Section 5.1 Practice Exercises. Vocabulary and Key Concepts
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1 Section 5.1 Practice Exercises Vocabulary and Key Concepts A(n) is used to show repeated multiplication of the base. 2. For b 0, the expression b 0 is defined to be. 3. For b 0, the expression b n is defined as. 4. A number expressed in the form a 10 n, where 1 a < 10 and n is an integer is said to be written in. Concept 1: Simplifying Expressions with Exponents 2. Write the expressions in expanded form and simplify. 3. Write the expressions in expanded form and simplify. For Exercises 4 9, write an example of each property. (Answers may vary.) 4. b n b m = b n + m 5. (ab) n = a n b n 6. (b n ) m = b nm b 0 = 1 (b 0) For Exercises 10 28, simplify. (See Example 1.) 10.
2 ( 5) ( 8) Exercise: Simplifying expressions with negative exponents PDF Transcript for Exercise: Simplifying expressions with negative exponents (10ab) 0
3 26. (13x) ab x 0 For Exercises 29 80, simplify and write the answer with positive exponents only. (See Examples 2 5.) 29. y 3 y x 4 x (y 2 ) (z 3 ) (3x 2 ) 4 Exercise: Simplifying an expression with exponents PDF Transcript for Exercise: Simplifying an expression with exponents 36. (2y 5 ) p q
4 a 2 a b 1 b Page (6xyz 2 ) ( 7ab 3 )
5 ( 3x 4 y 5 z 2 ) ( 6a 2 b 3 c) (4m 2 n)( m 6 n 3 ) 68. ( 6pq 3 )(2p 4 q) 69. (p 2 q) 3 (2pq 4 ) 2 Exercise: Simplifying expressions with exponents PDF Transcript for Exercise: Simplifying expressions with exponents 70. (mn 3 ) 2 (5m 2 n 2 ) 71.
6 Exercise: Simplifying rational expression with exponents PDF Transcript for Exercise: Simplifying rational expression with exponents
7 Section 5.2 Practice Exercises Vocabulary and Key Concepts A in the variable, x, is a single term or a sum of terms of the form ax n, where a is a real number and n is a nonnegative integer. 2. Given the term ax n, a is called the, and is called the degree of the term. 3. Given the term x, the coefficient of the term is and the degree is. 4. A monomial is a polynomial with exactly term(s). 5. A is a polynomial with exactly two terms. 6. A is a polynomial with exactly three terms. 7. The term with the highest degree is called the term and its coefficient is called the. 8. The degree of a polynomial is the degree of all of its terms. Review Exercises Page The degree of a nonzero constant such as 7 is. 10. If a term of a polynomial has more than one variable, then the degree of the term is the sum of the of the variables contained in the term. 11. A function is a function defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. For Exercises 2 6, simplify the expression (2ac 2 )(5a 1 c 4 ) ( )( ) 6. Concept 1: Polynomials: Basic Definitions For Exercises 7 12, write the polynomial in descending order. Then identify the leading coefficient and the degree.
8 7. a 2 6a 3 a 8. 2b b 4 + 5b x 2 x + 3x y + y 5 y 2 Exercise: Polynomials: Descending Order, Leading Coefficent, and Degree PDF Transcript for Exercise: Polynomials: Descending Order, Leading Coefficent, and Degree t s 2 For Exercises 13 18, write a polynomial in one variable that is described by the following. (Answers may vary.) 13. A monomial of degree A monomial of degree A trinomial of degree A trinomial of degree A binomial of degree A binomial of degree 2 Concept 2: Addition of Polynomials For Exercises 19 30, add the polynomials and simplify. (See Examples 1 and 2.) 19. ( 4m 2 + 4m) + (5m 2 + 6m) 20. (3n 3 + 5n) + (2n 3 2n) 21. (3x 4 x 3 x 2 ) + (3x 3 7x 2 + 2x)
9 22. (6x 3 2x 2 12) + (x 2 + 3x + 9) Exercise: Adding Polynomials PDF Transcript for Exercise: Adding Polynomials Add (9x 2 y 5xy + 1) to (8x 2 y + xy 15). 26. Add ( x 3 y 2 + 5xy) to (10x 3 y 2 + x 2 y 10). 27. Add ( 7a + 6a 2 + 1) to ( 8 4a 2a 2 ). 28. Add (1 12p + 8p 3 ) to (6p 2 + p 3 14) Concept 3: Subtraction of Polynomials For Exercises 31 36, write the opposite of the given polynomial. (See Example 3.) y x p 3 + 2p t 2 4t ab 2 + a 2 b rs 4r + 9s Page 336
10 For Exercises 37 46, subtract the polynomials and simplify. (See Examples 4 and 5.) 37. (13z 5 z 2 ) (7z 5 + 5z 2 ) 38. (8w 4 + 3w 2 ) (12w 4 w 2 ) 39. ( 3x 3 + 3x 2 x + 6) (1 x x 2 x 3 ) 40. ( 8x 3 + 6x + 7) ( 4 2x 5x 3 ) 41. ( 3xy 3 + 3x 2 y x + 6) ( xy 3 xy x + 1) 42. ( 8x 2 y 2 + 6xy 2 + 7xy) (5xy 2 2xy 4) Exercise: Subtracting Polynomials PDF Transcript for Exercise: Subtracting Polynomials Subtract (9x 2 5x + 1) from (8x 2 + x 15). (See Example 6.) 48. Subtract ( x 3 + 5x) from (10x 3 + x 2 10). 49. Find the difference of (3x 5 2x 3 + 4) and (x 4 + 2x 3 7). 50. Find the difference of (7x 10 2x 4 3x) and ( 4x 3 5x 4 + x + 5). Mixed Exercises For Exercises 51 74, add or subtract as indicated. Write the answers in descending order, if possible.
11 51. (8y 2 4y 3 ) (3y 2 8y 3 ) 52. ( 9y 2 8) (4y 2 + 3) 53. ( 2r 6r 4 ) + ( r 4 9r) 54. ( 8s 9 + 7s 2 ) + (7s 9 s 2 ) 55. (5xy + 13x 2 + 3y) (4x 2 8y) 56. (6p 2 q 2q) ( 2p 2 q + 13) 57. (11ab 23b 2 ) + (7ab 19b 2 ) 58. ( 4x 2 y + 9) + (8x 2 y 12) 59. [2p (3p + 5)] + (4p 6) (q 2) [4 (2q 3) + 5] [2m 2 (4m 2 + 1)] 62. [4n 3 (n 3 + 4)] + 3n (6x 3 5) ( 3x 3 + 2x) (2x 3 6x) Exercise: Mixed Exercise: Subtracting 3 Polynomials PDF Transcript for Exercise: Mixed PDF Transcript for Exercise: Subtracting 3 Polynomials 64. (9p 4 2) + (7p 4 + 1) (8p 4 10) 65. ( ab + 5a 2 b) [7ab 2 2ab (7a 2 b + 2ab 2 )] 66. (m 3 n 2 + 4m 2 n) [ 5m 3 n 2 4mn (7m 2 n 6mn)] 67. (8x 3 x 2 + 3) [5x 2 + x (4x 3 + x 2)] 68. (y 2 + 6y 6) [(2y 3 4y) (3y 2 + y + 1)]
12 69. Exercise: Mixed Exercise: Vertical Subtraction PDF Transcript for Exercise: Mixed PDF Transcript for Exercise: Vertical Subtraction Page 337 For Exercises 75 and 76, find the perimeter Concept 4: Polynomial Functions For Exercises 77 84, determine whether the given function is a polynomial function. If it is a polynomial function, state the degree. If not, state the reason why.
13 k(x) = 7x 4 0.3x + x q(x) = x 2 4x g(x) = g(x) = 4x 83. M(x) = x + 5x 84. N(x) = x 2 + x 85. Given P(x) = x 4 + 2x 5, find the function values. (See Example 7.) 1. P(2) 2. P( 1) 3. P(0) 4. P(1) 86. Given N(x) = x 2 + 5x, find the function values. 1. N(1) 2. N( 1) 3. N(2) 4. N(0) 87. Given find the function values. 1. H(0) 2. H(2) 3. H( 2) 4. H( 1) 88. Given find the function values. 1. K(0) 2. K(3) 3. K( 3) 4. K( 1)
14 89. A rectangular garden is designed to be 3 ft longer than it is wide. Let x represent the width of the garden. Find a function P that represents the perimeter in terms of x. (See Example 8.) Exercise: Polynomial Functions PDF Transcript for Exercise: Polynomial Functions 90. Pauline measures a rectangular conference room and finds that the length is 4 yd greater than twice the width. Let x represent the width. Find a function P that represents the perimeter in terms of x. 91. The cost in dollars of producing x calendars is C(x) = 5.40x The revenue for selling x calendars is R(x) = 12x. To calculate profit, subtract the cost from the revenue. 1. Write and simplify a function P that represents profit in terms of x. 2. Find the profit of producing and selling 50 calendars. 92. The cost in dollars of producing x lawn chairs is C(x) = 4.5x The revenue for selling x chairs is R(x) = 12.99x. To calculate profit, subtract the cost from the revenue. 1. Write and simplify a function P that represents profit in terms of x. 2. Find the profit of producing and selling 100 lawn chairs. Page The function defined by D(x) = 5.2x x approximates the average yearly dormitory charge for 4-yr universities x years since D(x) is the cost in dollars, and x represents the number of years since (See Example 9.) 1. Evaluate D(0) and D(18) and interpret their meaning in the context of this problem. 2. If this trend continues, what will the annual dormitory charge be in the year 2015?
15 Source: U.S. National Center for Education Statistics 94. The population of bacteria in a culture can be modeled by P(t) = 0.01t t + 10, where t is the time in hours after the culture was started and P(t) is the population in thousands. 1. Evaluate P(0) and P(14) and interpret their meaning in the context of this problem. 2. Predict the population of bacteria 24 hr after the culture was started. 95. The number of women, W, to be paid child support in the United States can be approximated by where t is the number of years since 2000, and W(t) is the yearly total measured in thousands. (Source: U.S. Bureau of the Census) 1. Evaluate W(0), W(5), and W(10). 2. Interpret the meaning of the function value W(10). 96. The total yearly amount of child support due (in billions of dollars) in the United States can be approximated by
16 where t is the number of years since 2000, and D(t) is the amount due (in billions of dollars). 1. Evaluate D(0), D(4), and D(8). 2. Interpret the meaning of the function value of D(8). Expanding Your Skills 97. A toy rocket is shot from ground level at an angle of 60 from the horizontal. See the figure. The x- and y-positions of the rocket (measured in feet) vary with time t according to 1. Evaluate x(0) and y(0), and write the values as an ordered pair. Interpret the meaning of these function values in the context of this problem. Match the ordered pair with a point on the graph. 2. Evaluate x(1) and y(1) and write the values as an ordered pair. Interpret the meaning of these function values in the context of this problem. Match the ordered pair with a point on the graph. 3. Evaluate x(2) and y(2), and write the values as an ordered pair. Match the ordered pair with a point on the graph.
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