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1© 2014 College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

Algebra 1 Unit 4 PracticeLeSSon 19-1 1. The size of a text file is 35 kilobytes. The size of a

video file is 312 kilobytes. How many times greater is the size of the video file than the size of the text file?

A. 324 B. 37

C. 317 D. 360

2. Arrange the expressions in order from least to greatest.

a. 42 ? 42 b. 22

8

2

c. 77

5

3 d. 33 ? 3

3. The formula for density is D 5 MV , where D is

density, M is mass, and V is volume. The density of an object is x4 kilograms per cubic meter. Its mass is x7 kilograms. What is the volume of the object?

4. Simplify the expression x x

x

95

25

15

? .

5. Write an expression containing multiplication and division that simplifies to y4.

6. Critique the reasoning of others. Nestor says that

the value of 66

66

5

2

8

3? is 615. Is he correct? If so,

explain why. If not, identify Nestor’s error and give the correct value.

LeSSon 19-2 7. Assume that x fi 0. For what value of y will 5xy

always be equal to 5? Explain your answer.

8. Simplify and write the expression a ba b93

4 3

5 7

2

2 2 without

negative powers.

9. For what value of n is 4m2n 5 m45 ?

A. 25 B. 15

C. 215

D. 5

10. For what value of a is b3 ? ba 5 1? Justify your answer.

11. Reason abstractly. Determine whether the statement below is always, sometimes, or never true. Explain your reasoning.

If x is a positive integer, then the value of a2x is negative.

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© 2014 College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

LeSSon 19-3 12. Simplify and write each expression without

negative powers.

a. x23

18

b. x y23

16

18

c. (a3b2c22)4(abc4)(ab)

13. Which expression is not equal to

xx

4

2

12

?

A. x B. 2x

C. ( )x 212 D.

xx

2

14. Write an expression involving at least one negative exponent and a power of a product that simplifies to mn3.

15. When a quotient is raised to a negative power, Brooke claims that you can invert the quotient and write it with a positive exponent. For example,

when asked to simplify

ab3

4

2

22

, Brooke begins by

writing

ba3 2

4

2

.

a. Simplify

ab3

4

2

22

by using Brooke’s method.

Then simplify without using Brooke’s method. How do your answers compare?

b. Does Brooke’s method always work? Explain why or why not.

16. Model with mathematics. The area of a rectangle is given by the formula A 5 ℓw, where ℓ is the length and w is the width. A rectangular patio has an area of (ab)2 square feet and a length of ab2 feet. Write a simplified expression that represents the width of the patio.

LeSSon 20-1 17. Kurt is cutting diagonal crossbars to stabilize a

rectangular wooden frame. If the frame has dimensions of 3 feet by 5 feet, what is the length of one crossbar? Give the exact answer using simplified radicals.

18. For each radical expression, write an equivalent expression with a fractional exponent.

a. 7

b. 193

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19. a. What is the value of 2713?

b. Make use of structure. How can you use your answer to part a to help you find the value of n

for which 27n2

5 9? Find the value of n and explain your reasoning.

20. Which of the following expressions is not

equivalent to y1634( ) ?

A. y16 34 B. y8 34

C. y834 D. y16

34

34

21. a. What is 1? What is 13 ? Explain your answers.

b. Let n be a positive integer. What is the value

of 1n1

? Explain your answer.

22. A cube-shaped box has a volume of 512 cubic inches. Celia has 2.5 square feet of wrapping paper. Does she have enough paper to cover the entire surface of the box? Explain your reasoning.

LeSSon 20-2 23. The perimeter of a rectangle is 8 8 feet and the

width is 4 2 feet. How many feet longer is the length of the rectangle than its width?

24. Write + +12 3 48 2 27 in simplest radical form. State whether the result is rational or irrational.

25. Find the value of a for which a5 5 3 52 5 . Explain how you found your answer.

26. Which is the sum of 2 50 and 8 ?

A. 12 2 B. 13 2

C. 13 5 D. 15 5

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27. Critique the reasoning of others. Identify and correct the error in each addition or subtraction problem.

a. 7 8 5 2 2 62 5

b. 9 5 5 9 101 5

c. 8 3 3 5 3 8 5 32 1 5 1

28. Ted is fencing in an area composed of a rectangle and a right triangle as shown below.

2 27�

243�

2 12� x

He still needs to buy fencing for the side labeled x. How much fencing does Ted need to buy for this side? Express the answer in simplest radical form.

LeSSon 20-3 29. Which of the following is 3 2

7 in simplest radical

form?

A. 3 2 B. 21 2

C. 3 147 D.

3 217

30. Jed has a rope that is 8 18 meters long. He cuts it into smaller pieces that are each 3 2 meters long. How many smaller pieces of rope does Jed now have?

31. a. Write32

1232

in simplest form. Is the

result rational or irrational?

b. What can you conclude from your answer to part a about whether the irrational numbers are closed under multiplication? Explain.

32. Lorraine solved the equation x3 24 12 6? 5 and found that x 5 4. Verify that Lorraine’s solution is correct.

33. Critique the reasoning of others. Deanna says

that 15 is in simplified form. Is she correct? If so,

explain why. If not, correct her mistake.

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LeSSon 21-1 34. Tell whether each sequence below is arithmetic,

geometric, or neither. If it is arithmetic, identify the common difference. If it is geometric, identify the common ratio.

a. 24, 4, 23

, 19

, …

b. 1, 4, 9, 16, 25, …

c. 3, 4.7, 6.3, 7.9, …

d. 2, 26, 18, 254, …

35. Model with mathematics. When school has been cancelled, a principal calls 4 teachers. These 4 teachers each call 4 other teachers who have not yet been called. Then those teachers each call 4 other teachers who have not yet been called, and so on.

a. The principal represents Stage 1. Make a tree diagram and a table of values to represent this situation.

b. Can this situation be represented by a geometric sequence? If so, identify the common ratio. If not, explain why not.

c. How many teachers will receive phone calls at Stage 4?

36. Consider the sequence 12, 3, x, ….

a. Find a value of x for which the sequence is arithmetic. Explain your answer.

b. Find a value of x for which the sequence is geometric. Explain your answer.

37. The terms in a geometric sequence alternate between positive and negative numbers. What must be true about this sequence?

A. The first term is negative.

B. The first term is greater than the second term.

C. The common ratio is between 0 and 1.

D. The common ratio is negative.

LeSSon 21-2 38. Write the first five terms of the geometric sequence

represented by the recursive formula below.

f

f n f n

(1) 2

( ) 12

( 1)

5

5 2

39. Ernie scores 50 points in Level 1 of a video game. In each subsequent level, he scores twice as many points as he did in the previous level.

a. Write a recursive formula that represents this situation.

b. Write an explicit formula that represents this situation.

c. Use either the recursive formula or the explicit formula to find the number of points that Eddie scores in Level 10. Why did you choose the formula you did?

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40. Write a geometric sequence in which every term is an odd integer. Write both the explicit and the recursive formulas for your sequence. Then identify the 9th term.

Use the geometric sequences below for Items 41 and 42.Sequence 1 Sequence 2

an 5 5 ? 2n21 aa a

25n n

1

1

5

5 2

41. Which statement is incorrect?

A. The terms in Sequence 2 increase more quickly than the terms in Sequence 1.

B. Both sequences have the same second term.

C. The explicit formula for Sequence 2 contains 2 raised to a power.

D. The common ratio for Sequence 2 is equal to the first term of Sequence 1.

42. Persevere in solving problems. How many terms in Sequence 1 are less than 500? Explain how you found your answer.

LeSSon 22-1 43. Rajiv bought a rare stamp for $125. A function

that models the value of Rajiv’s after t years is v(t) 5 125 ? (1.05)t. What is the value of Rajiv’s stamp after 20 years?

A. $131.25 B. $331.66

C. $2,625.00 D. $3,316.62

44. Attend to precision. The function f(t) 5 40,000 ? (1.3)t can be used to find the value of Sally’s house between 1970 and 2010, where t is the number of decades since 1970.

a. Identify the reasonable domain and range of the function. Explain your answers.

b. Sally wants to calculate the value of her house in 1995. What number should Sally substitute for t in the function? Explain.

c. Find the value of Sally’s house in 1995.

45. The function h(t) 5 5,000 ? (2.1)t models the value of Ms. Ruiz’s house, where t represents the number of decades since 1950. In what year did the value of Ms. Ruiz’s house first exceed $25,000? Explain how you can use a table to find the answer.

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46. The function h(t) 5 15,000 ? (1.5)t models the value of Sam’s house, where t represents the number of decades since 1960. The value of Kendra’s house has been doubling each decade since 1980. In 2010, the value of Sam’s house was greater than the value of Kendra’s house. Is it possible that the two houses had equal values in 1980? Explain.

LeSSon 22-2 47. Identify the constant factor for the exponential

function y 513

x

. How can you use the constant

factor to tell whether the function represents exponential growth or exponential decay?

48. Mia bought a new computer for $1,500. A function that models the value of Mia’s computer after t years is v(t) 5 1,500 ? (0.68)t. How much is Mia’s computer worth after 2.5 years?

49. Jane bought a new car for $30,000. A function that models the value of Jane’s car after t years is v(t) 5 30,000 ? (0.85)t. In how many years will the car be worth less than half of what Jane paid for it?

A. 2 B. 3

C. 4 D. 5

50. Compare the graph of an exponential growth function to the graph of an exponential decay function. Describe the similarities and differences.

51. Model with mathematics. Troy bought a book with 512 pages. The next day he read half the book. On each subsequent day, he read half of the remaining pages. The exponential decay function y 5 512(0.5)x gives the number of remaining pages x days after Troy bought the book.

a. How many pages did Troy have left to read after 6 days?

b. Blake says that the value of the exponential function can never be 0, so Troy will never finish reading the book. Do you agree with Blake? Explain why or why not.

LeSSon 22-3 52. Without graphing, tell which function increases

more slowly. Justify your answer.

f(x) 5 99x g(x) 5 9x

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53. Use a graphing calculator to graph the function

g(x) 5 14

12

x

.

a. Identify the values of a and b (from f(x) = abx), and describe their effects on the graph.

b. Graph f(x) 512

x

on the same screen as the

graph of g(x). Describe the similarities and differences between the graphs.

54. Which function increases the fastest?

A. y 5 14x B. y 5 23 · 17x

C. y 5 120x D. y 5 2275x

55. Make sense of problems. A health club with 100 members is trying to increase its membership. Judy has a plan that will increase membership by 25 members per month, so that the number of members y after x months is given by the function y 5 100 1 25x. Desmond has a plan that will increase membership by 10% each month, so that the number of members y after x months is given by the function y 5 100 ? 1.1x.

a. Whose plan will increase club membership more quickly? Use a graph to support your answer.

b. Whose plan would you recommend? Explain.

c. To keep the club from becoming overcrowded, the maximum club membership is 500 people. Does this additional information change your recommendation from part b? Explain why or why not.

LeSSon 23-1 56. On the coordinate grid below, p represents the

amount of money in Paola’s savings account, and v represents the amount in Vincent’s account. Whose account had a higher initial deposit, and how much was it? Use the graph to justify your answer.

x

y

200

400

600

800

40 800

v

p

Four students deposit money into accounts with interest that is compounded annually. The amount of money in each account after t years is given by the functions below. Use these functions for Items 57259.

Felicity: f(t) 5 500 · (1.02)t

Raisa: r(t) 5 800 · (1.01)t

Sanjay: s(t) 5 1,000 · (1.015)t

Megan: m(t) 5 200 · (1.025)t

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57. Identify the constant factor in Sanjay’s function and explain how it is related to his interest rate.

58. a. Write a function to represent the amount of money Felicity will have after m months if her interest were compounded monthly rather than annually.

b. Will Felicity earn more money when 2% annual interest is compounded annually or monthly? Explain.

59. Which shows the students’ names in order from greatest initial deposit to least initial deposit?

A. Megan, Felicity, Raisa, Sanjay

B. Felicity, Raisa, Megan, Sanjay

C. Sanjay, Megan, Raisa, Felicity

D. Sanjay, Raisa, Felicity, Megan

60. Use appropriate tools. The function t(x) 5 500 ∙ (1.01)x represents the amount of money in Tracy’s savings account after x years. The function j(x) 5 200 ∙ (1.03)x represents the amount of money in Julio’s savings account after x years. Explain how to use your graphing calculator to determine when the amount in Julio’s account will become greater than the amount in Tracy’s account. Round to the nearest whole year.

LeSSon 23-2The population of Arizona from 1970 to 2000 is shown in the table below. Use the table for Items 61263.

Arizona

Year Resident Population

1970 1,775,3991980 2,716,5461990 3,665,2282000 5,130,6322010 6,392,015

61. Use a graphing calculator to find a function that models Arizona’s population growth. Write the function using the variable n to represent the number of decades since 1970.

62. Use a graphing calculator to create a graph showing the data from the table and the function you wrote in Item 61. Make a sketch of the graph. Is the function a good fit for the data? Explain why or why not.

63. Before the 2012 population count was final, the Census Bureau predicted that Arizona’s population in 2012 would be 6,553,255.

a. Use the function from Item 61 to predict Arizona’s population in 2012. What number did you substitute into the function? Explain.

b. How does your prediction in part a compare to the prediction from the Census Bureau?

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64. Which function is the best model for the data in the table?

x y

0 151 42.52 1083 2644 688

A. y 5 16x 1 2.6 B. y 5 2.6 · 16x

C. y 5 2.6x 1 16 D. y 5 16 · 2.6x

65. Critique the reasoning of others. The function y 5 10,942(1.175)n represents the population of Nate’s hometown, where n is the number of decades since 1960. Nate wants to rewrite the function to show the growth per year. He rewrites the function as y 5 10,942(0.1175)n where n is now the number of years since 1960. Did Nate write the new function correctly? If so, explain why. If not, explain why not and write the correct function.

LeSSon 24-1 66. Copy and complete the table below.

Polynomial 8x2 1 2x3 29 1 23x 2 x2 13

x5

number of Terms

name

Leading Coefficient

Constant Term

Degree

67. Write a polynomial in standard form that has an even number of terms and whose degree is 4.

68. Attend to precision. Which shows the polynomial 3a 1 6a2 2 16 2 2a3 written in standard form?

A. 2a3 1 6a2 2 16 1 3a

B. 22a3 1 6a2 1 3a 2 16

C. 22a3 1 6a2 2 16 1 3a

D. 22a3 1 6a2 1 3a 2 16

69. a. Is the expression 34

x2 1 x5 1 2 a polynomial?

Explain why or why not.

b. Karina says that the expression 15

x4 1 7 2 2x2 is

not a polynomial because 15

is not a whole

number. Do you agree with Karina? Explain why or why not.

LeSSon 24-2 70. Add. Write your answers in standard form.

a. (2x2 1 x 1 4) 1 (6x2 1 x 2 4)

b. (5x2 1 x) 1 (7x3 2 3x 1 9)

c. (6x3 2 6x 1 1) 1 (24x3 1 x2 2 2)

d. x x x x12

6 12 34

8 92 21 2 1 2 1

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71. Write the perimeter of the triangle as a polynomial in standard form.

5x 2 53x

4x 1 2

72. Devon is fencing in a square garden. The length of each side of the garden is 2x 1 3 feet.

a. Show how Devon can use addition to find an expression that represents the total number of feet of fencing he needs for all four sides of the garden. Write the sum in standard form.

b. Compare the expression for the garden’s side length, 2x 1 3, with your answer to part a. What do you notice? Does this make sense? Explain.

73. Which sum is equal to 10x2 1 7?

A. (8x2 1 3x 1 1) 1 (2x2 2 3x 2 8)

B. (8x2 1 3x 2 1) 1 (2x2 2 3x 1 8)

C. (8x2 1 3x 2 1) 1 (2x2 1 3x 1 8)

D. (8x2 2 3x 2 1) 1 (2x2 2 3x 1 8)

74. Make use of structure. Write two polynomials whose sum is:

a. 3x4 1 2x2 1 6

b. x3 2 x 2 7

c. 4.6x4 2 1.5x2

LeSSon 24-3 75. Subtract. Write your answers in standard form.

a. (7x2 1 2x 1 9) 2 (5x2 1 8x 2 1)

b. (x2 2 3x 2 2) 2 (2x2 2 6x 1 2)

c.

1 2 2 2x x x x3

512

25

12

34 2 4 2

d. (23x3 1 4x2 2 7) 2 (29x2 1 10)

76. The perimeter of a rectangle is 12x 1 20 inches and the length is 4x 1 8 inches. Clark and Rachel were asked to find an expression for the width of this rectangle.

a. Clark began by writing (12x 1 20) 2 (4x 1 8). Find this difference and explain Clark’s reasoning.

b. What should Clark do next? Explain.

c. Rachel began by writing (4x 1 8) 1 (4x 1 8). Find this sum and explain Rachel’s reasoning.

d. What should Rachel do next? Explain.

e. Explain how to finish solving the problem to find an expression for the width of the rectangle.

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77. Donna is planning a rectangular flower garden. The total area of the garden will be 5x2 1 7x 1 12 square feet. A square area in the garden measuring x2 1 6x 1 9 square feet will contain flowers, and the rest of the garden will contain vegetables. Write an expression for the area of the garden that will contain vegetables.

78. Which difference is equal to 3x2 1 6x 2 6?

A. (7x2 1 2x 2 3) 2 (4x2 2 8x 1 3)

B. (7x2 1 2x 2 3) 2 (4x2 2 8x 2 3)

C. (7x2 2 2x 2 3) 2 (4x2 2 8x 1 3)

D. (7x2 2 2x 2 3) 2 (4x2 2 8x 2 3)

79. Reason abstractly. Is it possible for the difference

of two polynomials to be x1

? If so, give an example

of two polynomials whose difference is x1

. If not, explain why not.

LeSSon 25-1

80. Find each product. Write your answers in standard form.

a. (x 1 3)(x 2 7)

b. (2x 1 2)(x 1 9)

c. (x 2 1)(3x 1 1)

d. (x 2 5)(x 2 4)

81. Which expression represents the area of the rectangle?

5x 1 2 cm

7x 1 1 cm

A. 12x 1 3 cm2

B. 24x 1 6 cm2

C. 35x2 1 12x 1 2 cm2

D. 35x2 1 19x 1 2 cm2

82. Each product below contains an error. Explain how you can tell that the products are incorrect without multiplying. Then identify and correct each error.

a. (x 1 8)(x 1 7) 5 2x2 1 15x 1 56

b. (x 2 1)(x 2 12) 5 x2 2 13x 2 12

c. (2x 2 2)(x 1 5) 5 10x 2 10

83. Find the missing number in each product. Show that your answer is correct.

a. (x 1 5)(x 1 ) 5 x2 1 14x 1 45

b. (x 1 3)(x 2 ) 5 x2 2 3x 2 18

c. (2x 2 7)(x 1 ) 5 2x2 2 3x 2 14

d. (3x 2 1)(x 2 ) 5 3x2 2 25x 1 8

84. Make use of structure. As part of his math homework, Huong must show that (x 1 45)(x 2 80) 2 (x 2 80)(x 1 45) 5 0. Huong does not want to multiply the binomials because the numbers are large. Describe how Huong can show that the expression is equal to 0 without multiplying the binomials.

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LeSSon 25-2 85. Find each product. Write your answers in standard

form.

a. (x 1 5)(x 2 5)

b. (4x 1 2)(4x 2 2)

c. (x 1 7)2

d. (6x 2 5)2

86. Which product is equal to x2 2 6x 1 9?

A. (x 2 3)2 B. (3x 2 2)2

C. (x 1 3)(x 2 3) D. (3x 1 2)(3x 2 2)

87. Ginny says that the area of this quadrilateral is 9x2 1 42x 1 49 square units. What assumption is Ginny making?

3x 1 7

88. Critique the reasoning of others. Shirley says that the product (x 2 15)(x 1 15) is not a difference of two squares because the product is not in the form (a 1 b)(a 2 b). Explain to Shirley why she is incorrect.

LeSSon 25-3 89. Find each product. Write your answers in standard

form.

a. 5x(3x 1 1)

b. (x2 1 3)(2x 2 5)

c. (x 2 1)(4x2 1 10x 1 6)

d. 4x2(x 1 8)(x 2 8)

e. (x 2 2)(x 1 5)(7x 2 4)

90. Which product is equal to x3 2 5x?

A. (x 2 5)3 B. x(x 2 5)

C. x(x2 2 5) D. x(x2 2 5x)

91. The formula for the area of a triangle is A 5 12

bh.

Cole and Brenda are finding a polynomial that represents the area of the triangle below. Cole plans

to multiply 12

by x 1 1 and then multiply the result

by 2x 2 6. Brenda plans to multiply 12

by 2x 2 6

and then multiply the result by x 1 1. Explain why Brenda’s solution method might be better.

2x 2 6

x 1 1

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92. Express regularity in repeated reasoning. For parts a–d, find the degree of each polynomial. Then find their product and the degree of the product. Organize your results in a table.

a. Polynomial 1: 4x 2 3 Polynomial 2: x2 1 2

b. Polynomial 1: 5x4 2 2 Polynomial 2: x2 2 4x 1 6

c. Polynomial 1: 6x3

Polynomial 2: x3 1 x 2 2

d. Polynomial 1: 22x4 1 3x Polynomial 2: x 2 1

e. When two polynomials are multiplied, what is the relationship between the degree of each polynomial and the degree of their product?

LeSSon 26-1 93. For which polynomial is the GCF of the terms 3x?

A. 3x2 1 3x 1 3

B. 6x2 1 12x 1 36x

C. 9x3 1 3x2 2 12x

D. 3x3 1 6x2 1 x 1 3

94. Factor each polynomial.

a. 5x 2 30

b. 6x2 2 3x 1 21

c. 24x3 1 18x2 2 36x

d. 6x6 2 9x4 1 3x2

95. Model with mathematics. Adam is planning a rectangular patio that will have an area of 16x2 1 20x square feet. The length of the patio will be x 1 5 feet. Write an expression to represent the width of the patio.

96. Give an example of a polynomial with at least three terms that cannot be factored by factoring out the GCF.

97. The length of the side of a square is represented by the expression 2x 1 4. When Carlos is asked to write an expression for the perimeter of the square with the GCF factored out, he writes 4(2x 1 4). Is Carlos correct? If so, explain why. If not, explain Carlos’s error and give the correct answer.

LeSSon 26-2

98. Factor completely.

a. 4x2 2 25

b. 9x2 1 6x 1 1

c. x2 2 4x 1 4

d. 36x2 2 4

99. What factor would you need to multiply by (5x 2 1) to get 25x2 2 1?

A. 5 B. x

C. 5x 1 1 D. 5x 2 1

100. Sergio claims that x2 2 12x 2 36 is a perfect square trinomial. Explain how you can tell by examining the polynomial that Sergio is incorrect.

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101. Make sense of problems. Alison has a square carpet whose area is 9x2 1 12x 1 4 square feet. Karl has a square carpet whose side length is x 1 6 feet. Find a value of x for which the areas of the carpets are equal. What is the area of each carpet for this value of x? Explain how you found your answers.

LeSSon 27-1 102. Factor each trinomial. Write your answer as a

product of two binomials.

a. x2 2 x 2 20

b. x2 1 9x 1 18

c. x2 2 8x 1 12

d. x2 1 2x 2 15

103. Which trinomial cannot be factored?

A. x2 1 3x 2 4 B. x2 1 4x 1 3

C. x2 1 4x 2 3 D. x2 2 4x 1 3

104. For the trinomial x2 1 bx 2 8, give all values of b for which the trinomial can be factored. Explain how you know that you have found all possible answers.

105. Reason abstractly. Jackie says that if the factored form of a trinomial is (x 1 1)(x 1 c) for a positive number c, then c is the constant term of the trinomial and c must be a prime number because its only factors are 1 and c. Is Jackie correct? If so, explain why. If not, give a counterexample to disprove Jackie’s claim.

LeSSon 27-2 106. Factor each trinomial completely.

a. 5x2 1 13x 2 6

b. 3x2 2 2x 2 8

c. 10x2 1 17x 1 3

d. 6x2 2 16x 1 10

107. An architect represents the area of a rectangular window with the expression 28x2 1 5x 2 12. Factor this trinomial to find possible expressions for the length and the width of the window.

108. The trinomial 6x2 1 bx 1 12 can be factored. Which statement is true?

A. The value of b could be an even number.

B. The value of b cannot be greater than 72.

C. The value of b must be positive.

D. The value of b must be a multiple of 2, 3, or 6.

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109. Give an example of a trinomial for which one of the factors is 7x 2 5. Explain how you found the trinomial.

110. Critique the reasoning of others. Gordon says that when 3x2 1 15x 2 42 is factored completely, there are three factors. Holly says there are two factors. Who is correct? Explain. What error might the other student have made?

LeSSon 28-1 111. Simplify each expression.

a. x x xx

4 14 102

4 3 2

22 1

b. x

x x6302

1

1 2

c. 1 1

2

x xx

11 243

2

112. Taina correctly simplified the rational expression shown below.

x x xx

18 24 33

6 5 4

41 2

Which term appears in Taina’s simplified expression?

A. 8x2 B. 6x

C. 2x D. 21

113. Kevin says that for the rational expression x

x x1

5 621

1 1, x cannot equal 21, 22, or 23. Is

Kevin correct? If so, explain why. If not, describe Kevin’s error.

114. a. Make use of structure. Write a rational

expression that simplifies to xx

25

2

1. Explain

how you found your answer.

b. How many possible correct answers are there to part a? Explain.

115. A catering service charges $16 for each guest’s meal plus a flat fee of $500. Write a rational expression for the cost per guest for an event with g guests.

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LeSSon 28-2 116. Simplify by using long division.

a. x xx

8 26 154 3

21 1

1

b. (12x2 1 18x 1 5) 4 6x

c. 2 1

1 2

x xx x

5 16 13

2

2

117. Model with mathematics. The area of a rectangular flower bed is 2x2 1 x 1 20 square feet. The length of the flower bed is x2 1 2x 1 4 feet.

a. Write an expression for the width of the flower bed.

b. What are the length, width, and area of the flower bed when x 5 2?

118. Andy was asked to simplify each expression below using long division. For which expression should he have a remainder?

A. x x6 3 9

3

21 1

B. x x

x4 8 6

2

21 1

C. x x x

x9 15 27

3

3 21 1

D. x x x

x25 10 15

5

4 3 2

21 1

119. Identify and correct the error in each division.

a. )2 2 1

2 1

1

12

x x x

x x

xx

1 2 2 42 2

0 4

2 41

3

3 2

2

b. )2 1 2

2 1

2

12

2

x x x

x x

x

xxx

5 8 155

13 15

13 155

2

2

LeSSon 28-3 120. Multiply or divide.

a. 1

?1x

xxx2

3 62

3

b. 2 2

1? 2

x xx

x3 10

( 2)( 4)

2

22

c. 1

1 24

1 1

1

x xx x

x xx

2 63 4

2 7 34

2

2

2

d. 2 2

2 24

1 1

x xx x x x2 16 5 1

16 19 3

2

2 2

121. The figure shows a rectangular prism. The area of the rectangular face ABCD is x2 1 8x 1 15. The length AB is x 1 5 and the length of BF is x 1 2. Is it possible that BCFG is a square? Explain.

A B

CD

H G

FE

2

2

18

Name class date


122. Which product is equal to x 2 1?

A. 2 1

1 1?

1 2

1

x xx x

x xx

7 125 6

21

2

2

2

B. 1 1

1 1?

1 2

1

x xx x

x xx

7 125 6

21

2

2

2

C. 2 1

1 1?

1 2

1

x xx x

x xx

7 125 6

24

2

2

2

D. 1 1

1 1?

1 2

1

x xx x

x xx

7 125 6

24

2

2

2

123. Construct viable arguments. In the quotient a

x x( 7)3

( 7)2 214

1, a represents a real number.

Tony says that if the quotient is negative, then a must be negative. Is he correct? If so, explain why. If not, explain why not and correct Tony’s error.

LeSSon 28-4 124. Which pair of expressions has a least common

multiple that is the product of the expressions?

A. x 1 4 and x 2 4

B. x 2 4 and x2 2 6x 1 8

C. x 1 4 and x2 2 16

D. x 2 4 and (x 2 4)2

125. Roy subtracted 2

1

2

xx

xx

23

43

− as shown below.

Identify and correct Roy’s error.

22

1

2

xx

xx

23

43

2 1

2

x xx

2 43

1

2

xx

43

126. Add or subtract. Simplify your answers if possible.

a. x32

1 x32

b. xx 12

2 x112

c. x

x x6

5 622

1 1 2

x421

d. xx21622

1 x

x2

42

127. Becky and Shannon each added x6

56

12 as shown

below. Whose solution is correct? Explain.

Becky Shannon

x6

56

12

x6

11

56

?2

21

2

x6

56

2

21

2

x 56

2 1

2

x56

2

2

x6

56

12

x6

56

11

12

?2

2

x6

56

12

x ( 5)6

1 2

x 562

19

Name class date


128. Make sense of problems. On a hike, Rowena ran for a total of 5 miles and walked for a total of 5 miles. She ran at a rate that was twice as fast as her walking rate of r miles per hour. Write and simplify an expression for the total amount of time that Rowena walked and ran on her hike. What was the total time of her hike if she walked at a rate of 3 miles per hour?

name class date algebra 1 unit 4 practicespringboard algebra 1, unit 4 practice lesson 21-1 34. tell...

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