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Algebra 1 Unit 4 Practice
Lesson 19-1
Lesson 19-2
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?
7. Assume that x fi 0. For what value of y will 5xy
always be equal to 5? Explain your answer.
A.324
B.37
C.317
D.360
9a24b3
8. Simplify and write the expression 25 27 without
3a b
negative powers.
2. Arrange the expressions in order from least
to greatest.
28
2
2
a.4 ? 4 b. 2
2
c.
75
73
d.33 ? 3
9. For what value of n is 4m2n 5
M
, where D is
V
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?
3. The formula for density is D 5
4
?
m5
1
5
A. 25
B.
1
C. 2 5
D.5
10. For what value of a is b3 ? ba 5 1? Justify your
answer.
9
4. Simplify the expression
2
x 5 ?x 5
x
1
5
.
11. Reason abstractly. Determine whether the
statement below is always, sometimes, or never
true. Explain your reasoning.
5. Write an expression containing multiplication and
division that simplifies to y4.
If x is a positive integer, then the value of a2x is
negative.
6. Critique the reasoning of others. Nestor says that
65 6 8
the value of 2 ? 3 is 615. Is he correct? If so,
6 6
explain why. If not, identify Nestor’s error and give
the correct value.
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SpringBoard Algebra 1, Unit 4 Practice
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b. Does Brooke’s method always work? Explain
why or why not.
Lesson 19-3
12. Simplify and write each expression without
negative powers.
18
 2
a.  x 3   
 2 1
b.  x 3 y 6 


18
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.
c.(a3b2c22)4(abc4)(ab)
1
 x4  2
13. Which expression is not equal to  2  ?
x 
A. x
B.2x
1
C. ( x 2 ) 2 D.
x2
x
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.
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,
22
 a4 
when asked to simplify  2  , Brooke begins by
2
 3b 
 3b 2 
writing  4  .
 a 
18. For each radical expression, write an equivalent
expression with a fractional exponent.
a. 7
22
 a 
a.Simplify  2  by using Brooke’s method.
 3b 
Then simplify without using Brooke’s method.
How do your answers compare?
4
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SpringBoard Algebra 1, Unit 4 Practice
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Lesson 20-2
1
19. a. What is the value of 27 3?
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?
b. Make use of structure. How can you use your
answer to part a to help you find the value of n
2
for which 27 n 5 9? Find the value of n and
explain your reasoning.
24. Write 12 + 3 48 + 2 27 in simplest radical form.
State whether the result is rational or irrational.
20. Which of the following expressions is not
3
4
equivalent to (16 y ) ?
A. 4 16 y 3 3
C. 8 y 4
B. 8 4 y 3
3
3
D.16 4 y 4
25. Find the value of a for which 5 5 2 a 5 3 5 .
Explain how you found your answer.
21. a. What is 1? What is 3 1 ? Explain your answers.
b. Let n be a positive integer. What is the value
1
of 1n? Explain your answer.
26. Which is the sum of 2 50 and 8 ?
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.
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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.
date
 3   12 
in simplest form. Is the
31. a.Write 
 2   32 
result rational or irrational?
a. 7 8 2 5 2 5 2 6
b. 9 5 1 5 5 9 10
c. 8 3 2 3 1 5 3 5 8 1 5 3
b. What can you conclude from your answer to
part a about whether the irrational numbers are
closed under multiplication? Explain.
28. Ted is fencing in an area composed of a rectangle
and a right triangle as shown below.
2 27
x
2 12
32. Lorraine solved the equation 3 x ? 24 5 12 6 and
found that x 5 4. Verify that Lorraine’s solution is
correct.
243
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
form?
A. 3 2
C.
3 14
7
3 2
in simplest radical
7
B. 21 2
D.
33. Critique the reasoning of others. Deanna says
1
that
is in simplified form. Is she correct? If so,
5
explain why. If not, correct her mistake.
3 21
7
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?
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37. The terms in a geometric sequence alternate
between positive and negative numbers. What
must be true about this sequence?
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. The first term is negative.
B. The first term is greater than the second term.
2 1
a. 24, 4, , , …
3 9
C. The common ratio is between 0 and 1.
D. The common ratio is negative.
b. 1, 4, 9, 16, 25, …
Lesson 21-2
c. 3, 4.7, 6.3, 7.9, …
38. Write the first five terms of the geometric sequence
represented by the recursive formula below.
d.2, 26, 18, 254, …
 f (1) 5 2

1

 f (n) 5 2 f (n 2 1)
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.
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. The principal represents Stage 1. Make a tree
diagram and a table of values to represent this
situation.
a. Write a recursive formula that represents this
situation.
b. Can this situation be represented by a geometric
sequence? If so, identify the common ratio. If
not, explain why not.
b. Write an explicit formula that represents this
situation.
c. How many teachers will receive phone calls at
Stage 4?
36. Consider the sequence 12, 3, x, ….
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?
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.
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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.
date
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.
Use the geometric sequences below for Items 41 and 42.
Sequence 1
Sequence 2
an 5 5 ? 2n21
a1 5 2

an 5 5an21
b. Sally wants to calculate the value of her house in
1995. What number should Sally substitute for t
in the function? Explain.
41. Which statement is incorrect?
A. The terms in Sequence 2 increase more quickly
than the terms in Sequence 1.
c. Find the value of Sally’s house in 1995.
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.
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.
42. Persevere in solving problems. How many terms
in Sequence 1 are less than 500? Explain how you
found your answer.
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50. Compare the graph of an exponential growth
function to the graph of an exponential decay
function. Describe the similarities and differences.
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.
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.
Lesson 22-2
47. Identify the constant factor for the exponential
x
 1
function y 5   . How can you use the constant
 3
factor to tell whether the function represents
exponential growth or exponential decay?
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.
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?
Lesson 22-3
52. Without graphing, tell which function increases
more slowly. Justify your answer.
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
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f(x) 5 99x
g(x) 5 9x
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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.
53. Use a graphing calculator to graph the function
x
1  1
g(x) 5   .
4  2
a. Identify the values of a and b (from f(x) = abx),
and describe their effects on the graph.
x
 1
b.Graph f(x) 5   on the same screen as the
 2
graph of g(x). Describe the similarities and
differences between the graphs.
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.
54. Which function increases the fastest?
A. y 5 14x
B. y 5 23 · 17x
C. y 5 120x
D. y 5 2275x
y
800
v
600
400
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.
p
200
0
40
80
x
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.
a. Whose plan will increase club membership
more quickly? Use a graph to support your
answer.
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
b. Whose plan would you recommend? Explain.
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57. Identify the constant factor in Sanjay’s function
and explain how it is related to his interest rate.
date
Lesson 23-2
The population of Arizona from 1970 to 2000 is shown
in the table below. Use the table for Items 61263.
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.
Year
1970
1980
1990
2000
Arizona
Resident Population
1,775,399
2,716,546
3,665,228
5,130,632
2010
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.
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
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.
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.
© 2014 College Board. All rights reserved.
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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SpringBoard Algebra 1, Unit 4 Practice
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64. Which function is the best model for the data in
the table?
x
0
1
2
3
4
date
67. Write a polynomial in standard form that has an
even number of terms and whose degree is 4.
y
15
42.5
108
264
688
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
A. y 5 16x 1 2.6
B. y 5 2.6 · 16x
C. 22a3 1 6a2 2 16 1 3a
C. y 5 2.6x 1 16
D. y 5 16 · 2.6x
D. 22a3 1 6a2 1 3a 2 16
5
3
69. a.Is the expression x2 1 1 2 a polynomial?
x
4
Explain why or why not.
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.
1
b. Karina says that the expression x4 1 7 2 2x2 is
5
1
not a polynomial because is not a whole
5
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)
Lesson 24-1
66. Copy and complete the table below.
Polynomial
8x2 1 2x3
29 1 23x 2 x2
1 5
x
3
b.(5x2 1 x) 1 (7x3 2 3x 1 9)
Number of
Terms
Name
Leading
Coefficient
Constant
Term
Degree
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c.(6x3 2 6x 1 1) 1 (24x3 1 x2 2 2)
1 2
 3 2

d.  x 1 6 x 2 12 1  x 2 8 x 1 9
2
 4

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71. Write the perimeter of the triangle as a polynomial
in standard form.
date
Lesson 24-3
75. Subtract. Write your answers in standard form.
a.(7x2 1 2x 1 9) 2 (5x2 1 8x 2 1)
5x 2 5
3x
b.(x2 2 3x 2 2) 2 (2x2 2 6x 1 2)
4x 1 2
3

1  2
1
c.  x 4 1 x 2  2  x 4 2 x 2 2 3
5

2  5
2
72. Devon is fencing in a square garden. The length of
each side of the garden is 2x 1 3 feet.
d.(23x3 1 4x2 2 7) 2 (29x2 1 10)
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.
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. 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.
b. What should Clark do next? Explain.
73. Which sum is equal to 10x2 1 7?
c. Rachel began by writing (4x 1 8) 1 (4x 1 8).
Find this sum and explain Rachel’s reasoning.
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)
d. What should Rachel do next? Explain.
74. Make use of structure. Write two polynomials
whose sum is:
e. Explain how to finish solving the problem to
find an expression for the width of the rectangle.
a. 3x4 1 2x2 1 6
b. x3 2 x 2 7
c. 4.6x4 2 1.5x2
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SpringBoard Algebra 1, Unit 4 Practice
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81. Which expression represents the area of the
rectangle?
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.
7x 1 1 cm
5x 1 2 cm
A.12x 1 3 cm2
B.24x 1 6 cm2
C.35x2 1 12x 1 2 cm2
78. Which difference is equal to 3x2 1 6x 2 6?
D.35x2 1 19x 1 2 cm2
A.(7x2 1 2x 2 3) 2 (4x2 2 8x 1 3)
B.(7x2 1 2x 2 3) 2 (4x2 2 8x 2 3)
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.
C.(7x2 2 2x 2 3) 2 (4x2 2 8x 1 3)
D.(7x2 2 2x 2 3) 2 (4x2 2 8x 2 3)
a.(x 1 8)(x 1 7) 5 2x2 1 15x 1 56
79. Reason abstractly. Is it possible for the difference
1
of two polynomials to be ? If so, give an example
x
1
of two polynomials whose difference is . If not,
x
explain why not.
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
Lesson 25-1
b.(x 1 3)(x 2 ) 5 x2 2 3x 2 18
80. Find each product. Write your answers in standard
form.
a.(x 1 3)(x 2 7)
c.(2x 2 7)(x 1 ) 5 2x2 2 3x 2 14
b.(2x 1 2)(x 1 9)
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.
c.(x 2 1)(3x 1 1)
d.(x 2 5)(x 2 4)
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SpringBoard Algebra 1, Unit 4 Practice
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Lesson 25-2
Lesson 25-3
85. Find each product. Write your answers in standard
form.
89. Find each product. Write your answers in standard
form.
a.(x 1 5)(x 2 5)
a.5x(3x 1 1)
b.(4x 1 2)(4x 2 2)
b.(x2 1 3)(2x 2 5)
c.(x 1 7)2
c.(x 2 1)(4x2 1 10x 1 6)
d.(6x 2 5)2
d.4x2(x 1 8)(x 2 8)
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)
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)
1
91. The formula for the area of a triangle is A 5 bh.
2
Cole and Brenda are finding a polynomial that
represents the area of the triangle below. Cole plans
1
to multiply by x 1 1 and then multiply the result
2
1
by 2x 2 6. Brenda plans to multiply by 2x 2 6
2
and then multiply the result by x 1 1. Explain why
Brenda’s solution method might be better.
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
2x 2 6
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.
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96. Give an example of a polynomial with at least three
terms that cannot be factored by factoring out the
GCF.
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
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.
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?
Lesson 26-2
A.3x2 1 3x 1 3
B. 6x2 1 12x 1 36x
98. Factor completely.
C.9x 1 3x 2 12x
3
2
a.4x2 2 25
D.3x3 1 6x2 1 x 1 3
b. 9x2 1 6x 1 1
94. Factor each polynomial.
a.5x 2 30
c. x2 2 4x 1 4
b.6x2 2 3x 1 21
d.36x2 2 4
c.24x3 1 18x2 2 36x
99. What factor would you need to multiply by
(5x 2 1) to get 25x2 2 1?
d.6x6 2 9x4 1 3x2
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.
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.
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SpringBoard Algebra 1, Unit 4 Practice
Name
class
date
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.
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 27-2
106. Factor each trinomial completely.
Lesson 27-1
a.5x2 1 13x 2 6
102. Factor each trinomial. Write your answer as a
product of two binomials.
b.3x2 2 2x 2 8
a. x2 2 x 2 20
c.10x2 1 17x 1 3
b. x2 1 9x 1 18
d.6x2 2 16x 1 10
c. x2 2 8x 1 12
d. x2 1 2x 2 15
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.
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.
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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SpringBoard Algebra 1, Unit 4 Practice
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class
109. Give an example of a trinomial for which one of
the factors is 7x 2 5. Explain how you found the
trinomial.
date
112. Taina correctly simplified the rational expression
shown below.
18 x 6 1 24 x 5 2 3x 4
3x 4
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?
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 11
, x cannot equal 21, 22, or 23. Is
2
x 1 5x 1 6
Kevin correct? If so, explain why. If not, describe
Kevin’s error.
114. a. Make use of structure. Write a rational
x 22
expression that simplifies to
. Explain
x 15
how you found your answer.
Lesson 28-1
b. How many possible correct answers are there
to part a? Explain.
111. Simplify each expression.
a.
4 x 4 2 14 x 3 1 10 x 2
2x 2
b.
x 16
x 1 x 2 30
c.
x 2 1 11x 1 24
32 x
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.
2
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SpringBoard Algebra 1, Unit 4 Practice
Name
class
date
119. Identify and correct the error in each division.
Lesson 28-2
116. Simplify by using long division.
a.
2x 21
2
8 x 1 26 x 1 15
4x 1 3
)
3
a. x 212 2 x 2 2 x 1 4
22 x 3 1 2 x 2
014
b.(12x2 1 18x 1 5) 4 6x
c.
4
x 21
x1
2
5 x 2 16 x 1 1
x2 1 x 2 3
)
13x 2 15
x 25
2
b. x 252 x 1 8 x 2 15
2x 2 1 5 x
13x 215
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.
Lesson 28-3
120. Multiply or divide.
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.
6 x 2 1 3x 1 9
3
B.
4 x 2 1 8x 1 6
2x
C.
9 x 3 1 15 x 2 1 27 x
3x
a.
3x 1 6
x3
?
x 12
2x
b.
x 2 2 3x 2 10
? ( x 2 2 4)
( x 1 2)2
c.
2x 2 1 6x
2x 2 1 7 x 1 3
4
x 2 1 3x 2 4
x 14
d.
2 x 22x 21
1
4 2
2
6 x 2 5 x 21 6 x 119 x 1 3
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
25 x 4 1 10 x 3 1 15 x 2
.
D
5x 2
D
B
C
F
E
H
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G
SpringBoard Algebra 1, Unit 4 Practice
Name
class
122. Which product is equal to x 2 1?
date
x 14
2x
−
as shown below.
x 23 x 23
Identify and correct Roy’s error.
x 14
2x
2
x 23 x 23
125. Roy subtracted
A.
x 2 2 7 x 1 12 x 2 1 x 22
?
x 2 1 5x 1 6
x 11
B.
x 2 1 7 x 1 12 x 2 1 x 22
?
x 2 1 5x 1 6
x 11
C.
x 2 2 7 x 112 x 2 1 x 22
?
x 2 1 5x 1 6
x 14
x 14
x 23
D.
x 2 1 7 x 112 x 2 1 x 22
?
x 2 1 5x 1 6
x 14
126. Add or subtract. Simplify your answers if possible.
2x 2 x 1 4
x 23
a.
3x 3x
1
2
2
x
1
2
x 21 x 21
x 26
4
c. 2
2
x 1 5x 1 6 x 1 2
2x
2x
d. 2
1
x 2 16 x 2 4
b.
123. Construct viable arguments. In the quotient
a
3
, a represents a real number.
4
2
( x 1 7)
( x 1 7)2
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.
x
5
127. Becky and Shannon each added 6 1 26 as shown
below. Whose solution is correct? Explain.
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 x 2 16
2
D. x 2 4 and (x 2 4)2
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18
Becky
Shannon
x
5
1
26
6
x
5
1
26
6
x 21
5
1
?
6 21 26
5 21
x
1
?
26 21
6
5
2x
1
26
26
x 25
1
6
6
2x 1 5
26
x 1 (25)
6
5 2x
26
x 25
6
SpringBoard Algebra 1, Unit 4 Practice
Name
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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?
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SpringBoard Algebra 1, Unit 4 Practice