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he nineteenth-century physician and physicist, Jean Louis Marie Poiseuille (1799–1869) is given credit for discovering a formula associated with the circulation of blood through arteries. Poiseuille’s law, as it is known, can be used to determine the velocity of blood in an artery at a given distance from the center of the artery. The formula states that the flow of blood in an artery is faster toward the center of the blood vessel and is slower toward the outside. Blood flow can also be affected by a person’s blood pressure, the length of the blood vessel, and the viscosity of the blood itself. In later years, Poiseuille’s continued interest in blood circulation led him to experiments to show that blood pressure rises and falls when a person exhales and inhales. In modern medicine, physicians can use Poiseuille’s law to determine how much the radius of a blocked blood vessel must be widened to create a healthy flow of blood. In this chapter you will study polynomials, the fundamental expressions of algebra. Polynomials are to algebra what integers are to arithmetic. We use polynomials to represent quantities in general, such as perimeter, area, revenue, and the volume of blood flowing through an artery. In Exercise 85 of Section 4.4, you will see Poiseuille’s law represented by a polynomial.
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4.1
ADDITION AND SUBTRACTION OF POLYNOMIALS
We first used polynomials in Chapter 1 but did not identify them as polynomials. Polynomials also occurred in the equations and inequalities of Chapter 2. In this section we will define polynomials and begin a thorough study of polynomials.
In this section ●
Polynomials
Polynomials
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Value of a Polynomial
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Addition of Polynomials
In Chapter 1 we defined a term as an expression containing a number or the product of a number and one or more variables raised to powers. Some examples of terms are
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Subtraction of Polynomials
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Applications
4x 3, x 2y 3, 6ab, and 2. A polynomial is a single term or a finite sum of terms. The powers of the variables in a polynomial must be positive integers. For example, 4x 3 (15x 2) x (2) is a polynomial. Because it is simpler to write addition of a negative as subtraction, this polynomial is usually written as 4x 3 15x 2 x 2.
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The degree of a polynomial in one variable is the highest power of the variable in the polynomial. So 4x 3 15x 2 x 2 has degree 3 and 7w w 2 has degree 2. The degree of a term is the power of the variable in the term. Because the last term has no variable, its degree is 0.
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Many commuting students find it difficult to get help. Students are often stuck on a minor point that can be easily resolved over the telephone. So ask your instructor if he or she will answer questions over the telephone and during what hours.
Thirddegree term
Seconddegree term
Firstdegree term
Zerodegree term
A single number is called a constant and so the last term is the constant term. The degree of a polynomial consisting of a single number such as 8 is 0. The number preceding the variable in each term is called the coefficient of that variable or the coefficient of that term. In 4x 3 15x 2 x 2 the coefficient of x 3 is 4, the coefficient of x 2 is 15, and the coefficient of x is 1 because x 1 x.
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Identifying coefficients Determine the coefficients of x 3 and x 2 in each polynomial: b) 4x 6 x 3 x a) x 3 5x 2 6
Solution a) Write the polynomial as 1 x 3 5x 2 6 to see that the coefficient of x 3 is 1 and the coefficient of x 2 is 5. b) The x 2-term is missing in 4x 6 x 3 x. Because 4x 6 x 3 x can be written as 4x 6 1 x 3 0 x 2 x, the coefficient of x 3 is 1 and the coefficient of x 2 is 0. ■ For simplicity we generally write polynomials with the exponents decreasing from left to right and the constant term last. So we write x 3 4x 2 5x 1
rather than
4x 2 1 5x x 3.
When a polynomial is written with decreasing exponents, the coefficient of the first term is called the leading coefficient.
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Certain polynomials are given special names. A monomial is a polynomial that has one term, a binomial is a polynomial that has two terms, and a trinomial is a polynomial that has three terms. For example, 3x5 is a monomial, 2x 1 is a binomial, and 4x 6 3x 2 is a trinomial.
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Types of polynomials Identify each polynomial as a monomial, binomial, or trinomial and state its degree. b) x 43 x 2 c) 5x d) 12 a) 5x 2 7x 3 2
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Solution a) The polynomial 5x 2 7x 3 2 is a third-degree trinomial. b) The polynomial x 43 x 2 is a binomial with degree 43. c) Because 5x 5x1, this polynomial is a monomial with degree 1. d) The polynomial 12 is a monomial with degree 0.
Be active in class. Don’t be embarrassed to ask questions or answer questions. You can often learn more from a wrong answer than a right one. Your instructor knows that you are not yet an expert in algebra. Instructors love active classrooms and they will not think less of you for speaking out.
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Value of a Polynomial A polynomial is an algebraic expression. Like other algebraic expressions involving variables, a polynomial has no specific value unless the variables are replaced by numbers. A polynomial can be evaluated with or without the function notation discussed in Chapter 3.
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Evaluating polynomials a) Find the value of 3x4 x 3 20x 3 when x 1. b) Find the value of 3x4 x 3 20x 3 when x 2. c) If P(x) 3x 4 x 3 20x 3, find P(1).
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Solution a) Replace x by 1 in the polynomial: 3x4 x 3 20x 3 3(1)4 (1)3 20(1) 3 3 1 20 3 19 So the value of the polynomial is 19 when x 1. b) Replace x by 2 in the polynomial:
close-up To evaluate the polynomial in Example 3 with a calculator, first use Y to define the polynomial.
3x4 x3 20x 3 3(2)4 (2)3 20(2) 3 3(16) (8) 40 3 48 8 40 3 77 So the value of the polynomial is 77 when x 2. c) This is a repeat of part (a) using the function notation from Chapter 3. P(1), read “P of 1,” is the value of the polynomial P(x) when x is 1. To find P(1), replace x by 1 in the formula for P(x):
Then find y1(2) and y1(1).
P(x) 3x 4 x 3 20x 3 P(1) 3(1)4 (1)3 20(1) 3 19
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So P(1) 19. The value of the polynomial when x 1 is 19.
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Addition of Polynomials You learned how to combine like terms in Chapter 1. Also, you combined like terms when solving equations in Chapter 2. Addition of polynomials is done simply by adding the like terms. Addition of Polynomials
To add two polynomials, add the like terms. Polynomials can be added horizontally or vertically, as shown in the next example.
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Adding polynomials Perform the indicated operation. a) (x 2 6x 5) (3x 2 5x 9) b) (5a3 3a 7) (4a2 3a 7)
Solution a) We can use the commutative and associative properties to get the like terms next to each other and then combine them:
hint
(x 2 6x 5) (3x 2 5x 9) x 2 3x 2 6x 5x 5 9 2x 2 x 4 b) When adding vertically, we line up the like terms:
When we perform operations with polynomials and write the results as equations, those equations are identities. For example,
3a 7 4a 3a 7 3 5a 4a2 5a3
(x 1) (3x 5) 4x 6
2
is an identity. This equation is satisfied by every real number.
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Subtraction of Polynomials When we subtract polynomials, we subtract the like terms. Because a b a (b), we can subtract by adding the opposite of the second polynomial to the first polynomial. Remember that a negative sign in front of parentheses changes the sign of each term in the parentheses. For example, (x 2 2x 8) x 2 2x 8. Polynomials can be subtracted horizontally or vertically, as shown in the next example.
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Subtracting polynomials Perform the indicated operation. a) (x 2 5x 3) (4x 2 8x 9)
b) (4y 3 3y 2) (5y 2 7y 6)
Solution a) (x 2 5x 3) (4x 2 8x 9) x 2 5x 3 4x 2 8x 9 3x 2 13x 6
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Change signs. Add.
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b) To subtract 5y 2 7y 6 from 4y 3 3y 2 vertically, we line up the like terms as we do for addition:
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For subtraction, write the original problem and then rewrite it as addition with the signs changed. Many students have trouble when they write the original problem and then overwrite the signs. Vertical subtraction is essential for performing long division of polynomials in Section 4.5.
4y 3
3y 2 (5y 2 7y 6)
Now change the signs of 5y 2 7y 6 and add the like terms: 3y 2 4y 3 2 5y 7y 6 3 4y 5y2 4y 8
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CAUTION When adding or subtracting polynomials vertically, be sure to line up the like terms.
In the next example we combine addition and subtraction of polynomials.
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Adding and subtracting Perform the indicated operations: (2x 2 3x) (x 3 6) (x4 6x 2 9)
Solution Remove the parentheses and combine the like terms: (2x 2 3x) (x 3 6) (x4 6x 2 9) 2x 2 3x x 3 6 x4 6x 2 9 ■ x4 x 3 8x 2 3x 15
Applications Polynomials are often used to represent unknown quantities. In certain situations it is necessary to add or subtract such polynomials.
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Profit from prints Trey pays $60 per day for a permit to sell famous art prints in the Student Union Mall. Each print costs him $4, so the polynomial 4x 60 represents his daily cost in dollars for x prints sold. He sells the prints for $10 each. So the polynomial 10x represents his daily revenue for x prints sold. Find a polynomial that represents his daily profit from selling x prints. Evaluate the profit polynomial for x 30.
Solution Because profit is revenue minus cost, we can subtract the corresponding polynomials to get a polynomial that represents the daily profit: 10x (4x 60) 10x 4x 60 6x 60 His daily profit from selling x prints is 6x 60 dollars. Evaluate this profit polynomial for x 30: 6x 60 6(30) 60 120
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M A T H
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The message we hear today about healthy eating is “more fiber and less fat.” But healthy eating can be challenging. Jill Brown, Registered Dietitian and owner of Healthy Habits Nutritional Counseling and Consulting, provides nutritional counseling for NUTRITIONIST people who are basically healthy but interested in improving their eating habits. On a client’s first visit, an extensive nutrition, medical, and family history is taken. Behavior patterns are discussed, and nutritional goals are set. Ms. Brown strives for a realistic rather than an idealistic approach. Whenever possible, a client should eat foods that are high in vitamin content rather than take vitamin pills. Moreover, certain frame types and family history make a slight and slender look difficult to achieve. Here, Ms. Brown might use the Harris-Benedict formula to determine the daily basal energy expenditure based on age, gender, and size. In addition to diet, exercise is discussed and encouraged. Finally, Ms. Brown provides a support system and serves as a “nutritional coach.” By following her advice, many of her clients lower their blood pressure, reduce their cholesterol, have more energy, and look and feel better. In Exercises 101 and 102 of this section you will use the Harris-Benedict formula to calculate the basal energy expenditure for a female and a male.
WARM-UPS
True or false? Explain your answer.
In the polynomial 2x 2 4x 7 the coefficient of x is 4. False The degree of the polynomial x 2 5x 9x 3 6 is 2. False In the polynomial x 2 x the coefficient of x is 1. True The degree of the polynomial x 2 x is 2. True A binomial always has a degree of 2. False The polynomial 3x 2 5x 9 is a trinomial. True Every trinomial has degree 2. False x 2 7x 2 6x 2 for any value of x. True (3x 2 8x 6) (x 2 4x 9) 4x 2 4x 3 for any value of x. True 10. (x 2 4x) (x 2 3x) 7x for any value of x. False 1. 2. 3. 4. 5. 6. 7. 8. 9.
4. 1
EXERCISES
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Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What is a term? A term is a single number or the product of a number and one or more variables raised to powers.
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2. What is a polynomial? A polynomial is a single term or a finite sum of terms. 3. What is the degree of a polynomial? The degree of a polynomial in one variable is the highest power of the variable in the polynomial.
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Determine the coefficients of x 3 and x 2 in each polynomial. See Example 1. 7. 3x 3 7x 2 8. 10x 3 x 2 10, 1 9. x4 6x 2 9 0, 6 10. x 5 x 3 3 1, 0 3 2 x 7x x3 x2 1 1 11. 4 12. 2x 1 3 2 2 4 Identify each polynomial as a monomial, binomial, or trinomial and state its degree. See Example 2. 13. 1 14. 5 15. m 3 Monomial, 0 Monomial, 0 Monomial, 3 8 16. 3a 17. 4x 7 18. a 6 Monomial, 8 Binomial, 1 Binomial, 1 10 2 6 3 19. x 3x 2 20. y 6y 9 Trinomial, 10 Trinomial, 6 6 21. x 1 22. b2 4 Binomial, 6 Binomial, 2 3 2 23. a a 5 24. x 2 4x 9 Trinomial, 3 Trinomial, 2
60. (345.56x 347.4) (56.6x 433) Add or subtract the polynomials as indicated. See Examples 4 and 5. 61. Add: 62. Add: 3a 4 2w 8 a6 w3
Evaluate each polynomial as indicated. See Example 3. 25. Evaluate 2x 2 3x 1 for x 1. 6 26. Evaluate 3x 2 x 2 for x 2. 16 27. Evaluate 3x 3 x 2 3x 4 for x 3. 85 28. Evaluate 2x 4 3x 2 5x 9 for x 2. 43 29. If P(x) 3x 4 2x 3 7, find P(2). 71 30. If P(x) 2x 3 5x 2 12, find P(5). 137 31. If P(x) 1.2x 3 4.3x 2.4, find P(1.45). 4.97665 32. If P(x) 3.5x 4 4.6x 3 5.5, find P(2.36).
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Perform the indicated operation. See Example 4. 33. (x 3) (3x 5) 34. (x 2) (x 3) 4x 8 2x 1 35. (q 3) (q 3) 36. (q 4) (q 6) 2q 2q 10 37. (3x 2) (x2 4) 38. (5x 2 2) (3x 2 1) x x 2 2x 3 39. (4x 1) (x 3 5x 6) x 9x 7 40. (3x 7) (x 2 4x 6) x x 1 2 2 41. (a 3a 1) (2a 4a 5) 3a 7a 4 42. (w 2 2w 1) (2w 5 w2) 2w 4 43. (w 2 9w 3) (w 4w 2 8) 3w 8w 5 44. (a 3 a2 5a) (6 a 3a2 ) a 4a 6 6 2 2 45. (5.76x 3.14x 7.09) (3.9x 1.21x 5.6) 9.66x 1.93x 1.49
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46. (8.5x 2 3.27x 9.33) (x 2 4.39x 2.32) 9.5 x 11.65 Perform the indicated operation. See Example 5. 47. (x 2) (5x 8) 48. (x 7) (3x 1) 4x 6 2x 49. (m 2) (m 3) 50. (m 5) (m 9) 5 4 51. (2z 2 3z) (3z 2 5z) 52. (z 2 4z) (5z 2 3z) z 2z 53. (w 5 w 3) (w4 w2) w w 54. (w 6 w 3) (w 2 w) w w 55. (t 2 3t 4) (t 2 5t 9) 2t t 56. (t 2 6t 7) (5t 2 3t 2) 57. (9 3y y 2) (2 5y y 2 ) 58. (4 5y y 3) (2 3y y 2 ) y 59. (3.55x 879) (26.4x 455.8)
4. What is the value of a polynomial? The value of a polynomial is the number obtained when the variable is replaced by a number. 5. How do we add polynomials? Polynomials are added by adding the like terms. 6. How do we subtract polynomials? Polynomials are subtracted by subtracting like terms.
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4a 2 63. Subtract: 3x 11 5x 7
3w 5 64. Subtract: 4x 3 2x 9
2x 4 65. Add: ab ab
2x 6 66. Add: s6 s1
2a 67. Subtract: 3m 1 2m 6
2s 7 68. Subtract: 5n 2 3n 4
m 7 69. Add: 2x 2 x 3 2x 2 x 4
8n 6 70. Add: x 2 4x 6 3x 2 x 5
4x 1 71. Subtract: 3a3 5a2 7 2a3 4a2 2a
2x 3x 11 72. Subtract: 2b3 7b2 9 b3 4b 2
73. Subtract: x 2 3x 6 x2 3
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b b 4b 74. Subtract: x4 3x 2 2 3x4 2x 2
3 75. Add: y 3 4y 2 6y 5 y 3 3y 2 2y 9
2 x 2 76. Add: q 2 4q 9 3q 2 7q 5
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Perform the operations indicated. 77. Find the sum of 2m 9 and 3m 4. 78. Find the sum of 3n 2 and 6m 3. 79. Find the difference when 7y 3 is subtracted from 9y 2. 2y 1 80. Find the difference when 2y 1 is subtracted from 3y 4. 5y 3 81. Subtract x 2 3x 1 from the sum of 2x 2 x 3 and x 2 5x 9. 2x 7x 5 82. Subtract 2y 2 3y 8 from the sum of 3y 2 2y 6 and 7y 2 8y 3. 6y 3y 11 Perform the indicated operations. See Example 6. 83. (4m 2) (2m 4) (9m 1) m 3 84. (5m 6) (8m 3) (5m 3) 8m 12 85. (6y 2) (8y 3) (9y 2) 11y 3 86. (5y 1) (8y 4) (y 3) 14y
c) Find the cost of labor for 500 transmissions and the cost of materials for 500 transmissions. a) 0.4x 450x 1350 b) $326,350 c) $275,550, $50,800 93. Perimeter of a triangle. The shortest side of a triangle is x meters, and the other two sides are 3x 1 and 2x 4 meters. Write a polynomial that represents the perimeter and then evaluate the perimeter polynomial if x is 4 meters. 6x 3 meters, 27 meters 94. Perimeter of a rectangle. The width of a rectangular playground is 2x 5 feet, and the length is 3x 9 feet. Write a polynomial that represents the perimeter and then evaluate this perimeter polynomial if x is 4 feet. 10x 8 feet, 48 feet
87. (x 2 5x 4) (6x 2 8x 9) (3x 2 7x 1) 2x 6 88. (8x 2 5x 12) (3x 2 9x 18) (3x 2 9x 4) 13 10 89. (6z4 3z 3 7z 2) (5z 3 3z 2 2) (z4 z 2 5) 5z 8z 3z 7 90. (v 3 v2 1) (v4 v2 v 1) (v3 3v2 6) 6 Solve each problem. See Example 7. 91. Profitable pumps. Walter Waterman, of Walter’s Water Pumps in Winnipeg has found that when he produces x water pumps per month, his revenue is x 2 400x 300 dollars. His cost for producing x water pumps per month is x 2 300x 200 dollars. Write a polynomial that represents his monthly profit for x water pumps. Evaluate this profit polynomial for x 50. 100x 500 dollars, $5500 92. Manufacturing costs. Ace manufacturing has determined that the cost of labor for producing x transmissions is 0.3x 2 400x 550 dollars, while the cost of materials is 0.1x 2 50x 800 dollars. a) Write a polynomial that represents the total cost of materials and labor for producing x transmissions. b) Evaluate the total cost polynomial for x 500.
2x 5 ft
3x 9 ft
FIGURE FOR EXERCISE 94 95. Before and after. Jessica traveled 2x 50 miles in the morning and 3x 10 miles in the afternoon. Write a polynomial that represents the total distance that she traveled. Find the total distance if x 20. 5x 40 miles, 140 miles 96. Total distance. Hanson drove his rig at x mph for 3 hours, then increased his speed to x 15 mph and drove for 2 more hours. Write a polynomial that represents the total distance that he traveled. Find the total distance if x 45 mph. 5x 30 miles, 255 miles 97. Sky divers. Bob and Betty simultaneously jump from two airplanes at different altitudes. Bob’s altitude t seconds after leaving the plane is 16t 2 6600 feet. Betty’s altitude t seconds after leaving the plane is 16t 2 7400 feet. Write a polynomial that represents the difference between their altitudes t seconds after leaving the planes. What is the difference between their altitudes 3 seconds after leaving the planes? 800 feet, 800 feet
Total 0.5
16t 2 7400 ft 16t 2 6600 ft
Labor Materials
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500 1000 Number of transmissions
FIGURE FOR EXERCISE 97
FIGURE FOR EXERCISE 92
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98. Height difference. A red ball and a green ball are simultaneously tossed into the air. The red ball is given an initial velocity of 96 feet per second, and its height t seconds after it is tossed is 16t 2 96t feet. The green ball is given an initial velocity of 30 feet per second, and its height t seconds after it is tossed is 16t 2 80t feet. a) Find a polynomial that represents the difference in the heights of the two balls. 16t feet b) How much higher is the red ball 2 seconds after the balls are tossed? 32 feet c) In reality, when does the difference in the heights stop increasing? when green ball hits ground
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101. Harris-Benedict for females. The Harris-Benedict polynomial 655.1 9.56w 1.85h 4.68a represents the number of calories needed to maintain a female at rest for 24 hours, where w is her weight in kilograms, h is her height in centimeters, and a is her age. Find the number of calories needed by a 30-year-old 54-kilogram female who is 157 centimeters tall. 1321.39 calories 102. Harris-Benedict for males. The Harris-Benedict polynomial 66.5 13.75w 5.0h 6.78a
Height (feet)
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represents the number of calories needed to maintain a male at rest for 24 hours, where w is his weight in kilograms, h is his height in centimeters, and a is his age. Find the number of calories needed by a 40-year-old 90-kilogram male who is 185 centimeters tall. 1957.8 calories
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103. Discussion. Is the sum of two natural numbers always a natural number? Is the sum of two integers always an integer? Is the sum of two polynomials always a polynomial? Explain. Yes, yes, yes 104. Discussion. Is the difference of two natural numbers always a natural number? Is the difference of two rational numbers always a rational number? Is the difference of two polynomials always a polynomial? Explain. No, yes, yes 105. Writing. Explain why the polynomial 24 7x3 5x2 x has degree 3 and not degree 4. 106. Discussion. Which of the following polynomials does not have degree 2? Explain. a) r 2 b) 2 4 c) y 2 4 d) x 2 x 4 e) a2 3a 9 b and d
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FIGURE FOR EXERCISE 98 99. Total interest. Donald received 0.08(x 554) dollars interest on one investment and 0.09(x 335) interest on another investment. Write a polynomial that represents the total interest he received. What is the total interest if x 1000? 0.17x 74.47 dollars, $244.47 100. Total acid. Deborah figured that the amount of acid in one bottle of solution is 0.12x milliliters and the amount of acid in another bottle of solution is 0.22(75 x) milliliters. Find a polynomial that represents the total amount of acid? What is the total amount of acid if x 50? x 16.5 milliliters, 11.5 milliliters
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MULTIPLICATION OF POLYNOMIALS
section ●
Multiplying Monomials with the Product Rule
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Multiplying Polynomials
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The Opposite of a Polynomial
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Applications
You learned to multiply some polynomials in Chapter 1. In this section you will learn how to multiply any two polynomials.
Multiplying Monomials with the Product Rule To multiply two monomials, such as x3 and x5, recall that
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So 3 factors
5 factors
x 3 x 5 (x x x )(x x x x x) x8. 8 factors 3
The exponent of the product of x and x5 is the sum of the exponents 3 and 5. This example illustrates the product rule for multiplying exponential expressions. Product Rule
If a is any real number and m and n are any positive integers, then am an amn.
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Multiplying monomials Find the indicated products. b) (2ab)(3ab) a) x2 x4 x
c) 4x2y2 3xy5
d) (3a)2
Solution a) x2 x4 x x2 x4 x1 x7 Product rule b) (2ab)(3ab) (2)(3) a a b b 6a2b2 Product rule c) (4x2y2)(3xy5) (4)(3)x2 x y2 y5 12x3y7 Product rule d) (3a)2 3a 3a 9a2
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As soon as possible after class, find a quiet place and work on your homework. The longer you wait the harder it is to remember what happened in class.
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CAUTION Be sure to distinguish between adding and multiplying monomials. You can add like terms to get 3x4 2x4 5x4, but you cannot combine the terms in 3w5 6w2. However, you can multiply any two monomials: 3x4 2x4 6x8 and 3w5 6w2 18w7.
Multiplying Polynomials To multiply a monomial and a polynomial, we use the distributive property.
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Multiplying monomials and polynomials Find each product. b) (y2 3y 4)(2y) a) 3x2(x3 4x)
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Solution a) 3x2(x3 4x) 3x2 x3 3x2 4x 3x5 12x3
Distributive property
b) (y2 3y 4)(2y) y2(2y) 3y (2y) 4(2y) 2y3 (6y2) (8y) 2y3 6y2 8y
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When doing homework or taking notes, use a pencil with an eraser. Everyone makes mistakes. If you get a problem wrong, don’t start over. Check your work for errors and use the eraser.It is better to find out where you went wrong than to simply get the right answer.
c) a(b c)
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Distributive property
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Distributive property
Note in part (c) that either of the last two binomials is the correct answer. The last ■ one is just a little simpler to read. Just as we use the distributive property to find the product of a monomial and a polynomial, we can use the distributive property to find the product of two binomials and the product of a binomial and a trinomial.
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Multiplying polynomials Use the distributive property to find each product. a) (x 2)(x 5) b) (x 3)(x2 2x 7)
Solution a) First multiply each term of x 5 by x 2: (x 2)(x 5) (x 2)x (x 2)5 x2 2x 5x 10 x2 7x 10
Distributive property Distributive property Combine like terms.
b) First multiply each term of the trinomial by x 3: (x 3)(x2 2x 7) (x 3)x2 (x 3)2x (x 3)(7) x3 3x2 2x2 6x 7x 21 x 5x x 21 3
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Distributive property
Distributive property
Combine like terms.
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Products of polynomials can also be found by arranging the multiplication vertically like multiplication of whole numbers.
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Multiplying vertically Find each product. a) (x 2)(3x 7)
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Solution a) 3x 7 x2 6x 14 2 3x 7x 3x2 x 14
Many students find vertical multiplication easier than applying the distributive property twice horizontally. However, you should learn both methods because horizontal multiplication will help you with factoring by grouping in Section 5.2.
b) (x y)(a 3) xy a3 3x 3y
b) ← 2 times 3x 7 ← x times 3x 7 Add.
ax ay ax ay 3x 3y
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These examples illustrate the following rule.
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Multiplication of Polynomials
To multiply polynomials, multiply each term of one polynomial by every term of the other polynomial, then combine like terms.
The Opposite of a Polynomial Note the result of multiplying the difference a b by 1: 1(a b) a b b a Because multiplying by 1 is the same as taking the opposite, we can write (a b) b a. So a b and b a are opposites or additive inverses of each other. If a and b are replaced by numbers, the values of a b and b a are additive inverses. For example, 3 7 4 and 7 3 4. CAUTION
E X A M P L E
5
The opposite of a b is a b, not a b.
Opposite of a polynomial Find the opposite of each polynomial. a) x 2 b) 9 y2
c) a 4
d) x 2 6x 3
Solution a) (x 2) 2 x b) (9 y2 ) y2 9 c) (a 4) a 4 d) (x 2 6x 3) x2 6x 3
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Applications E X A M P L E
6
Multiplying polynomials A parking lot is 20 yards wide and 30 yards long. If the college increases the length and width by the same amount to handle an increasing number of cars, then what polynomial represents the area of the new lot? What is the new area if the increase is 15 yards?
Solution If x is the amount of increase, then the new lot will be x 20 yards wide and x 30 yards long as shown in Fig. 4.1. Multiply the length and width to get the area:
x
(x 20)(x 30) (x 20)x (x 20)30 x 2 20x 30x 600 x 2 50x 600
30 yd
The polynomial x 2 50x 600 represents the area of the new lot. If x 15, then 20 yd
x 2 50x 600 (15)2 50(15) 600 1575.
FIGURE 4.1
If the increase is 15 yards, then the area of the lot will be 1575 square yards.
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4.2
WARM-UPS
Multiplication of Polynomials
(4-13)
219
True or false? Explain your answer. 3x3 5x4 15x12 for any value of x.
1. 2. 3. 4. 5. 6. 7.
False 3x 2x 5x for any value of x. False (3y3)2 9y6 for any value of y. True 3x(5x 7x2) 15x3 21x2 for any value of x. False 2x(x2 3x 4) 2x3 6x2 8x for any number x. True 2(3 x) 2x 6 for any number x. True (a b)(c d) ac ad bc bd for any values of a, b, c, and d. True 8. (x 7) 7 x for any value of x. True 9. 83 37 (37 83) True 10. The opposite of x 3 is x 3 for any number x. False
4. 2
2
7
9
EXERCISES 19. (5y)2 25y 21. (2x3)2 4x
Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What is the product rule for exponents? 2. Why is the sum of two monomials not necessarily a monomial? The sum of two monomials can be a binomial if the terms are not like terms. 3. What property of the real numbers is used when multiplying a monomial and a polynomial? To multiply a monomial and a polynomial we use the distributive property. 4. What property of the real numbers is used when multiplying two binomials? To multiply two binomials we use the distributive property twice. 5. How do we multiply any two polynomials? To multiply any two polynomials we multiply each term of the first polynomial by every term of the second polynomial. 6. How do we find the opposite of a polynomial? To find the opposite of a polynomial, change the sign of each term in the polynomial.
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Find each product. See Example 1. 7. 3x2 9x3 8. 5x7 3x5 27x 15x 10. 3y12 5y15 11. 6x2 5x2 15y x 13. (9x10)(3x7) 14. (2x2)(8x9) 27x 16x 16. 12sq 3s 17. 3wt 8w7t6 qs 24t
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Find each product. See Example 2. 23. 4y2(y5 2y) 6t3(t5 3t2) 3y(6y 4) 9y(y2 1) (y2 5y 6)(3y) 3 (x3 5x2 1)7x2 7x x(y2 x2) ab(a2 b2) (3ab3 a2b2 2a3b)5a (3c2d d3 1)8cd2 24 1 33. t2v(4t3v2 6tv 4v) 2 1 34. m2n3(6mn2 3mn 12) 3
24. 25. 26. 27. 28. 29. 30. 31. 32.
Use the distributive property to find each product. See Example 3.
9. 2a3 7a8 14a 12. 2x 2 8x 5 16x 15. 6st 9st 54s 18. h8k3 5h 5h
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20. (6x)2 36x 22. (3y5)2 9y
35. (x 1)(x 2) x 3x 2 37. (x 3)(x 5) x 2x 15 39. (t 4)(t 9) t 13t 36 41. (x 1)(x2 2x 2) x 3x 4x
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36. (x 6)(x 3) x 9x 18 38. ( y 2)( y 4) y y 8 40. (w 3)(w 5) w 8w 15 42. (x 1)(x2 x 1) x
Textbook Table of Contents
220
(4-14)
Chapter 4
Polynomials and Exponents
43. (3y 2)(2y2 y 3) 44. (4y 3)(y2 3y 1) 6y 7 6 4y 15y 13 3 45. (y2z 2y4)(y2z 3z2 y4) 2y 3 y 3y 3 2 4 2 46. (m 4mn )(6m n 3m6 m2n4) 18m 23m 3m 4m Find each product vertically. See Example 4. 47. 2a 3 48. 2w 6 a5 w5 2a 7a 49. 7x 30 2x 5
15
14x 95x 51. 5x 2 4x 3 20x 7x 53. m 3n 2a b
2w 4w 50. 5x 7 3x 6
150
30
15x 51x 52. 4x 3 2x 6
6
8x 18x 54. 3x 7 a 2b
2am 6an mb 55. x2 3x 2 x6
42
18
3ax 7a 6xb 56. x2 3x 5 x7
14b
x 10x 26 58. 3x2 5x 2 5x 6
35
15x 7x 60. a2 b2 a2 b2
12
59. x y xy x y 61. x2 xy y2 xy
a b 62. 4w2 2wv v2 2w v
9x 16x 57. 2a3 3a2 4 2a 3 a
3nb
83. (5x 6)(5x 6) 25x 36 85. 2x2(3x5 4x2) 6x 8x 87. (m 1)(m2 m 1) m 1 89. (3x 2)(x2 x 9) 90. (5 6y)(3y2 y 7)
12
a
12
20x
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x y 8w v Find the opposite of each polynomial. See Example 5. 63. 3t u u 3t 64. 3t u 3t u 65. 3x y 3x y 66. x 3y 3y x 67. 3a2 a 6 6 2 68. 3b b 6 6 69. 3v2 v 6 6 70. 3t2 t 6 Perform the indicated operation. 71. 3x(2x 9) 72. 1(2 3x) x 27x 3x 2 73. 2 3x(2x 9) 74. 6 3(4x 8) 27x 2 30 12x 75. (2 3x) (2x 9) 76. (2 3x) (2x 9) 5 11 6 2 3 2 77. (6x ) 36x 78. (3a b) 79. 3ab3(2a2b7) a 80. 4xst 8xs 81. (5x 6)(5x 6) 82. (5x 6)(5x 6) 25x 60x 36 25x 60x 36
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84. (2x 9)(2x 9) 4x 81 86. 4a3(3ab3 2ab3) 4a b 88. (a b)(a2 ab b2) a b
Solve each problem. See Example 6. 91. Office space. The length of a professor’s office is x feet, and the width is x 4 feet. Write a polynomial that represents the area. Find the area if x 10 ft. x 4x square feet, 140 square feet 92. Swimming space. The length of a rectangular swimming pool is 2x 1 meters, and the width is x 2 meters. Write a polynomial that represents the area. Find the area if x is 5 meters. 2x 3x 2 square meters, 63 square meters 93. Area of a truss. A roof truss is in the shape of a triangle with a height of x feet and a base of 2x 1 feet. Write a polynomial that represents the area of the triangle. What is the area if x is 5 feet? 1
x ft
2x 1 ft
FIGURE FOR EXERCISE 93 94. Volume of a box. The length, width, and height of a box are x, 2x, and 3x 5 inches, respectively. Write a polynomial that represents its volume. 6x 10x
3x 5
2x
x
FIGURE FOR EXERCISE 94 95. Number pairs. If two numbers differ by 5, then what polynomial represents their product? x 5x 96. Number pairs. If two numbers have a sum of 9, then what polynomial represents their product? 9x x 97. Area of a rectangle. The length of a rectangle is 2.3x 1.2 meters, and its width is 3.5x 5.1 meters. What polynomial represents its area? 8.05 x
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Total revenue (thousands of dollars)
4.3
98. Patchwork. A quilt patch cut in the shape of a triangle has a base of 5x inches and a height of 1.732x inches. What polynomial represents its area? 4.33x2 square inches
Multiplication of Binomials
(4-15)
221
10 9 8 7 6 5 4 3 2 1 0 2
4
6
8 10 12 14 16 18 Price (dollars)
FIGURE FOR EXERCISE 100 101. Periodic deposits. At the beginning of each year for 5 years, an investor invests $10 in a mutual fund with an average annual return of r. If we let x 1 r, then at the end of the first year (just before the next investment) the value is 10x dollars. Because $10 is then added to the 10x dollars, the amount at the end of the second year is (10x 10)x dollars. Find a polynomial that represents the value of the investment at the end of the fifth year. Evaluate this polynomial if r 10%.
1.732 x 5x
102. Increasing deposits. At the beginning of each year for 5 years, an investor invests in a mutual fund with an average annual return of r. The first year, she invests $10; the second year, she invests $20; the third year, she invests $30; the fourth year, she invests $40; the fifth year, she invests $50. Let x 1 r as in Exercise 101 and write a polynomial in x that represents the value of the investment at the end of the fifth year. Evaluate this polynomial for r 8%.
FIGURE FOR EXERCISE 98 99. Total revenue. If a promoter charges p dollars per ticket for a concert in Tulsa, then she expects to sell 40,000 1000p tickets to the concert. How many tickets will she sell if the tickets are $10 each? Find the total revenue when the tickets are $10 each. What polynomial represents the total revenue expected for the concert when the tickets are p dollars each? 30,000, $300,000, 40,000p 1000p 100. Manufacturing shirts. If a manufacturer charges p dollars each for rugby shirts, then he expects to sell 2000 100p shirts per week. What polynomial represents the total revenue expected for a week? How many shirts will be sold if the manufacturer charges $20 each for the shirts? Find the total revenue when the shirts are sold for $20 each. Use the bar graph to determine the price that will give the maximum total revenue. 2000p 100p
4.3
GET TING MORE INVOLVED 103. Discussion. Name all properties of the real numbers that are used in finding the following products: a) 2ab3c2 5a2bc b) (x2 3)(x2 8x 6) 104. Discussion. Find the product of 27 and 436 without using a calculator. Then use the distributive property to find the product (20 7)(400 30 6) as you would find the product of a binomial and a trinomial. Explain how the two methods are related.
MULTIPLICATION OF BINOMIALS
In Section 4.2 you learned to multiply polynomials. In this section you will learn a rule that makes multiplication of binomials simpler.
In this section
The FOIL Method
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The FOIL Method
●
Multiplying Binomials Quickly
We can use the distributive property to find the product of two binomials. For example,
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(x 2)(x 3) (x 2)x (x 2)3 x2 2x 3x 6 x2 5x 6
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Distributive property Distributive property Combine like terms.
222
(4-16)
Chapter 4
Polynomials and Exponents
There are four terms in x 2 2x 3x 6. The term x2 is the product of the first term of each binomial, x and x. The term 3x is the product of the two outer terms, 3 and x. The term 2x is the product of the two inner terms, 2 and x. The term 6 is the product of the last term of each binomial, 2 and 3. We can connect the terms multiplied by lines as follows: L
F
(x 2)(x 3)
F First terms O Outer terms I Inner terms L Last terms
I O
If you remember the word FOIL, you can get the product of the two binomials much faster than writing out all of the steps above. This method is called the FOIL method. The name should make it easier to remember.
E X A M P L E
1
Using the FOIL method Find each product. a) (x 2)(x 4) c) (a b)(2a b)
b) (2x 5)(3x 4) d) (x 3)( y 5)
Solution
helpful
L
F
hint
F
O
I
L
a) (x 2)(x 4) x 4x 2x 8 x 2 2x 8 Combine the like terms. 2
You may have to practice FOIL a while to get good at it. However, the better you are at FOIL, the easier you will find factoring in Chapter 5.
I O
b) (2x 5)(3x 4) 6x2 8x 15x 20 6x2 7x 20 Combine the like terms. c) (a b)(2a b) 2a2 ab 2ab b2 2a2 3ab b2 d) (x 3)( y 5) xy 5x 3y 15 There are no like terms to combine.
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FOIL can be used to multiply any two binomials. The binomials in the next example have higher powers than those of Example 1.
E X A M P L E
study
2
Using the FOIL method Find each product. a) (x3 3)(x3 6)
tip
Solution a) (x3 3)(x3 6) x6 6x3 3x3 18 x6 3x3 18 b) (2a2 1)(a2 5) 2a4 10a2 a2 5 2a4 11a2 5
Remember that everything we do in solving problems is based on principles (which are also called rules, theorems, and definitions). These principles justify the steps we take. Be sure that you understand the reasons. If you just memorize procedures without understanding, you will soon forget the procedures.
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Multiplying Binomials Quickly The outer and inner products in the FOIL method are often like terms, and we can combine them without writing them down. Once you become proficient at using FOIL, you can find the product of two binomials without writing anything except the answer.
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b) (2a2 1)(a2 5)
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4.3
E X A M P L E
3
Combine like terms: 3x 4x 7x. Combine like terms: 10x x 9x.
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Combine like terms: 6a 6a 0.
Area of a garden Sheila has a square garden with sides of length x feet. If she increases the length by 7 feet and decreases the width by 2 feet, then what trinomial represents the area of the new rectangular garden?
x2
Solution The length of the new garden is x 7 and the width is x 2 as shown in Fig. 4.2. ■ The area is (x 7)(x 2) or x2 5x 14 square feet.
x7
FIGURE 4.2
@ http://ellerbruch.nmu.edu/cs255/npaquett/binomials.html
WARM-UPS
True or false? Answer true only if the equation is true for all values of the variable or variables. Explain your answer.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
4. 3
223
4
x
x
(4-17)
Using FOIL to find a product quickly Find each product. Write down only the answer. a) (x 3)(x 4) b) (2x 1)(x 5) c) (a 6)(a 6)
Solution a) (x 3)(x 4) x2 7x 12 b) (2x 1)(x 5) 2x2 9x 5 c) (a 6)(a 6) a2 36
E X A M P L E
Multiplication of Binomials
(x 3)(x 2) x2 6 False (x 2)( y 1) xy x 2y 2 True (3a 5)(2a 1) 6a2 3a 10a 5 True (y 3)( y 2) y2 y 6 True (x2 2)(x2 3) x4 5x2 6 True (3a2 2)(3a2 2) 9a2 4 False (t 3)(t 5) t2 8t 15 True (y 9)( y 2) y2 11y 18 False (x 4)(x 7) x2 4x 28 False It is not necessary to learn FOIL as long as you can get the answer.
False
EXERCISES
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Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What property of the real numbers do we usually use to find the product of two binomials? We use the distributive property to find the product of two binomials. 2. What does FOIL stand for? FOIL stands for first, outer, inner, and last. 3. What is the purpose of FOIL? The purpose of FOIL is to provide a faster method for finding the product of two binomials.
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4. What is the maximum number of terms that can be obtained when two binomials are multiplied? The maximum number of terms obtained in multiplying binomials is four. Use FOIL to find each product. See Example 1. 5. (x 2)(x 4) 6. (x 3)(x 5) 7. (a 3)(a 2) 8. (b 1)(b 2) 9. (2x 1)(x 2)
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224 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
(4-18)
Chapter 4
Polynomials and Exponents
(2y 5)(y 2) (2a 3)(a 1) (3x 5)(x 4) 3 (w 50)(w 10) w (w 30)(w 20) w (y a)(y 5) y (a t)(3 y) 3a (5 w)(w m) 5w (a h)(b t) ab (2m 3t)(5m 3t) (2x 5y)(x y) 2x (5a 2b)(9a 7b) (11x 3y)(x 4y)
55. 56. 57. 58.
(h 7w)(h 7w) h (h 7q)(h 7q) h (2h2 1)(2h2 1) 4 (3h2 1)(3h2 1) 9
Perform the indicated operations. 1 1 8 59. 2a 4a 2 2
2 1 60. 3b 6b 3 3 1 1 1 1 61. x x 2 3 4 2 62.
Use FOIL to find each product. See Example 2. 23. (x 2 5)(x 2 2) 24. (y2 1)(y 2 2) 25. (h 3 5)(h 3 5) 26. (y6 1)( y6 4) 27. (3b3 2)(b3 4) 28. (5n4 1)(n4 3) 29. (y2 3)( y 2) 30. (x 1)(x2 1) x x 31. (3m3 n2)(2m3 3n2) 32. (6y4 2z2)(6y4 3z2) 33. (3u 2v 2)(4u2v 6) 34. (5y3w 2 z)(2y3w 2 3z)
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Find each product. Try to write only the answer. See Example 3. 35. (b 4)(b 5) b 36. (y 8)(y 4) 37. (x 3)(x 9) x 38. (m 7)(m 8) 39. (a 5)(a 5) a 40. (t 4)(t 4) t 41. (2x 1)(2x 1) 42. (3y 4)(3y 4) 43. (z 10)(z 10) 44. (3h 5)(3h 5) 45. (a b)(a b) a 46. (x y)(x y) x 47. (a 1)(a 2) a 48. (b 8)(b 1) b 49. (2x 1)(x 3) 50. (3y 5)(y 3) 51. (5t 2)(t 1) 5 52. (2t 3)(2t 1) 53. (h 7)(h 9) h 54. (h 7w)(h 7w)
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63. 64. 65. 66. 67. 68. 69. 70.
3t 42t 2 2
1
1
1
2x 4(3x 1)(2x 5) 4xy3(2x y)(3x y) 24 (x 1)(x 1)(x 3) x (a 3)(a 4)(a 5) a (3x 2)(3x 2)(x 5) (x 6)(9x 4)(9x 4) 81x (x 1)(x 2) (x 3)(x 4) (k 4)(k 9) (k 3)(k 7) k
15
Solve each problem. 71. Area of a rug. Find a trinomial that represents the area of a rectangular rug whose sides are x 3 feet and 2x 1 feet. 2x 5x 3 square feet
x+3
2x – 1
FIGURE FOR EXERCISE 71 72. Area of a parallelogram. Find a trinomial that represents the area of a parallelogram whose base is 3x 2 meters and whose height is 2x 3 meters. 6x 13x 6 square meters 73. Area of a sail. The sail of a tall ship is triangular in shape with a base of 4.57x 3 meters and a height of 2.3x 1.33 meters. Find a polynomial that represents the area of the triangle. 5.2555x 0.41095x 1.995 square meters 74. Area of a square. A square has a side of length 1.732x 1.414 meters. Find a polynomial that represents its area. 2.9998x 4.898 1.999 square meters
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4.4
(4-19)
Special Products
225
GET TING MORE INVOLVED 76. Exploration. Find the area of each of the four regions shown in the figure. What is the total area of the four regions? What does this exercise illustrate?
75. Exploration. Find the area of each of the four regions shown in the figure. What is the total area of the four regions? What does this exercise illustrate? 12 ft h h ft h 7h 12 ft (h 3)(h 4) h 7h 12 h ft
4 ft
a
b
h ft
h ft
b
b
3 ft
3 ft
a
a
h ft
4 ft
a
FIGURE FOR EXERCISE 75
4.4
In this section ●
The Square of a Binomial
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Product of a Sum and a Difference
●
Higher Powers of Binomials
●
Applications to Area
helpful
SPECIAL PRODUCTS
In Section 4.3 you learned the FOIL method to make multiplying binomials simpler. In this section you will learn rules for squaring binomials and for finding the product of a sum and a difference. These products are called special products.
To compute (a b)2, the square of a binomial, we can write it as (a b)(a b) and use FOIL: (a b)2 (a b)(a b) a2 ab ab b2 a2 2ab b2
To visualize the square of a sum, draw a square with sides of length a b as shown. b
a
a2
ab
b
ab
b2
FIGURE FOR EXERCISE 76
The Square of a Binomial
hint
a
b
So to square a b, we square the first term (a2), add twice the product of the two terms (2ab), then add the square of the last term (b2). The square of a binomial occurs so frequently that it is helpful to learn this new rule to find it. The rule for squaring a sum is given symbolically as follows. The Square of a Sum
The area of the large square is (a b)2. It comes from four terms as stated in the rule for the square of a sum.
1
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Using the rule for squaring a sum Find the square of each sum. a) (x 3)2 ▲
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E X A M P L E
(a b)2 a2 2ab b2
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b) (2a 5)2
Textbook Table of Contents
(4-20)
helpful
Chapter 4
Polynomials and Exponents
Solution a) (x 3)2 x2 2(x)(3) 32 x 2 6x 9
hint
↑ Square of first
You can mentally find the square of a number that ends in 5. To find 252 take the 2 (from 25), multiply it by 3 (the next integer) to get 6, and then annex 25 for 625. To find 352, find 3 4 12 and annex 25 for 1225. To see why this works, write (30 5)2 302 2 30 5 52 30(30 2 5) 25 30 40 25 1200 25 1225. 2 Now find 45 and 552 mentally.
226
↑ Twice the product
↑ Square of last
b) (2a 5)2 (2a)2 2(2a)(5) 52 4a2 20a 25
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CAUTION
Do not forget the middle term when squaring a sum. The equation (x 3)2 x2 6x 9 is an identity, but (x 3)2 x2 9 is not an identity. For example, if x 1 in (x 3)2 x2 9, then we get 42 12 9, which is false. When we use FOIL to find (a b)2, we see that (a b)2 (a b)(a b) a2 ab ab b2 a2 2ab b2. So to square a b, we square the first term (a2), subtract twice the product of the two terms (2ab), and add the square of the last term (b2). The rule for squaring a difference is given symbolically as follows. The Square of a Difference
(a b)2 a2 2ab b2
E X A M P L E
helpful
2
Using the rule for squaring a difference Find the square of each difference. a) (x 4)2 b) (4b 5y)2
Solution a) (x 4)2 x2 2(x)(4) 42 x2 8x 16 b) (4b 5y)2 (4b)2 2(4b)(5y) (5y)2 16b2 40by 25y2
hint
Many students keep using FOIL to find the square of a sum or difference. However, learning the new rules for these special cases will pay off in the future.
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Product of a Sum and a Difference If we multiply the sum a b and the difference a b by using FOIL, we get (a b)(a b) a2 ab ab b2 a2 b2.
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The inner and outer products have a sum of 0. So the product of a sum and a difference of the same two terms is equal to the difference of two squares.
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4.4
Special Products
(4-21)
227
The Product of a Sum and a Difference
(a b)(a b) a2 b2
E X A M P L E
helpful
3
Product of a sum and a difference Find each product. a) (x 2)(x 2) b) (b 7)(b 7)
Solution a) (x 2)(x 2) x2 4 b) (b 7)(b 7) b2 49 c) (3x 5)(3x 5) 9x2 25
hint
You can use (a b)(a b) a2 b2 to perform mental arithmetic tricks like 19 21 (20 1)(20 1) 400 1 399 What is 29 31? 28 32?
E X A M P L E
study
c) (3x 5)(3x 5)
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Higher Powers of Binomials To find a power of a binomial that is higher than 2, we can use the rule for squaring a binomial along with the method of multiplying binomials using the distributive property. Finding the second or higher power of a binomial is called expanding the binomial because the result has more terms than the original.
4
Higher powers of a binomial Expand each binomial. a) (x 4)3
b) (y 2)4
Solution a) (x 4)3 (x 4)2(x 4) (x 2 8x 16)(x 4) (x 2 8x 16)x (x 2 8x 16)4 x3 8x 2 16x 4x2 32x 64 x3 12x2 48x 64 b) (y 2)4 (y 2)2(y 2)2 (y2 4y 4)(y2 4y 4) (y2 4y 4)(y2) (y2 4y 4)(4y) (y2 4y 4)(4) y4 4y3 4y2 4y3 16y2 16y 4y2 16y 16 ■ y4 8y3 24y2 32y 16
tip
Correct answers often have more than one form. If your answer to an exercise doesn’t agree with the one in the back of this text, try to determine if it is simply a different form of the answer. For example, 1x 2
and x look different but they 2
are equivalent expressions.
Applications to Area E X A M P L E
5
Area of a pizza A pizza parlor saves money by making all of its round pizzas one inch smaller in radius than advertised. Write a trinomial for the actual area of a pizza with an advertised radius of r inches.
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Solution A pizza advertised as r inches has an actual radius of r 1 inches. The actual area is (r 1)2: (r 1)2 (r 2 2r 1) r 2 2r . ■ So r 2 2r is a trinomial representing the actual area.
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228
(4-22)
Chapter 4
Polynomials and Exponents
WARM-UPS
True or false? Explain your answer.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
4. 4
(2 3)2 22 32 False (x 3)2 x 2 6x 9 for any value of x. True (3 5)2 9 25 30 True (2x 7)2 4x 2 28x 49 for any value of x. True (y 8)2 y2 64 for any value of y. False The product of a sum and a difference of the same two terms is equal to the difference of two squares. True (40 1)(40 1) 1599 True 49 51 2499 True (x 3)2 x2 3x 9 for any value of x. False The square of a sum is equal to a sum of two squares. False
EXERCISES
Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What are the special products?
2. What is the rule for squaring a sum? 3. Why do we need a new rule to find the square of a sum when we already have FOIL? It is faster to do by the new rule than with FOIL. 4. What happens to the inner and outer products in the product of a sum and a difference? In (a b)( ) the inner and outer products have a sum of zero. 5. What is the rule for finding the product of a sum and a difference? 6. How can you find higher powers of binomials? Higher powers of binomials are found by using the distributive property.
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Square each binomial. See Example 1. 7. (x 1)2 8. (y 2)2 x 2x 1 y 4 4 10. (z 3)2 9. (y 4)2 y 8 z 6z 9 12. (2m 7)2 11. (3x 8)2 9x 48 64 4m 28m 14. (x z)2 13. (s t)2 s 2st x 2xz 15. (2x y)2 16. (3t v)2 4x 4xy 9t 6tv 18. (3z 5k)2 17. (2t 3h)2 4t 12 9z 30kz
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Square each binomial. See Example 2. 19. (a 3)2 20. (w 4)2 a 6a 9 w 8w 16 2 21. (t 1) 22. (t 6)2 t t 1 t t 36 2 24. (5a 6)2 23. (3t 2) 9t 12t 4 25a 60a 36 2 2 26. (r w) 25. (s t) s 2st r rw 2 27. (3a b) 28. (4w 7)2 9a 6ab 16w 56w 2 29. (3z 5y) 30. (2z 3w)2 9z 30yz 4z 12wz Find each product. See Example 3. 31. (a 5)(a 5) 32. (x 6)(x 6) a 25 x 36 33. (y 1)(y 1) 34. (p 2)(p 2) y 1 p 4 35. (3x 8)(3x 8) 36. (6x 1)(6x 1) 9x 36x 1 37. (r s)(r s) 38. (b y)(b y) r s b y 39. (8y 3a)(8y 3a) 40. (4u 9v)(4u 9v) 64y 9a 16u 81v 42. (3y2 1)(3y2 1) 41. (5x 2 2)(5x 2 2) 25x 4 9y 1 Expand each binomial. See Example 4. 43. (x 1)3 44. (y 1)3 45. (2a 3)3 46. (3w 1)3 47. (a 3)4
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48. (2b 1)4 16 49. (a b)4 a 50. (2a 3b)4 Find each product. 51. (a 20)(a 20) a 400 53. (x 8)(x 7) x 15x 56 55. (4x 1)(4x 1) 16x 1 57. (9y 1)2 81y 18y 1 59. (2t 5)(3t 4) 6t 7t 61. (2t 5)2 4t 20t 25 63. (2t 5)(2t 5) 4t 25 2 65. (x 1)(x 2 1) x 1 3 67. (2y 9)2 4y 36y 81 3 2 2 69. (2x 3y ) 4x 12x 9y
1 1 71. x 2 3
t + 3 cm
2
v k(R r)(R r), where k is a constant. Rewrite the formula using a special product rule. v k(R r
R
In Exercises 81–90, solve each problem. 81. Shrinking garden. Rose’s garden is a square with sides of length x feet. Next spring she plans to make it rectangular by lengthening one side 5 feet and shortening the other side by 5 feet. What polynomial represents the new area? By how much will the area of the new garden differ from that of the old garden? x 25 square feet, 25 square feet smaller
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85. Poiseuille’s law. According to the nineteenth-century physician Poiseuille, the velocity (in centimeters per second) of blood r centimeters from the center of an artery of radius R centimeters is given by
73. (0.2x 0.1)2 0.04x 0.04x 0.01 74. (0.1y 0.5)2 0.01y 0.1y 0.25 75. (a b)3 a a 3 76. (2a 3b)3 8a 36a 77. (1.5x 3.8)2 2.25x 11.4x 14.44 78. (3.45a 2.3)2 11.9025a 15.87a 5.29 79. (3.5t 2.5)(3.5t 2.5) 12.25t 6.25 80. (4.5h 5.7)(4.5h 5.7) 20.25h 32.49
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FIGURE FOR EXERCISE 84
2 1 72. y 3 2
(4-23)
82. Square lot. Sam lives on a lot that he thought was a square, 157 feet by 157 feet. When he had it surveyed, he discovered that one side was actually 2 feet longer than he thought and the other was actually 2 feet shorter than he thought. How much less area does he have than he thought he had? 4 square feet 83. Area of a circle. Find a polynomial that represents the area of a circle whose radius is b 1 meters. Use the value 3.14 for . 3.14b 6.28b 3.14 square meters 84. Comparing dart boards. A toy store sells two sizes of circular dartboards. The larger of the two has a radius that is 3 inches greater than that of the other. The radius of the smaller dartboard is t inches. Find a polynomial that represents the difference in area between the two dartboards. 6t 9 square centimeters
52. (1 x)(1 x) 1 x 54. (x 9)(x 5) x x 45 56. (9y 1)(9y 1) 81y 1 58. (4x 1)2 16x 8x 1 60. (2t 5)(3t 4) 6t t 20 62. (2t 5)2 4t 20t 25 64. (3t 4)(3t 4) 9t 16 3 66. (y 1)(y3 1) y 1 4 68. (3z 8)2 9z 48z 64 5 3 2 70. (4y 2w )
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FIGURE FOR EXERCISE 85 86. Going in circles. A promoter is planning a circular race track with an inside radius of r feet and a width of w feet.
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89. Investing in treasury bills. An investment advisor uses the polynomial P(1 r)10 to predict the value in 10 years of a client’s investment of P dollars with an average annual return r. The accompanying graph shows historic average annual returns for the last 20 years for various asset classes (T. Rowe Price, www.troweprice.com). Use the historical average return to predict the value in 10 years of an investment of $10,000 in U.S. treasury bills? $20,230.06
w r
FIGURE FOR EXERCISE 86 The cost in dollars for paving the track is given by the formula C 1.2[(r w)2 r 2]. Use a special product rule to simplify this formula. What is the cost of paving the track if the inside radius is 1000 feet and the width of the track is 40 feet? C 1.2 [2rw w 87. Compounded annually. P dollars is invested at annual interest rate r for 2 years. If the interest is compounded annually, then the polynomial P(1 r)2 represents the value of the investment after 2 years. Rewrite this expression without parentheses. Evaluate the polynomial if P $200 and r 10%. Pr Pr 2 88. Compounded semiannually. P dollars is invested at annual interest rate r for 1 year. If the interest is compounded 2
semiannually, then the polynomial P1 represents r 2
the value of the investment after 1 year. Rewrite this expression without parentheses. Evaluate the polynomial if P $200 and r 10%. Pr P Pr
4.5
Average annual return (percent)
230
20 16
16.7%
12
10.3% 7.3%
8
3.4%
4 0
Large Long-term U.S. Inflation company corporate treasury stocks bonds bills
FIGURE FOR EXERCISES 89 AND 90 90. Comparing investments. How much more would the investment in Exercise 89 be worth in 10 years if the client invests in large company stocks rather than U.S. treasury bills? $26,619.83
GET TING MORE INVOLVED 91. Writing. What is the difference between the equations (x 5)2 x 2 10x 25 and (x 5)2 x2 25? The first is an identity and the second is a conditional equation. 92. Writing. Is it possible to square a sum or a difference without using the rules presented in this section? Why should you learn the rules given in this section? A sum or difference can be squared with the distributive property, FOIL, or the special product rules. It is easier with the special product rules.
DIVISION OF POLYNOMIALS
You multiplied polynomials in Section 4.2. In this section you will learn to divide polynomials.
In this section ●
Dividing Monomials Using the Quotient Rule
Dividing Monomials Using the Quotient Rule
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Dividing a Polynomial by a Monomial
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Dividing a Polynomial by a Binomial
In Chapter 1 we used the definition of division to divide signed numbers. Because the definition of division applies to any division, we restate it here. Division of Real Numbers
If a, b, and c are any numbers with b 0, then
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Division of Polynomials
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If a b c, we call a the dividend, b the divisor, and c (or a b) the quotient. You can find the quotient of two monomials by writing the quotient as a fraction and then reducing the fraction. For example,
tip
Establish a regular routine of eating, sleeping, and exercise. The ability to concentrate depends on adequate sleep, decent nutrition,and the physical well-being that comes with exercise.
x5 x x x x x x 5 x 2 2 x 3. x x x You can be sure that x 3 is correct by checking that x 3 x 2 x 5. You can also divide x 2 by x 5, but the result is not a monomial: 1 x2 1 x x x 2 x 5 5 3 x x x x x x x Note that the exponent 3 can be obtained in either case by subtracting 5 and 2. These examples illustrate the quotient rule for exponents. Quotient Rule
Suppose a 0, and m and n are positive integers. am If m n, then n a mn. a am 1 If n m, then n . a a nm Note that if you use the quotient rule to subtract the exponents in x 4 x 4, you get the expression x 44, or x 0, which has not been defined yet. Because we must have x 4 x 4 1 if x 0, we define the zero power of a nonzero real number to be 1. We do not define the expression 00. Zero Exponent
For any nonzero real number a, a0 1.
E X A M P L E
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Using the definition of zero exponent Simplify each expression. Assume that all variables are nonzero real numbers. b) (3xy)0 c) a0 b0 a) 50
Solution a) 50 1
b) (3xy)0 1
c) a0 b0 1 1 2
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With the definition of zero exponent the quotient rule is valid for all positive integers as stated.
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Using the quotient rule in dividing monomials Find each quotient. y9 12b2 a) 5 b) c) 6x3 (2x9) y 3b7 ▲
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x8y2 d) x2y2
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helpful
Polynomials and Exponents
Solution y9 a) 5 y95 y4 y Use the definition of division to check that y4 y5 y9. 12b2 12 b2 1 4 b) 7 7 4 72 5 3b b b 3 b Use the definition of division to check that 4 12b7 12b2. 5 3b7 b b5
hint
Recall that the order of operations gives multiplication and division an equal ranking and says to do them in order from left to right. So without parentheses, 6x 3 2x 9 actually means 6x 3 9 x. 2
6x 3 3 c) 6x 3 (2x 9) 2x 9 x6 Use the definition of division to check that 3 6x 9 9 6x 3. 2x x6 x6 x8y2 x 8 y2 d) x 6 y0 x 6 x 2 y2 x 2 y2 Use the definition of division to check that x 6 x 2 y 2 x 8y2.
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We showed more steps in Example 2 than are necessary. For division problems like these you should try to write down only the quotient.
Dividing a Polynomial by a Monomial We divided some simple polynomials by monomials in Chapter 1. For example, 6x 8 1 (6x 8) 3x 4. 2 2 We use the distributive property to take one-half of 6x and one-half of 8 to get 3x 4. So both 6x and 8 are divided by 2. To divide any polynomial by a monomial, we divide each term of the polynomial by the monomial.
E X A M P L E
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Dividing a polynomial by a monomial Find the quotient for (8x6 12x4 4x 2) (4x 2).
tip
Solution 8x6 12x 4 4x 2 8x6 12x 4 4x 2 2 4x 2 4x 2 4x 2 4x 4 2 2x 3x 1 The quotient is 2x 4 3x 2 1. We can check by multiplying. 4x 2(2x 4 3x 2 1) 8x 6 12x 4 4x 2.
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Because division by zero is undefined, we will always assume that the divisor is nonzero in any quotient involving variables. For example, the division in Example 3 is valid only if 4x2 0, or x 0.
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Play offensive math,not defensive math. A student who says, “Give me a question and I’ll see if I can answer it,” is playing defensive math. The student is taking a passive approach to learning. A student who takes an active approach and knows the usual questions and answers for each topic is playing offensive math.
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Division of Polynomials
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233
Dividing a Polynomial by a Binomial Division of whole numbers is often done with a procedure called long division. For example, 253 is divided by 7 as follows: 36 ← Quotient Divisor → 72 53 ← Dividend 21 43 42 1 ← Remainder Note that 36 7 1 253. It is always true that (quotient)(divisor) (remainder) dividend. To divide a polynomial by a binomial, we perform the division like long division of whole numbers. For example, to divide x2 3x 10 by x 2, we get the first term of the quotient by dividing the first term of x 2 into the first term of x 2 3x 10. So divide x 2 by x to get x, then multiply and subtract as follows: 1 Divide: 2 Multiply: 3
Subtract:
x x 2x2 x 3 0 1 x2 2x 5x
x2 x x x (x 2) x2 2x 3x 2x 5x
Now bring down 10 and continue the process. We get the second term of the quotient (below) by dividing the first term of x 2 into the first term of 5x 10. So divide 5x by x to get 5: 1 Divide: 2 Multiply:
3
Subtract:
x5 x 2x2 x 3 0 1 x2 2x ↓ 5x 10 5x 10 0
5x x 5 Bring down 10. 5(x 2) 5x 10 10 (10) 0
So the quotient is x 5, and the remainder is 0. In the next example we must rearrange the dividend before dividing.
E X A M P L E
helpful
4
Dividing a polynomial by a binomial Divide 2x 3 4 7x 2 by 2x 3, and identify the quotient and the remainder.
Solution Rearrange the dividend as 2x3 7x2 4. Because the x-term in the dividend is missing, we write 0 x for it:
hint
x2 2x 3 2x 32 x 3 x 72 0x 4 3 2x 3x2 4x 2 0 x 4x 2 6x 6x 4 6x 9 13
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2x 3 (2x) x 2
7x 2 (3x2) 4x 2 0 x 6x 6x 4 (9) 13
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The quotient is x2 2x 3, and the remainder is 13. Note that the degree of the remainder is 0 and the degree of the divisor is 1. To check, we must verify that (2x 3)(x 2 2x 3) 13 2x3 7x2 4.
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CAUTION To avoid errors, always write the terms of the divisor and the dividend in descending order of the exponents and insert a zero for any term that is missing.
If we divide both sides of the equation dividend (quotient)(divisor) (remainder) by the divisor, we get the equation dividend remainder quotient . divisor divisor This fact is used in expressing improper fractions as mixed numbers. For example, if 19 is divided by 5, the quotient is 3 and the remainder is 4. So 4 19 4 3 3. 5 5 5 We can also use this form to rewrite algebraic fractions.
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Rewriting algebraic fractions 3x Express in the form x2
remainder quotient . divisor
Solution Use long division to get the quotient and remainder: 3 3x 0 x 2 3x 6 6 Because the quotient is 3 and the remainder is 6, we can write 6 3x 3 . x2 x2
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To check, we must verify that 3(x 2) 6 3x.
CAUTION When dividing polynomials by long division, we do not stop until the remainder is 0 or the degree of the remainder is smaller than the degree of the divisor. For example, we stop dividing in Example 5 because the degree of the remainder 6 is 0 and the degree of the divisor x 2 is 1.
WARM-UPS
True or false? Explain your answer.
1. y10 y2 y5 for any nonzero value of y. False 2.
7x 2 7
3.
7x 2 7
x 2 for any value of x. False
x 2 for any value of x. True
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4. If 3x2 6 is divided by 3, the quotient is x2 6.
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False
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WARM-UPS
Division of Polynomials
(4-29)
235
(continued)
5. If 4y2 6y is divided by 2y, the quotient is 2y 3. True 6. The quotient times the remainder plus the dividend equals the divisor. False 7. (x 2)(x 1) 3 x2 3x 5 for any value of x. True 8. If x2 3x 5 is divided by x 2, then the quotient is x 1. True 9. If x2 3x 5 is divided by x 2, the remainder is 3. True 10. If the remainder is zero, then (divisor)(quotient) dividend. True
@ http://www.sosmath.com/algebra/factor/fac01/fac01.html 4. 5
EXERCISES 9x 2y2 25. 3x 5y2
Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What rule is important for dividing monomials? The quotient rule is used for dividing monomials. 2. What is the meaning of a zero exponent? The zero power of a nonzero real number is 1. 3. How many terms should you get when dividing a polynomial by a monomial? When dividing a polynomial by a monomial the quotient should have the same number of terms as the polynomial. 4. How should the terms of the polynomials be written when dividing with long division? The terms of a ploynomial should be written in descending order of the exponents. 5. How do you know when to stop the process in long division of polynomials? The long division process stops when the degree of the remainder is less than the degree of the divisor. 6. How do you handle missing terms in the dividend polynomial when doing long division? Insert a term with zero coefficient for each missing term when doing long division. Simplify each expression. See Example 1. 7. 90 1 8. m0 1 9. (2x3)0 1 3 0 0 0 10. (5a b) 1 11. 2 5 3 1 12. 4 0 8 0 2 0 13. (2x y) 1 14. (a2 b2)0 1
Find the quotients. See Example 3. 3x 6 27. x 2 3 5y 10 28. 2 5 x5 3x4 x3 x 3 29. x2 6y6 9y4 12y2 30. 2y 3 3y2 8x2y2 4x 2y 2xy2 31. 4 2xy 9ab2 6a3b3 32. 3 a 3ab2 33. (x2y3 3x3y2) (x2y) y 34. (4h5k 6h2k2) (2h2k) Find the quotient and remainder for each division. Check by using the fact that dividend (divisor)(quotient) remainder. See Example 4. 35. (x2 5x 13) (x 3) x 2, 7 36. (x2 3x 6) (x 3) x, 6 37. (2x) (x 5) 2, 10 38. (5x) (x 1) 5, 5 39. (a3 4a 3) (a 2) a 2a 8, 13 40. (w3 2w2 3) (w 2) w 4w 8, 13 41. (x2 3x) (x 1) x 4, 4 42. (3x2) (x 1) 3x 3, 3 43. (h3 27) (h 3) h 3h 9, 0 44. (w3 1) (w 1) w w 1, 0 45. (6x2 13x 7) (3x 2) 2x 3, 1 46. (4b2 25b 3) (4b 1) b 6, 9 47. (x3 x2 x 2) (x 1) x 1, 1 48. (a3 3a2 4a 4) (a 2) a 2, 0
Find each quotient. Try to write only the answer. See Example 2. x8 15. 2 x x 6a7 17. 2a12
y9 16. 3 y y 30b2 1 18. 3b6
19. 12x5 (3x9)
4 x
20. 6y5 (3y10)
21. 6y2 (6y)
y
22. 3a2b (3ab) 4h2k4 24. 3 2hk
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Write each expression in the form remainder quotient . divisor See Example 5. 2x 3x 49. 50. x5 x1 x 3x 51. 52. x3 x1 a5 x1 53. 1 54. 1 x a 2y 1 3x 1 55. 56. x y x2 x2 57. x 58. x x1 x1 x2 4 59. x2 x2 1 60. x1 x3 61. x2 x3 1 62. x 1 x1 x3 3 2x2 4 63. 64. x x 2x Find each quotient. 65. 6a3b (2a2b)
3a
Solve each problem. 85. Area of a rectangle. The area of a rectangular billboard is x 2 x 30 square meters. If the length is x 6 meters, find a binomial that represents the width. x 5 meters
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GAS FOR LESS NEXT EXIT TEXACO x 6 meters
FIGURE FOR EXERCISE 85 86. Perimeter of a rectangle. The perimeter of a rectangular backyard is 6x 6 yards. If the width is x yards, find a binomial that represents the length. 2x 3 yards
66. 14x7 (7x2) 2x
x yards
67. 8w4t7 (2w9t3) 68. 9y7z4 (3y3z11)
?
(3a 12) (3) (6z 3z 2) (3z) 2 z (3x 2 9x) (3x) x 3 (5x 3 15x 2 25x) (5x) x 3x 5 4 3 2 2 (12x 4x 6x ) (2x ) x 2x 3 (9x 3 3x 2 15x) (3x) 3x 5 (t 2 5t 36) (t 9) t 4 (b2 2b 35) (b 5) b 7 (6w2 7w 5) (3w 5) 2w 1 (4z2 23z 6) (4z 1) z 6 (8x 3 27) (2x 3) 4x 6x 9 (8y 3 1) (2y 1) 4y 2y 1 (t 3 3t 2 5t 6) (t 2) t t 3 (2u3 13u2 8u 7) (u 7) 2u u 1 (6v2 4 9v v 3) (v 4) (14y 8y2 y3 12) (6 y)
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FIGURE FOR EXERCISE 86 GET TING MORE INVOLVED 87. Exploration. Divide x3 1 by x 1, x4 1 by x 1, and x5 1 by x 1. What is the quotient when x9 1 is divided by x 1? 88. Exploration. Divide a3 b3 by a b and a4 b4 by a b. What is the quotient when a8 b8 is divided by a b? x 89. Discussion. Are the expressions 150 , 10x 5x, and (10x) x (5x) equivalent? Before you answer, review the order of operations in Section 1.5 and evaluate each expression for x 3. 10x 5x is not equivalent to the other two.
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4.6 In this section
Positive Integral Exponents
237
POSITIVE INTEGRAL EXPONENTS
The product rule for positive integral exponents was presented in Section 4.2, and the quotient rule was presented in Section 4.5. In this section we review those rules and then further investigate the properties of exponents.
The Product and Quotient Rules
The Product and Quotient Rules
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Raising an Exponential Expression to a Power
The rules that we have already discussed are summarized below.
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Power of a Product
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Power of a Quotient
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Summary of Rules
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(4-31)
The following rules hold for nonnegative integers m and n and a 0. a m a n a mn am n a mn if m n a am 1 if n m n nm a a a0 1
Product rule Quotient rule
Zero exponent
CAUTION The product and quotient rules apply only if the bases of the expressions are identical. For example, 32 34 36, but the product rule cannot be applied to 52 34. Note also that the bases are not multiplied: 32 34 96.
Note that in the quotient rule the exponents are always subtracted, as in x7 x4 x3
and
y5 1 8 3. y y
If the larger exponent is in the denominator, then the result is placed in the denominator.
E X A M P L E
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Using the product and quotient rules Use the rules of exponents to simplify each expression. Assume that all variables represent nonzero real numbers. b) (3x)0(5x2)(4x) a) 23 22 2 8x (3a2b)b9 c) 5 d) 2x (6a5)a3b2
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Solution a) Because the bases are both 2, we can use the product rule: Product rule 23 22 25 32 Simplify. 0 2 2 b) (3x) (5x )(4x) 1 5x 4x Definition of zero exponent 20x 3 Product rule 2 8x 4 c) 5 3 Quotient rule 2x x
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d) First use the product rule to simplify the numerator and denominator:
study
tip
(3a2b)b9 3a2b10 5 3 2 (6a )a b 6a8b2
Keep track of your time for one entire week. Account for how you spend every half hour. Add up your totals for sleep, study, work, and recreation. You should be sleeping 50–60 hours per week and studying 1–2 hours for every hour you spend in the classroom.
b8 6 2a
Product rule
Quotient rule
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Raising an Exponential Expression to a Power When we raise an exponential expression to a power, we can use the product rule to find the result, as shown in the following example: (w4)3 w4 w4 w4 w12
Three factors of w4 because of the exponent 3 Product rule
By the product rule we add the three 4’s to get 12, but 12 is also the product of 4 and 3. This example illustrates the power rule for exponents. Power Rule
If m and n are nonnegative integers and a 0, then (a m)n a mn. In the next example we use the new rule along with the other rules.
2
Using the power rule Use the rules of exponents to simplify each expression. Assume that all variables represent nonzero real numbers. (23)4 27 3(x 5)4 b) c) a) 3x 2(x 3)5 5 9 2 2 15x 22
Solution a) 3x 2(x 3)5 3x 2x15 3x17 3 4 7 212 27 (2 ) 2 b) 214 25 2 9 219 214 25 32 5 4 3(x ) 3x 20 1 c) 22 15x 15x 22 5x 2
Power rule Product rule Power rule and product rule
Product rule Quotient rule Evaluate 25.
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Power of a Product Consider an example of raising a monomial to a power. We will use known rules to rewrite the expression. (2x)3 2x 2x 2x 222xxx 23x3
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Definition of exponent 3 Commutative and associative properties Definition of exponents
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Positive Integral Exponents
239
Note that the power 3 is applied to each factor of the product. This example illustrates the power of a product rule. Power of a Product Rule
If a and b are real numbers and n is a positive integer, then (ab)n a nb n.
E X A M P L E
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Using the power of a product rule Simplify. Assume that the variables are nonzero. b) (3m)3 a) (xy3)5
c) (2x 3y 2z7)3
Solution a) (xy 3)5 x 5(y 3)5 Power of a product rule x 5y15 Power rule 3 3 3 b) (3m) (3) m Power of a product rule 27m3 (3)(3)(3) 27 c) (2x 3y 2z7)3 23(x 3)3(y 2)3(z7)3 8x9y6z 21
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Power of a Quotient
helpful
Raising a quotient to a power is similar to raising a product to a power: x 3 x x x Definition of exponent 3 5 5 5 5 xxx Definition of multiplication of fractions 555 x3 3 Definition of exponents 5
hint
Note that these rules of exponents are not absolutely necessary. We could simplify every expression here by using only the definition of exponent. However, these rules make it a lot simpler.
The power is applied to both the numerator and denominator. This example illustrates the power of a quotient rule. Power of a Quotient Rule
If a and b are real numbers, b 0, and n is a positive integer, then
a b
E X A M P L E
4
Solution 2 2 22 a) 3 5x (5x 3)2 4 6 25x
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an n. b
Using the power of a quotient rule Simplify. Assume that the variables are nonzero. 2 2 3x4 3 a) 3 b) 3 5x 2y
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12a5b c) 4a2b7
Power of a quotient rule
(5x 3)2 52(x 3)2 25x 6
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240
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Chapter 4
Polynomials and Exponents
3x4 3 2y
33x12 23y9 27x12 8y 9
b)
3
Power of a quotient and power of a product rules
Simplify.
c) Use the quotient rule to simplify the expression inside the parentheses before using the power of a quotient rule.
12a5b 4a2b7
3a3 b6
3
27a9 b18
3
Use the quotient rule first.
Power of a quotient rule
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Summary of Rules
helpful
The rules for exponents are summarized in the following box.
hint
Note that the rules of exponents show how exponents behave with respect to multiplication and division only. We studied the more complicated problem of using exponents with addition and subtraction in Section 4.4 when we learned rules for (a b)2 and (a b)2.
Rules for Nonnegative Integral Exponents
The following rules hold for nonzero real numbers a and b and nonnegative integers m and n. 1. a0 1 Definition of zero exponent 2. am an amn Product rule am 3. n amn for m n, a am 1 for n m n a anm
Quotient rule
4. (am)n amn Power rule 5. (ab)n an bn Power of a product rule
n
a 6. b
an n b
Power of a quotient rule
@ http://www.cne.gmu.edu/modules/dau/algebra/exponents/exp1_frm.html WARM-UPS
True or false? Assume that all variables represent nonzero real numbers. A statement involving variables is to be marked true only if it is an identity. Explain your answer.
1. 30 1 False 4. (2x)4 2x4 False 7. (ab3)4 a4b12 2y3 10. 9
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4y6 81
2. 25 28 413 False 5. (q3)5 q8 False a12 True 8. 4 a3 False a
3. 23 33 65 False 6. (3x 2)3 27x6 False 6w4 9. 9 2w5 False 3w
True
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4. 6
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Positive Integral Exponents
241
EXERCISES
Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.
Simplify. See Example 4.
2a 16a 37. b 2x y 39. 4y 6x y z 4 41. 3x y z 35.
1. What is the product rule for exponents?
9r 38. t 3y 40. 2zy 10rs t 42. 2rs t
3
x4 4
36.
2 4
2. What is the quotient rule for exponents?
3 2
3
n and
5
3
2
45. (3 6)2 49. (2 3)
For all exercises in this section, assume that the variables represent nonzero real numbers. Simplify the exponential expressions. See Example 1.
51.
8. x6 x7 x 10. (3r4)(6r 2)
r
2a3 13. 4a7 2a 5b 3a7b3 15. 15a6b8
0
3t9 14. 6t18
18. 2 10 2
9
19. (x )
17. 23 52 200
x
20. (y )
2 5 2
1
58.
61. (5x4)3
62. (4z3)3
125x
5 6
27. (xy2)3 x 29. (2t5)3
28. (wy2)6 w 30. (3r3)3
31. (2x2y5)3
32. (3y2z3)3
(a4b2c5)3 33. a3b4c
(2ab2c3)5 34. 34 (2a bc)
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64z
64. 2a4b5 2a9b2 20a5b13 66. 4 5a b13
2x x4
4 7
27y
3 16x
4
68. (ab)3(3ba2)4
9y 10a c 72. 5a b 5x yz 74. 5x yz 70.
3y8 y5
2
5
5
Simplify. See Example 3.
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7
3 4 5
5y3(y5)2 26. 5 10y (y2)6
1
2
60. 2y3(3y)
8a b 71. 4c 8x y 73. 16x y
(r4)2 24. 53 (r )
3 3 3
Simplify each expression. 59. 3x4 5x7 15x
69.
22. (y2)6 3y5
2x
(t 2)5 23. 34 (t ) 3x(x5)2 25. 6x3(x 2)4
56. 102 104 1,000,000
2 3
y
3
55. 23 24 128
67. (xt2)(2x2t)4 2 4
4
343
3
54. 103 33 27,000
9u4v9 65. 3u5v8
3
21. 2x 2 (x2)5
52.
63. 3y z 9yz
Simplify. See Example 2. 2 3
3
5 12
3xy 5xy 16. 20x3y14 8
2
50. (3 4)3
53. 52 23 200
57.
12. x y x y (x y) 3 6
5
48. 33 43 91
1
3
3 2
11. a3b4 ab6(ab)0 a 2
46. 52 32 16
81
47. 23 33 19
The power of a quotient rule says that
6
2 7
Simplify each expression. Your answer should be an integer or a fraction. Do not use a calculator. 43. 32 62 45 44. (5 3)2 4
6. What is the power of a quotient rule?
9. (3u8)(2u2)
9 4 3
6 4 3
5. What is the power of a product rule? The power of a product rule says that (
7. 22 25 128
2
2 4 9 2
4. What is the power rule for exponents?
4
8
2
3. Why must the bases be the same in these rules? These rules do not make sense without identical bases.
3
y2 2
5
5 4
5
2
3
2
32c b
3 5
z
Solve each problem. 75. Long-term investing. Sheila invested P dollars at annual rate r for 10 years. At the end of 10 years her investment was worth P(1 r)10 dollars. She then reinvested this money for another 5 years at annual rate r. At the end of the second time period her investment was worth
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Polynomials and Exponents
P(1 r)10 (1 r)5 dollars. Which law of exponents can be used to simplify the last expression? Simplify it. P(1 r) 76. CD rollover. Ronnie invested P dollars in a 2-year CD with an annual rate of return of r. After the CD rolled over two times, its value was P((1 r)2)3. Which law of exponents can be used to simplify the expression? Simplify it. P(1 r)
4.7
GET TING MORE INVOLVED 77. Writing. When we square a product, we square each factor in the product. For example, (3b)2 9b2. Explain why we cannot square a sum by simply squaring each term of the sum. 78. Writing. Explain why we define 20 to be 1. Explain why 20 1.
NEGATIVE EXPONENTS AND SCIENTIFIC NOTATION
We defined exponential expressions with positive integral exponents in Chapter 1 and learned the rules for positive integral exponents in Section 4.6. In this section you will first study negative exponents and then see how positive and negative integral exponents are used in scientific notation.
In this section ●
Negative Integral Exponents
●
Rules for Integral Exponents
Negative Integral Exponents
●
Converting from Scientific Notation
●
Converting to Scientific Notation
If x is nonzero, the reciprocal of x is written as 1. For example, the reciprocal of 23 is x written as 13. To write the reciprocal of an exponential expression in a simpler way,
●
Computations with Scientific Notation
2
1
we use a negative exponent. So 23 23. In general we have the following definition. Negative Integral Exponents
If a is a nonzero real number and n is a positive integer, then 1 an . (If n is positive, n is negative.) an
E X A M P L E
1
Simplifying expressions with negative exponents Simplify. a) 25
b) (2)5
2 3 c) 2 3
Solution
calculator
1 1 a) 25 5 2 32 1 b) (2)5 5 Definition of negative exponent (2) 1 1 32 32
close-up You can evaluate expressions with negative exponents on a calculator as shown here.
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23 c) 2 23 32 3 1 1 3 2 2 3 1 1 1 9 9 8 9 8 1 8
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Negative Exponents and Scientific Notation
243
In simplifying 52, the negative sign preceding the 5 is used after 5 is squared and the reciprocal is found. So 52 (52) 215. CAUTION
To evaluate an, you can first find the nth power of a and then find the reciprocal. However, the result is the same if you first find the reciprocal of a and then find the nth power of the reciprocal. For example, 1 2 1 1 1 1 1 or 32 . 32 2 3 9 3 3 9 3
helpful
So the power and the reciprocal can be found in either order. If the exponent is 1, we simply find the reciprocal. For example,
hint
Just because the exponent is negative, it doesn’t mean the expression is negative.
1 51 , 5
1
4,
3 5
1
and
5 . 3
Because 32 32 1, the reciprocal of 32 is 32, and we have 1 2 32. 3
1
Note that (2)3 while (2)4 .
1
1 4
8
16
These examples illustrate the following rules. Rules for Negative Exponents
If a is a nonzero real number and n is a positive integer, then 1 n an , a
1 a1 , a
E X A M P L E
2
1 n an, and a
a b
3
b) 101 101
Solution 3 3 4 a) 4 3
2 c) 103
3
64 27 1 1 2 1 b) 101 101 10 10 10 5 2 1 c) 2 2 103 2 1000 2000 103 103
calculator close-up You can use a calculator to demonstrate that the product rule for exponents holds when the exponents are negative numbers.
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Rules for Integral Exponents Negative exponents are used to make expressions involving reciprocals simpler looking and easier to write. Negative exponents have the added benefit of working in conjunction with all of the rules of exponents that you learned in Section 4.6. For example, we can use the product rule to get x2 x3 x2(3) x5 and the quotient rule to get y3 5 y35 y2. y ▲
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b n . a
Using the rules for negative exponents Simplify. 3 a) 4
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Polynomials and Exponents
With negative exponents there is no need to state the quotient rule in two parts as we did in Section 4.6. It can be stated simply as am n amn a for any integers m and n. We list the rules of exponents here for easy reference.
helpful
Rules for Integral Exponents
hint
The following rules hold for nonzero real numbers a and b and any integers m and n. Definition of zero exponent 1. a0 1 m n mn Product rule 2. a a a m a 3. n amn Quotient rule a Power rule 4. (am)n amn
The definitions of the different types of exponents are a really clever mathematical invention. The fact that we have rules for performing arithmetic with those exponents makes the notation of exponents even more amazing.
5. (ab)n an bn n a an 6. n b b
E X A M P L E
3
Power of a product rule Power of a quotient rule
The product and quotient rules for integral exponents Simplify. Write your answers without negative exponents. Assume that the variables represent nonzero real numbers. a) b3b5 m6 c) 2 m
helpful
b) 3x3 5x 2 4y5 d) 12y3
Solution a) b3b5 b35 b2
hint
Example 3(c) could be done using the rules for negative exponents and the old quotient rule: 1 m6 m2 m2 m6 m4 It is always good to look at alternative methods.The more tools in your toolbox the better.
3
b) 3x
Product rule Simplify.
5x 15x1 15 x 2
m6 c) 2 m6(2) m m4 1 4 m
Product rule Definition of negative exponent
Quotient rule Simplify. Definition of negative exponent
4y5 y 5(3) y8 d) 12y3 3 3
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In the next example we use the power rules with negative exponents.
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4.7
E X A M P L E
4
(4-39)
245
The power rules for integral exponents Simplify each expression. Write your answers with positive exponents only. Assume that all variables represent nonzero real numbers. 4x5 2 b) (10x3)2 c) a) (a3)2 y2 Solution a) (a3)2 a3 2 Power rule a6 1 6 Definition of negative exponent a b) (10x3)2 102(x3)2 Power of a product rule 102x(3)(2) Power rule x6 2 Definition of negative exponent 10 x6 100 5 2 (4x5)2 4x c) Power of a quotient rule 2 (y 2)2 y
calculator
close-up You can use a calculator to demonstrate that the power rule for exponents holds when the exponents are negative integers.
helpful
Negative Exponents and Scientific Notation
hint
The exponent rules in this section apply to expressions that involve only multiplication and division. This is not too surprising since exponents, multiplication, and division are closely related. Recall that a3 a a a and a b a b1.
42x10 y 4
Power of a product rule and power rule
1 42 x10 4 y 1 2 x10 y4 4 x10y4 16
a 1 Because a . b b Definition of negative exponent
Simplify.
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Converting from Scientific Notation Many of the numbers occurring in science are either very large or very small. The speed of light is 983,569,000 feet per second. One millimeter is equal to 0.000001 kilometer. In scientific notation, numbers larger than 10 or smaller than 1 are written by using positive or negative exponents. Scientific notation is based on multiplication by integral powers of 10. Multiplying a number by a positive power of 10 moves the decimal point to the right: 10(5.32) 53.2 102(5.32) 100(5.32) 532 103(5.32) 1000(5.32) 5320
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Multiplying by a negative power of 10 moves the decimal point to the left: 1 101(5.32) (5.32) 0.532 10 1 102(5.32) (5.32) 0.0532 100 1 3 10 (5.32) (5.32) 0.00532 1000
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Polynomials and Exponents
So if n is a positive integer, multiplying by 10n moves the decimal point n places to the right and multiplying by 10n moves it n places to the left. A number in scientific notation is written as a product of a number between 1 and 10 and a power of 10. The times symbol indicates multiplication. For example, 3.27 109 and 2.5 104 are numbers in scientific notation. In scientific notation there is one digit to the left of the decimal point. To convert 3.27 109 to standard notation, move the decimal point nine places to the right: 3.27 109 3,270,000,000
calculator close-up On a graphing calculator you can write scientific notation by actually using the power of 10 or press EE to get the letter E, which indicates that the following number is the power of 10.
9 places to the right
Of course, it is not necessary to put the decimal point in when writing a whole number. To convert 2.5 104 to standard notation, the decimal point is moved four places to the left: 2.5 104 0.00025 4 places to the left
In general, we use the following strategy to convert from scientific notation to standard notation. Note that if the exponent is not too large, scientific notation is converted to standard notation when you press ENTER.
1. Determine the number of places to move the decimal point by examining the exponent on the 10. 2. Move to the right for a positive exponent and to the left for a negative exponent.
5
Converting scientific notation to standard notation Write in standard notation. b) 8.13 105 a) 7.02 106
Solution a) Because the exponent is positive, move the decimal point six places to the right: 7.02 106 7020000. 7,020,000 b) Because the exponent is negative, move the decimal point five places to the left. 8.13 105 0.0000813
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Converting to Scientific Notation To convert a positive number to scientific notation, we just reverse the strategy for converting from scientific notation. Strategy for Converting to Scientific Notation
1. Count the number of places (n) that the decimal must be moved so that it will follow the first nonzero digit of the number. 2. If the original number was larger than 10, use 10n. 3. If the original number was smaller than 1, use 10n.
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E X A M P L E
Strategy for Converting from Scientific Notation to Standard Notation
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Negative Exponents and Scientific Notation
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247
Remember that the scientific notation for a number larger than 10 will have a positive power of 10 and the scientific notation for a number between 0 and 1 will have a negative power of 10.
E X A M P L E
6
Converting numbers to scientific notation Write in scientific notation. a) 7,346,200 b) 0.0000348
c) 135 1012
Solution a) Because 7,346,200 is larger than 10, the exponent on the 10 will be positive:
calculator
7,346,200 7.3462 106
close-up
b) Because 0.0000348 is smaller than 1, the exponent on the 10 will be negative: 0.0000348 3.48 105
To convert to scientific notation, set the mode to scientific. In scientific mode all results are given in scientific notation.
c) There should be only one nonzero digit to the left of the decimal point: 135 1012 1.35 102 1012 1.35 1010
Convert 135 to scientific notation. Product rule
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Computations with Scientific Notation An important feature of scientific notation is its use in computations. Numbers in scientific notation are nothing more than exponential expressions, and you have already studied operations with exponential expressions in this section. We use the same rules of exponents on numbers in scientific notation that we use on any other exponential expressions.
E X A M P L E
7
Using the rules of exponents with scientific notation Perform the indicated computations. Write the answers in scientific notation. a) (3 106)(2 108) 4 105 b) 8 102 c) (5 107)3
calculator close-up With a calculator’s built-in scientific notation, some parentheses can be omitted as shown below. Writing out the powers of 10 can lead to errors.
Solution a) (3 106)(2 108) 3 2 106 108 6 1014 4 105 4 105 1 5(2) b) Quotient rule 2 2 10 8 10 8 10 2 (0.5)107
5 101 107 Write 0.5 in scientific notation. 5 106 Product rule 7 3 3 7 3 c) (5 10 ) 5 (10 ) Power of a product rule 21 125 10 Power rule 2 21 1.25 10 10 125 1.25 102 19 1.25 10 Product rule ▲
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Try these computations with your calculator.
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E X A M P L E
Polynomials and Exponents
8
Converting to scientific notation for computations Perform these computations by first converting each number into scientific notation. Give your answer in scientific notation. a) (3,000,000)(0.0002) b) (20,000,000)3(0.0000003)
Solution a) (3,000,000)(0.0002) 3 106 2 104 Scientific notation 6 102 Product rule 3 7 3 b) (20,000,000) (0.0000003) (2 10 ) (3 107) Scientific notation 8 1021 3 107 Power of a product rule 24 1014 2.4 101 1014 24 2.4 101 2.4 1015 Product rule
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True or false? Explain your answer.
1 1 2. 5 False 5 32 1 4. 1 True 3 3 6. 0.000036 3.6 105 True 8. 0.442 103 4.42 104 True
1 1. 102 True 100 3. 32 21 63 5. 7. 9. 10.
False
23.7 2.37 101 False 25 107 2.5 108 True (3 109)2 9 1018 True (2 105)(4 104) 8 1020
False
@ http://www.napanet.net/education/redwood/departments/mathhandbook/Default.htm#exponents 4.7
EXERCISES
Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What does a negative exponent mean? A 2. What is the correct order for evaluating the operations indicated by a negative exponent? The operations can be evaluated in any order. 3. What is the new quotient rule for exponents? m
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4. How do you convert a number from scientific notation to standard notation? Convert from scientific notation by multiplying by the appropriate power of 10. 5. How do you convert a number from standard notation to scientific notation? Convert from standard notation by counting the number of places the decimal must move so that there is one nonzero digit to the left of the decimal point.
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6. Which numbers are not usually written in scientific notation? Numbers between 1 and 10 are not written in scientific notation. Variables in all exercises represent positive real numbers. Evaluate each expression. See Example 1. 7. 31
8. 33
10. (3)4 52 13. 102
11. 42 4
9. (2)4
12. 24
34 14. 2 6
Simplify. See Example 2. 5 3 4 2 9 16. 15. 17. 61 61 16 2 3 1 0 1 18. 21 41 19. 3 1250 20. 25 104 5 1 32 23 2 1 22. 702 21. 3 4 2 102 72
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4.7
Negative Exponents and Scientific Notation
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249
Simplify. Write answers without negative exponents. See Example 3. 23. x1x 2 x 24. y3y5
68. 5,670,000,000 69. 525 109 70. 0.0034 108
25. 2x2 8x6
26. 5y5(6y7)
Perform the computations. Write answers in scientific notation. See Example 7.
27. 3a2(2a3)
28. (b3)(b5)
u5 1 29. u3 u8 8t3 31. 2t5 6x 5 33. 2
w4 1 30. w6 w10 22w4 32. 11w3 51y6 34. 17y9
71. (3 105)(2 1015) 72. (2 109)(4 1023) 4 108 73. 2 1030 9 104 74. 3 106 3 1020 75. 6 108
Simplify each expression. Write answers without negative exponents. See Example 4. 35. (x2)5
36. (y2)4
37. (a3)3
38. (b5)2 b
39. (2x3)4
40. (3y1)2
41. (4x2y3)2 2x1 43. y 3
1 108 76. 4 107 77. (3 1012)2 78. (2 105)3 79. (5 104)3 80. (5 1014)1 81. (4 1032)1 82. (6 1011)2 3.6
42. (6s2t4)1
2
a2 44. 3 3b
2a 45. ac
3
3w 46. wx
3 4
Perform the following computations by first converting each number into scientific notation. Write answers in scientific notation. See Example 8. 83. (4300)(2,000,000)
2 2
2
4 3
Simplify. Write answers without negative exponents. 47. 21 31
48. 21 31
49. (2 31)1
50. (21 3)1
84. 85. 86. 87.
(40,000)(4,000,000,000) (4,200,000)(0.00005) 2.1 (0.00075)(4,000,000) (300)3(0.000001)5
51. (x2)3 3x7(5x1)
88. (200)4(0.0005)3
52. (ab1)2 ab(ab3)
(4000)(90,000) 89. 3 0.00000012 (30,000)(80,000) 90. 2 (0.000006)(0.002)
a3b2 b6a2 53. 1 b5 a
54.
x3y1 2x
3
2
6x9y3 3x3
Write each number in standard notation. See Example 5. 55. 9.86 109 9,860,000,000 56. 4.007 104 40,070 3
91. 92. 93. 94.
5
57. 1.37 10 0.00137 6 59. 1 10 0.000001 61. 6 105 600,000
58. 9.3 10 0.000093 1 60. 3 10 0.3 62. 8 106 8,000,000
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Write each number in scientific notation. See Example 6. 63. 9000 9 64. 5,298,000 65. 0.00078 7.8 66. 0.000214 67. 0.0000085
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95.
2
Perform the following computations with the aid of a calculator. Write answers in scientific notation. Round to three decimal places. (6.3 106)(1.45 104) (8.35 109)(4.5 103) (5.36 104) (3.55 105) (8.79 108) (6.48 109) (3.5 105)(4.3 106) 3.4 108
(3.5 108)(4.4 104) 96. 2.43 1045 97. (3.56 1085)(4.43 1096) 98. (8 1099) (3 1099)
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Polynomials and Exponents
Solve each problem. 99. Distance to the sun. The distance from the earth to the sun is 93 million miles. Express this distance in feet. (1 mile 5280 feet.) 4.910 10 feet 93 million miles
Sun
Earth
FIGURE FOR EXERCISE 99 100. Speed of light. The speed of light is 9.83569 108 feet per second. How long does it take light to travel from the sun to the earth? See Exercise 99. 8.3 minutes 101. Warp drive, Scotty. How long does it take a spacecraft traveling at 2 1035 miles per hour (warp factor 4) to travel 93 million miles. 4.65 10 hours 102. Area of a dot. If the radius of a very small circle is 2.35 108 centimeters, then what is the circle’s area? 103. Circumference of a circle. If the circumference of a circle is 5.68 109 feet, then what is its radius?
10 4 10 3 Ag
102
V
101 Hg
1 Pb
Cr Ba
101
102
102 100 1 104 Concentration in waste (percent)
FIGURE FOR EXERCISE 106 107. Present value. The present value P that will amount to A dollars in n years with interest compounded annually at annual interest rate r, is given by P A (1 r )n. Find the present value that will amount to $50,000 in 20 years at 8% compounded annually. $10,727.41 108. Investing in stocks. U.S. small company stocks have returned an average of 14.9% annually for the last 50 years (T. Rowe Price, www.troweprice.com). Use the present value formula from the previous exercise to find the amount invested today in small company stocks that would be worth $1 million in 50 years, assuming that small company stocks continue to return 14.9% annually for the next 50 years. $963.83
Value (millions of dollars)
108 Ra
106 104
Au
102 1
Cu
U
102 103 106 109 100 1 Concentration in ore (percent)
FIGURE FOR EXERCISE 105
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Metal price (dollars/pound)
104. Diameter of a circle. If the diameter of a circle is 1.3 1012 meters, then what is its radius? 6.5 10 meters 105. Extracting metals from ore. Thomas Sherwood studied the relationship between the concentration of a metal in commercial ore and the price of the metal. The accompanying graph shows the Sherwood plot with the locations of several metals marked. Even though the scales on this graph are not typical, the graph can be read in the same manner as other graphs. Note also that a concentration of 100 is 100%. a) Use the figure to estimate the price of copper (Cu) and its concentration in commercial ore. $1 per pound and 1% b) Use the figure to estimate the price of a metal that has a concentration of 106 percent in commercial ore. $1,000,000 per pound. c) Would the four points shown in the graph lie along a straight line if they were plotted in our usual coordinate system? No
106. Recycling metals. The accompanying graph shows the prices of various metals that are being recycled and the minimum concentration in waste required for recycling. The straight line is the line from the figure for Exercise 105. Points above the line correspond to metals for which it is economically feasible to increase recycling efforts. a) Use the figure to estimate the price of mercury (Hg) and the minimum concentration in waste required for recycling mercury. $1 per pound and 1% b) Use the figure to estimate the price of silver (Ag) and the minimum concentration in waste required for recycling silver. $100 per pound and 0.1%
Metal price (dollars/pound)
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1 Amount after 50 years, $1 million
0.5 Present value, P
0 0
10
20 30 Years
40 50
FIGURE FOR EXERCISE 108
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GET TING MORE INVOLVED
Collaborative Activities
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251
110. Discussion. Which of the following expressions is not equal to 1? Explain your answer. a) 11 b) 12 c) (11)1 d) (1)1 2 e) (1) e
3
109. Exploration. a) If w 0, then what can you say about w? b) If (5)m 0, then what can you say about m? c) What restriction must be placed on w and m so that w m 0? a) w 0 b) m is odd c) w 0 and m odd
COLLABORATIVE ACTIVITIES Area as a Model of FOIL
Grouping: Pairs Topic: Multiplying polynomials
Sometimes we can use drawings to represent mathematical operations. The area of a rectangle can represent the process we use when multiplying binomials. The rectangle below represents the multiplication of the binomials (x 3) and (x 5): 3x
x
2
5x
x
5
(x + 3) x
3.
The areas of the inner rectangles are x2, 3x, 5x, and 15.
15
3
For problem 3, student B uses FOIL and A uses the diagram.
5
The area of the red rectangle equals the sum of the areas of the four inner rectangles.
(x + 5)
(x 5)(x 4) ?
(x + 5) x x
4 (x + 4)
Area of red rectangle: (x 3)(x 5) x 3x 5x 15 2
4. StudentAdraws a diagram to find the product (x 3)(x 7). Student B finds (x 3)(x 7) using FOIL.
x 2 8x 15 1. a. With your partner, find the areas of the inner rectangles to find the product (x 2)(x 7) below:
5. Student B draws a diagram to find the product (x 2)(x 1). Student A finds (x 2)(x 1) using FOIL. Thinking in reverse: Work together to complete the product that is represented by the given diagram.
2 (x + 2) x
6. x
9
7
(x + ?)
(x + 7) x2
(x 2)(x 7) ? b. Find the same product (x 2)(x 7) using FOIL.
(x + ?)
For problem 2, student A uses FOIL to find the given product while student B finds the area with the diagram.
7. 10
2.
(x + ?) x2 8
(x 8)(x 1) ?
(x + 8)
(x + ?) x x
Extension: Make up a FOIL problem, then have your partner draw a diagram of it.
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252
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Chapter 4
Polynomials and Exponents
C H A P T E R
W R A P - U P
4
SUMMARY Polynomials
Examples
Term
A number or the product of a number and one or more variables raised to powers
5x3, 4x, 7
Polynomial
A single term or a finite sum of terms
2x 5 9x 2 11
Degree of a polynomial The highest degree of any of the terms
Degree of 2x 9 is 1. Degree of 5x 3 x 2 is 3.
Adding, Subtracting, and Multiplying Polynomials
Examples
Add or subtract polynomials
Add or subtract the like terms.
(x 1) (x 4) 2x 3 (x2 3x) (4x2 x) 3x2 2x
Multiply monomials
Use the product rule for exponents.
2x5 6x8 12x13
Multiply polynomials
Multiply each term of one polynomial by every term of the other polynomial, then combine like terms.
x 2 2x 5 x1 2 x 2x 5 x 3 2x 2 5x x 3 x 2 3x 5
Binomials
Examples
FOIL
A method for multiplying two binomials quickly
(x 2)(x 3) x2 x 6
Square of a sum
(a b)2 a2 2ab b2
(x 3)2 x2 6x 9
Square of a difference
(a b)2 a2 2ab b2
(m 5)2 m2 10m 25
Product of a sum and a difference
(a b)(a b) a2 b2
(x 2)(x 2) x2 4
Dividing Polynomials
Examples
Dividing monomials
Use the quotient rule for exponents
8x5 (2x2) 4x3
Divide a polynomial by a monomial
Divide each term of the polynomial by the monomial.
3x5 9x x4 3 3x
x 7 ← Quotient If the divisor is a binomial, use Divisor → x 2x2 x 5 4 ← Dividend long division. x2 2x (divisor)(quotient) (remainder) dividend 7x 4 7x 14 10 ← Remainder
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Divide a polynomial by a binomial
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Chapter 4
Rules of Exponents
Summary
(4-47)
Examples
The following rules hold for any integers m and n, and nonzero real numbers a and b. Zero exponent
a0 1
20 1, (34)0 1
Product rule
am an amn
a2 a3 a5 3x6 4x9 12x15
Quotient rule
am n amn a
y3 x8 x2 x6, 3 y0 1 y c7 1 9 c2 2 c c
(am)n amn
Power rule
(22)3 26, (w5)3 w15
Power of a product rule (ab)n anbn
a b
Power of a quotient rule
n
(2t)3 8t3
an n b
x 3
Negative Exponents
3
x3 27
Examples
Negative integral exponents
If n is a positive integer and a is a nonzero real 1 number, then an n a
Rules for negative exponents
If a is a nonzero real number and n is a positive
n
1 1 1 integer, then an , a1 , and n an. a a a Scientific Notation
1 5 1 5 32 , 2 x 3 x 2 3 3 3 1 , 51 3 2 5 1 8 w8 w
Examples
Converting from scientific notation
1. Find the number of places to move the decimal point by examining the exponent on the 10. 2. Move to the right for a positive exponent and to the left for a negative exponent. 1. Count the number of places (n) that the decimal point must be moved so that it will follow the first nonzero digit of the number. 2. If the original number was larger than 10, use 10n. 3. If the original number was smaller than 1, use 10n.
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Converting into scientific notation (positive numbers)
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5.6 103 5600 9 104 0.0009
304.6 3.046 102 0.0035 3.5 103
Textbook Table of Contents
253
254
(4-48)
Chapter 4
Polynomials and Exponents
ENRICHING YOUR MATHEMATICAL WORD POWER
1. term a. an expression containing a number or the product of a number and one or more variables b. the amount of time spent in this course c. a word that describes a number d. a variable a
c. an equation with no solution d. a polynomial with five terms
b
7. dividend a. a in ab b. b in ab c. the result of ab d. what a bank pays on deposits
a
2. polynomial a. four or more terms b. many numbers c. a sum of four or more numbers d. a single term or a finite sum of terms
8. divisor a. a in ab b. b in ab c. the result of ab d. two visors b
For each mathematical term, choose the correct meaning.
d
3. degree of a polynomial a. the number of terms in a polynomial b. the highest degree of any of the terms of a polynomial c. the value of a polynomial when x 0 d. the largest coefficient of any of the terms of a polynomial b 4. leading coefficient a. the first coefficient b. the largest coefficient c. the coefficient of the first term when a polynomial is written with decreasing exponents d. the most important coefficient c 5.
monomial a. a single polynomial b. one number c. an equation that has only one solution d. a polynomial that has one term d
9.
quotient a. a in ab b. b in ab c. ab d. the divisor plus the remainder
c
10. binomial a. a polynomial with two terms b. any two numbers c. the two coordinates in an ordered pair d. an equation with two variables a 11. integral exponent a. an exponent that is an integer b. a positive exponent c. a rational exponent d. a fractional exponent a 12. scientific notation a. the notation of rational exponents b. the notation of algebra c. a notation for expressing large or small numbers with powers of 10 d. radical notation c
6. FOIL a. a method for adding polynomials b. first, outer, inner, last
REVIEW EXERCISES 4.1 Perform the indicated operations. 1. (2w 6) (3w 4) 5w 2 2. (1 3y) (4y 6) y 5 3. (x2 2x 5) (x2 4x 9) x 4. (3 5x x2) (x2 7x 8) 5. (5 3w w2) (w2 4w 9) 6. (2t2 3t 4) (t2 7t 2) 7. (4 3m m2) (m2 6m 5) 8. (n3 n2 9) (n4 n3 5) n
4 5
13. x 5(x 3) x 15 15. 5x 3(x2 5x 4)
14. x 4(x 9) 3x 36 16. 5 4x2(x 5)
17. 3m2(5m3 m 2)
18. 4a4(a2 2a 4)
4
study
2 1 4
Note how the review exercises are arranged according to the sections in this chapter. If you are having trouble with a certain type of problem, refer back to the appropriate section for examples and explanations.
4.2 Perform the indicated operations. 10. 3h3t2 2h2t5 12. (12b3)2 144
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Chapter 4
19. (x 5)(x2 2x 10) 20. (x 2)(x2 2x 4)
61. (x3 x2 11x 10) (x 1) 62. (y3 9y2 3y 6) (y 1)
21. (x2 2x 4)(3x 2) 22. (5x 3)(x2 5x 4)
27. (4y 3)(5y 2)
24. (w 5)(w 12) w 60 26. (5r 1)(5r 2) r 2 28. (11y 1)(y 2)
29. (3x 5)(2x 1)
30. (x 7)(2x 7)
25. (2t 3)(t 9)
2
2
3
2x 63. x3 6 x 3
2
2x 65. 1x
3
2 1 x
2
4.4 Perform the indicated operations. Try to write only the
x2 3 67. x1
answers. 31. (z 7)(z 7) z 49 33. (y 7)2 y 14y 49 35. (w 3)2 w 6w 9 37. (x2 3)(x2 3)
32. (a 4)(a 4) a 16 34. (a 5)2 a 10a 25 36. (a 6)2 a 12a 36 38. (2b2 1)(2b2 1)
39. (3a 1) 9a 6a 41. (4 y)2 16 y
40. (1 3c) 1 6c 42. (9 t)2 81 18t
2
x
x
x 3
6
72. (3a5)(5a3) 30k3y9 74. 15k3y2 76. (y5)8 y 78. (3a4b6)4 3y 3 80. 2 3a4b8 4 82. 3 6a b12
4.7 Simlify each expression. Use only positive exponents in answers. 83. 25 4m 1
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84. 24
32
86. 51 50
8
1
a8 89. a a 12 b2 92. 6 b b
Find the quotient and remainder. 55. (3m3 9m2 18m) (3m) m 56. (8x3 4x2 18x) (2x) 9, 0 57. (b2 3b 5) (b 2) b 58. (r2 5r 9) (r 3) r 2, 3 59. (4x2 9) (2x 1) 2x 1, 60. (9y3 2y) (3y 2) 2, 4
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x
1
For the following exercises, assume that all of the variables represent positive real numbers.
1
(a 1) (1 a) 1 (t 3) (3 t) 1 (m4 16) (m 2) m 2m (x4 1) (x 1) x x x
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8x3y5 4x2y4 2xy3 50. 2xy2 51. 52. 53. 54.
19
remainder quotient . divisor 3x 64. x4 12 3 x 4 3x 66. 5x 15 5 x x2 3x 1 68. x3 19 x 6 1 x 3 2x2 70. x3
71. 2y10 3y20 6 10b5c3 5 73. 5 2b c9 75. (b5)6 b 77. (2x3y2)3 8x 2a 3 79. b 6x2y5 3 8x 81. 3z6
44. 6x4y2 (2x2y2) 3x 9h5t9r2 t 46. 7 3h t6r2 h 7y 48. y 7 1 2x
2x 9, 1 10
4.6 Simplify each expression.
4.5 In Exercises 43–54, find each quotient. 43. 10x5 (2x3) 5x 6a5b7c6 a 45. 3 3a b9c6 b 3x 9 3 47. 3 9x3 6x2 3x x 49. 3x
2 x
1
x2 69. x1
2
1
255
Write each expression in the form
4.3 Perform the indicated operations. 23. (q 6)(q 8)
(4-49)
Review Exercises
95. (2x3)3
x
1 1000
85. 103
87. x5x8
88. a3a9
a10 90. 4 a
a3 91. 7 a
93. (x3)4
94. (x5)10
96. (3y5)2
a 97. 3b3
2 3
a 98. 5b
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256
(4-50)
Chapter 4
Polynomials and Exponents
Convert each number in scientific notation to a number in standard notation, and convert each number in standard notation to a number in scientific notation. 99. 5000 100. 0.00009 10 9 10 102. 5.7 108 101. 3.4 105 340,000 0.000000057 103. 0.0000461 104. 44,000 4.61 10 4.4 10 6 106. 5.5 109 105. 5.69 10 0.00000569 5,500,000,000
140. (2x2 7x 4)(x 3) 141. (x2 4x 12) (x 2) 142. (a2 3a 10) (a 5) Solve each problem. 143. Roundball court. The length of a basketball court is 44 feet larger than its width. Find polynomials that represent its perimeter and area. The actual width of a basketball court is 50 feet. Evaluate these polynomials to find the actual perimeter and area of the court.
Perform each computation without using a calculator. Write answers in scientific notation. 107. (3.5 108)(2.0 1012) 7 10 108. (9 1012)(2 1017) 1.8 109. (2 104)4 1.6 110. (3 105)3 111. (0.00000004)(2,000,000,000) 8 112. (3,000,000,000) (0.000002) 1.5 10 113. (0.0000002)5 3.2 114. (50,000,000,000)3 1.25
(5x2 8x 8) (4x2 x 3) 5 (4x2 6x 8) (9x2 5x 7) (2x2 2x 3) (3x2 x 9) 5x (x2 3x 1) (x2 2x 1) 2x 19x (x 4)(x2 5x 1) x
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135. 136. 137. 138. 139.
1000
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Weekly revenue (hundreds of dollars)
(k 5)(k 4) k k 20 (t 7z)(t 6z) t 13tz 42z (4y2 9)0 1 (9y3c4)2 81y c (3x 5)(2x 6) 6x 8x 30 2 (9x 2)(9x2 2) 81x 4 (4v 3)2 16v 24v 9 (k 10)3 k k 300k 5w3r2 2w4r8 10w 6h3y5 4 7 2h y2 81y h
FIGURE FOR EXERCISE 143 144. Badminton court. The width of a badminton court is 24 feet less than its length. Find polynomials that represent its perimeter and area. The actual length of a badminton court is 44 feet. Evaluate these polynomials to find the perimeter and area of the court. P 4x 48, A x 24x, P 128 ft, A 880 ft 145. Smoke alert. A retailer of smoke alarms knows that at a price of p dollars each, she can sell 600 15p smoke alarms per week. Find a polynomial that represents the weekly revenue for the smoke alarms. Find the revenue for a week in which the price is $12 per smoke alarm. Use the bar graph to find the price per smoke alarm that gives the maximum weekly revenue. R 15p 600p, $5040, $20
MISCELL ANEOUS Perform the indicated operations. 115. (x 3)(x 7) 116. x 10x 21 117. (t 3y)(t 4y) 118. t 7ty 12y 120. 119. (2x3)0 (2y)0 2 122. 121. (3ht6)3 27h 123. (2w 3)(w 6) 124. 2w 9w 18 125. (3u 5v)(3u 5v) 126. 9u v 2 127. (3h 5) 128. 9h 25 130. 129. (x 3)3 x 9x x 27 132. 131. (7s2t)(2s3t5) 14s k4m2 4 134. 133. 22 2k m
w 44 ft
w ft
70 60 50 40 30 20 10 0
4
8
12 16 20 24
28 32 36
Price (dollars)
FIGURE FOR EXERCISE 145 146. Boom box sales. A retailer of boom boxes knows that at a price of q dollars each, he can sell 900 3q boom boxes per month. Find a polynomial that represents the monthly revenue for the boom boxes? How many boom boxes will he sell if the price is $300 each? R q 900q, 0
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Chapter 4
Test
(4-51)
257
CHAPTER 4 TEST Perform the indicated operations. 1. (7x3 x2 6) (5x2 2x 5) 2. (x2 3x 5) (2x2 6x 7) 6y3 9y2 3. 3y 4. (x 2) (2 x) 1 5. (x3 2x2 4x 3) (x 3) 6. 3x2(5x3 7x2 4x 1) 15x
study
Before you take an in-class exam on this chapter, work the sample test given here. Set aside one hour to work this test and use the answers in the back of this book to grade yourself. Even though your instructor might not ask exactly the same questions, you will get a good idea of your test readiness.
Find the products. 7. (x 5)(x 2) x 10 9. (a 7)2 a 14a 49 11. (b 3)(b 3) b 9 13. (4x2 3)(x2 2) 4x x 6
tip
8. (3a 7)(2a 5) 6a 35 10. (4x 3y)2 16x 24xy 9y 12. (3t2 7)(3t2 7) 9t 49 14. (x 2)(x 3)(x 4) x 3 x 24
25. (3s3t2)2
26. (2x6y)3
Convert to scientific notation. 27. 5,433,000 5.433 10
28. 0.0000065
Perform each computation by converting to scientific notation. Give answers in scientific notation. 29. (80,000)(0.000006) 30. (0.0000003)4 4.8 10
Write each expression in the form
Solve each problem.
remainder quotient . divisor 2x x2 3x 5 16. 15. x3 x2 6 15 x x 3 x 2 Use the rules of exponents to simplify each expression. Write answers without negative exponents. 17. 5x3 7x5 x 18. 3x3y (2xy4 6 5 5 19. 4a b (2a b) ab 20. 3x2 5x7 5 2a 2a 6a7b6c2 21. 22. 3 2 b b 2a b8c2 7 6t w6 23. 24. 4 9 2t w
31. Find the quotient and remainder when x2 5x 9 is divided by x 3. x 2, 3 32. Subtract 3x2 4x 9 from x2 3x 6. x x 15 33. The width of a pool table is x feet, and the length is 4 feet longer than the width. Find polynomials that represent the area and the perimeter of the pool table. Evaluate these polynomials for a width of 4 feet. x x ft x 8 ft, 32 ft 34. If a manufacturer charges q dollars each for footballs, then he can sell 3000 150q footballs per week. Find a polynomial that represents the revenue for one week. Find the weekly revenue if the price is $8 for each football. R q 3000q, $14,400
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MAKING CONNECTIONS CHAPTERS 1–4 1. 3. 5. 7. 8. 9. 11.
16 (2) 2. (2)3 1 2 (5) 3(5) 1 4. 210 215 15 10 2 2 6. 210 25 32 42 (172 85) (85 172) (5 3)2 10. 52 32 (30 1)(30 1) 12. (30 1)2
Perform the indicated operations. 13. (x 3)(x 5) 14. (x2 8x 15) (x 5) 15. x 3(x 5)
16. (x2 8x 15)(x 5)
17. 5t3v 3t2v6
18. (10t3v2) (2t2v)
19. (6y3 8y2) (2y2) 20. (y2 3y 9) (3y2 2y 6) Solve each equation. 21. 2x 1 0
22. x 7 0
23. 2x 3 0
24. 3x 7 5
25. 8 3x x 20
Solve the problem. 27. Average cost. Pineapple Recording plans to spend $100,000 to record a new CD by the Woozies and $2.25 per CD to manufacture the disks. The polynomial 2.25n 100,000 represents the total cost in dollars for recording and manufacturing n disks. Find an expression that represents the average cost per disk by dividing the total cost by n. Find the average cost per disk for n 1000, 100,000, and 1,000,000. What happens to the large initial investment of $100,000 if the company sells one million CDs?
6
Average cost (dollars)
Simplify each expression.
4 3 2 1 0
26. 4 3(x 2) 0
3
5
0
0.5 1 Number of disks (millions)
FIGURE FOR EXERCISE 27
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