. 76. If n 2p 50 and E 3p 300, find P. 77. In Problem 75, if the price was $2, what was the profit P? 78. In Problem 76, if the price was $10, what was the profit P? WRITE ON 79. Will the product of two monomials always be a monomial? Explain. 80. If you multiply a monomial and a binomial, will you ever get a trinomial? Explain. 81. Will the product of two binomials (after combining like terms) always be a trinomial? Explain. 82. Multiply: (x 1)(x 1) (y 2)(y 2) (z 3)(z 3) What is the pattern? MASTERY TEST If you know how to do these problems, you have learned your lesson! Multiply: 83. (7x 4)(5x 2) 84. (8a3)(5a5) 85. (x 7)(x 3) 86. (x 2)(x 8) 87. (3x 4)(3x 1) 88. (4x 3y)(3x 2y) 89. (5x 2y)(2x 3y) 90. (x L)(x 3L) 91. Simplify 6(x 3y). 92. Simplify (x 3 5x)(4x 5).