Algebra 1

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Algebra Part 1/2 Chapter 1: Tools of Algebra

Chapter 2: Functions, Equations, and Graphs

Chapter 3: Linear Systems

Chapter 4: Matrices

Chapter 5: Quadratic Equations and Functions

Chapter 6: Polynomials and Polynomial Functions

Chapter 7: Radical Functions and Rational Exponents

Part 2/2 Chapter 8: Exponential and Logarithmic Functions

Chapter 9: Rational Functions

Chapter 10: Quadratic Relations

Chapter 11: Sequences and Series

Chapter 12: Probability and Statistics

Chapter 13: Periodic Functions and Trigonometry

Chapter 14: Trigonometric Identities and Equations


Tools of Algebra

1. Evaluate 2a 5b 3a for a = 2 and b = 5.

23

27

35

15

2. Solve 4|3w + 1| 7 = 17.

,

,

,

,

3. One side of a triangle is 4 cm longer than the shortest side and 2 cm shorter than the longest side. The perimeter is 38 cm. Find the dimensions of the triangle.

8, 12, 18

9 , 13 , 15

9.3, 13.3, 15.3

11, 13, 14

4. Solve = b for x. Find any restrictions.

x = nb + 3; n 0

x = n(b + 3) 3; n 0

x = + 3; n 0

x = nb 6; n 0 5. The expression 200 16t2 models the height in feet of an object that falls from an initial height of 200 feet t seconds after being dropped. Find the height of the object after 2.5 seconds.

60 feet

80 feet

120 feet

100 feet


6. Simplify the expression |3| |5|.

8

8

2

2

7. Graph the solution of 7x + 3 3 or 6x 8 > 16.

8. Suppose you take a true/false quiz that contains five true/false questions. Let an even digit

represent a correct answer and an odd digit represent an incorrect answer. In the five trials shown,

how often did you get at least 3 questions correct? (Note: Assume there is a 0 before the number 4908 in the first trial.)

2

3

4

5

9. Graph the solution of 7x + 4 11 and 10 3x > 1.


10. Replace with , or = to make the sentence 12 18 true.

<

>

=

11. Solve 3(x 5) 15 = 48.

20

26

4

14

12. Solve bx cx = 7 for x.

x = 7(b c)

x = c

x =

x =

13. Graph the solution of 2x 5 > 3 and 3x + 7 19.

14. Solve 11x 7 = 7x + 13.

x =

x = 5

x = 2

x = 7


15. Solve |8 m| + 4 = 21.

9, 25

25, 9

25, 9

9, 25

16. Brown is the dominant eye color for human beings. If a father and mother each carry a gene for brown eyes and a gene for blue eyes, what is the probability of their having a child with brown eyes?

1

17. A jar contains 12 blue marbles, 8 red marbles, and 20 green marbles. You pick one marble from the jar. Find the theoretical probability, P(blue or green).

18. Simplify 9a (6a 8b) by combining like terms.

3a 8b

3a + 8b

3a + 8b

3a 8b 19. Jean has $20 and is going shopping to buy 3 scarves and as many pairs of earrings as possible.

How many pairs of earrings can she buy if scarves cost $2.55 each and earrings cost $2.99 a pair?

Jean can buy less than 4 pairs of earrings.

Jean can buy at most 4 pairs of earrings.

Jean can buy more than 4 pairs of earrings.

Jean can buy at least 4 pairs of earrings.


20. Solve V = pir2h for h.

h =

h =

h =

h =

Tools of Algebra

1. Evaluate 2a 5b 3a for a = 2 and b = 5.

CORRECT: 23 2. Solve 4|3w + 1| 7 = 17.

CORRECT: , 3. One side of a triangle is 4 cm longer than the shortest side and 2 cm shorter than the longest

side. The perimeter is 38 cm. Find the dimensions of the triangle.

CORRECT: 9 , 13 , 15

4. Solve = b for x. Find any restrictions.

CORRECT: x = nb + 3; n 0 5. The expression 200 16t2 models the height in feet of an object that falls from an initial height of 200 feet t seconds after being dropped. Find the height of the object after 2.5 seconds.

CORRECT: 100 feet 6. Simplify the expression |3| |5|.

CORRECT: 2 7. Graph the solution of 7x + 3 3 or 6x 8 > 16.

CORRECT: 8. Suppose you take a true/false quiz that contains five true/false questions. Let an even digit represent a correct answer and an odd digit represent an incorrect answer. In the five trials shown,

how often did you get at least 3 questions correct? (Note: Assume there is a 0 before the number


4908 in the first trial.)

CORRECT: 3 9. Graph the solution of 7x + 4 11 and 10 3x > 1.

CORRECT: 10. Replace with , or = to make the sentence 12 18 true.

CORRECT: > 11. Solve 3(x 5) 15 = 48.

CORRECT: 26 12. Solve bx cx = 7 for x.

CORRECT: x = 13. Graph the solution of 2x 5 > 3 and 3x + 7 19.

CORRECT: 14. Solve 11x 7 = 7x + 13.

CORRECT: x = 5 15. Solve |8 m| + 4 = 21.

CORRECT: 9, 25 16. Brown is the dominant eye color for human beings. If a father and mother each carry a gene

for brown eyes and a gene for blue eyes, what is the probability of their having a child with brown eyes?

CORRECT: 17. A jar contains 12 blue marbles, 8 red marbles, and 20 green marbles. You pick one marble from the jar. Find the theoretical probability, P(blue or green).

CORRECT: 18. Simplify 9a (6a 8b) by combining like terms.

CORRECT: 3a + 8b 19. Jean has $20 and is going shopping to buy 3 scarves and as many pairs of earrings as possible. How many pairs of earrings can she buy if scarves cost $2.55 each and earrings cost $2.99 a pair?

CORRECT: Jean can buy at most 4 pairs of earrings.


20. Solve V = pir2h for h.

CORRECT: h =

Functions, Equations, and Graphs

1. What are the domain and range of the function below?

Domain: {2, 1}; Range: {3, 1, 0, 1, 2}

Domain: {3, 1, 0, 1, 2}; Range: {2, 1}

Domain: {all real numbers}; Range: {2, 1}

Domain: {2, 1}; Range: {all real numbers}

2. What is the vertex of the graph of y = |x + 3|?

(3, 0)

(0, 3)

(0, 3)

(3, 0)


3. What is the graph of y > |x 3|?

4. What point has coordinates of (5, 4)?


N

M

L

O

5. What is the graph of 0.4x + 0.1y 1?


6. In chemistry Charles Law states that if the pressure remains constant, the volume of an ideal

gas varies directly with the temperature in degrees Kelvin. Suppose a gas has a volume of 100 mL at 250K. Which proportion could you use to find the volume v of the gas if the temperature

decreases to 150K?

=

=

=

=

7. What is the graph of y x?


8. What is the graph of y = |x| |2x|?

not here

9. The value of y varies directly with x. If y = 64 when x is 16, what is x when yis 288?

72

864

1,152

96


10. What is the slope of this line?

11. Which line, having the given slope and passing through the given point, represents a direct variation?

m = ; (9, 6)

m = ; (6, 9)

m = ; (9, 6)

m = ; (9, 6)

12. What is an equation of the line whose slope is and which contains the point (3, )?

3x 6y = 13

3x 5y = 7

3x 6y = 5

6x 3y = 13


13. What is an equation of the inequality?

y |x 1| 2

y > |x + 1| 2

y |x + 1| 2

y |x 1| 2

14. Identify the set of linear equations and graph for y = | x + 6| + 3.

y = x 3, if x 4; y = x + 9, if x > 4

y = x 9, if x 4; y = x + 3, if x < 4


y = x + 9, if x 4; y = x 3, if x > 4

y = x + 3, if x 4; y = x 9, if x < 4 15. The length of an object's shadow varies directly with its height. A boy who is 4 ft 6 in. tall casts a shadow that is 6 ft 3 in. long. What is the height of a tree that casts a shadow that is 25 ft long?

18 ft

20 ft

16 ft

216 ft

16. What is an equation of a line that is parallel to the line whose equation is y = x + 1?

y = x + 5

y = x

y = x 1

y = x + 3


17. What is the graph of the equation 5x 2y = 6?


18. The capacity of the gas tank in a car is 16.5 gallons. The car uses 10.5 gallons to travel 178.5

miles. After driving 85 more miles, only 1 gallon of gasoline remains in the tank. Which linear equation and graph model the amount of gas gleft in the tank after driving m miles?

g = m + 16.5

g = m + 16.5

g = m + 16.5


g = m + 16.5 19. This graph represents a translation of the graph of y = |x|. What is the equation of this graph?

y = |x 3|

y = |x 3|

y = |x + 3|

y = |x + 3| 20. What is an inequality for this graph?

y 3x > 5

y 3x 5

y 3x 5


y 3x < 5

Functions, Equations, and Graphs

1. In chemistry Charles Law states that if the pressure remains constant, the volume of an ideal

gas varies directly with the temperature in degrees Kelvin. Suppose a gas has a volume of 100 mL at 250K. Which proportion could you use to find the volume v of the gas if the temperature

decreases to 150K?

CORRECT: = 2. What is an equation for a direct variation whose graph passes through the point (5, 3)?

CORRECT: y = x

3. Which statement describes the relationship between the graph of the equationy = | | and the

graph of the equation y = | | + 4?

CORRECT: The graph of y = | | was shifted 4 units up. 4. What is the slope of the line through the points P(5, 6) and Q(12, 5)?

CORRECT: 5. What is the graph of the equation 4x y = 5?

CORRECT:

6. Which graph represents the equation y = |x|?

CORRECT:


7. What is a point-slope equation of the line through the points (1, 5) and (2, 4)?

CORRECT: y + 5 = 9(x + 1) 8. What is the equation of y = |x| translated 1 unit up and 6 units to the left?

CORRECT: y = |x + 6| + 1 9. What is the graph of y = |3x|?

CORRECT: 10. What is the graph of y > |x 3|?

CORRECT: 11. What are the domain and range of this function?

CORRECT: Domain: x > 0; Range: y > 0 12. Which statement about 5y = 2x is true?

CORRECT: The equation represents a direct variation whose constant of variation is .


13. What is the graph of y > + 1?

CORRECT: 14. What is the constant of variation for the function given in the table?

x y

20 50

42 105

64 160

81 202.5

CORRECT: 2.5 15. Find f(5) if f(x) = 8x 1.

CORRECT: 39 16. You can use a linear model to represent the relationship between the hours worked h and the money earned m. When h = 0, then m = 0. If a person earned $39 after working 6 hours, how much will the person earn after working 25 hours?

CORRECT: $162.50 17. What are the ordered pairs of the relation? Find the domain and range.

CORRECT: {(2, 5), (1, 2), (0, 1), (1, 2), (2, 5)}; domain {2, 1, 0, 1, 2}; range {5, 2, 1}

18. Find an equation for the linear model of the situation below and use it to make a prediction. A

train is traveling north at a constant rate. At 3:00 P.M. it is 55 miles north of a city. At 4:15 P.M. it is 80 miles north of the city. If d represents the distance in miles, and t represents the time in hours, how many miles north of the city will the train be at 5:45 P.M.?


CORRECT: d = 64t + 55; d = 231 miles 19. What is an inequality for this graph?

CORRECT: y 3x 5

20. What is an equation of the line whose slope is and which contains the point (3, )?

CORRECT: 3x 6y = 13

Linear Systems

1. Identify the graph that shows the solution to the system of inequalities.

CORRECT:

2. Solve the system .

CORRECT: (1, 5, 2)


3. Use elimination to find the solution of the system .

CORRECT: (1, 3, 2) 4. A small fish market sells only tuna and salmon. A tuna costs the fish market $0.75 per pound to buy and $2.53 per pound to clean and package. A salmon costs the fish market $3.00 per pound to buy and $2.75 per pound to clean and package. The fish market makes $2.50 per pound profit for each tuna it sells and $2.80 per pound profit for each salmon it sells. The fish market owner can

spend only $159.00 per day to buy fish and $197.34 per day to clean and package the fish. The graph below represents the feasible region for this linear programming model. What are the coordinates of the vertices of the feasible region, and what are the vales of t and s that maximize the objective function?

; P = 2.50t + 2.80s

CORRECT: (0, 0), (0, 53), (71.76, 0), (28, 46); t = 28 and s = 46. 5. What equivalent system of two equations do you get when you use substitution to eliminate y and solve this system of three equations?

CORRECT: 6. How could you solve this system using substitution?

CORRECT: Substitute 2r + 3 for t in the equation 5r 4t = 6. 7. Identify the graph of the system and its solution.


CORRECT: infinite number of solutions 8. Graph the system of inequalities.

CORRECT: 9. Which system has no solution because exactly two of the planes are parallel?

CORRECT: 10. This spreadsheet shows the monthly revenue R and monthly expenses E for a new business. In what month will the revenue equal the expenses if the linear models for R and E are R = 4100x + 300 and E = 1900x + 19900? Assume July is month 1.

CORRECT: March


11. Plot the point (3, 1, 2) in a three-dimensional coordinate system.

CORRECT: 12. A factory can produce two products, x and y, with a profit P approximated byP = 12x + 15y. The production of y can exceed x by no more than 200 units. Moreover, production levels are limited by the formula x + 2y 1000. What production levels yield maximum profit?

CORRECT: x = 1000; y = 0 13. What is the solution of this system?

CORRECT: (2, 3) 14. The objective function for this graph is P = 5x + y. What values of x and ymaximize this function?

CORRECT: (1000, 0): The maximum is 5000. 15. Tasty Bakery sells three kinds of muffins: chocolate chip muffins at 40 cents each, oatmeal

muffins at 45 cents each, and cranberry muffins at 50 cents each. Charles buys some of each kind and chooses three times as many cranberry muffins as chocolate chip muffins. If he spends $8.85

on 19 muffins, how many chocolate chip muffins does he buy?

CORRECT: 3 16. One rental car agency charges $35 per day plus 11 cents per mile to rent a mid-size car.

Another agency charges $42 per day plus 6 cents per mile to rent an equivalent type of car. Let c represent the cost of renting a car from either agency, and m represent the number of miles

traveled daily. Which system of equations represents the cost of renting from each agency? For what value of mare the costs of rental equal?


CORRECT: ; 140 miles 17. Which graph represents this system and its solution?

CORRECT: (3, 4) 18. The equation P = 5x + 3y represents the objective function for this linear programming model. What is the vertex of the feasible region shown below that results in the maximum value of P?

CORRECT: (12, 14)

19. What steps can you use to solve the system using the method of elimination?

CORRECT: Multiply 4x + 5y = 53 by 2 and multiply 7x + 2y = 59 by 5. Then add.

20. Classify the system without graphing.

CORRECT: independent


Matrices

1. Find .

2. What is the determinant of ?

10

20

10

38

3. Identify a23 in .

23

16

21

3

4. Find BA, if A = , and B = , if it is defined. If it is not defined, choose undefined.

undefined

5. Which product is undefined?

A = , B = , C = , D =


AB

BA

DB

CD

6. Find the inverse of , if it exists.

does not exist

7. Solve X = .


8. Write the coordinates of the image after a translation 3 units right and 2 units down.

9. Find 4 .

10. What are the dimensions of ? Identify a12.

3 2, 4

2 3, 3


2 3, 4

3 2, 3

11. Write the coordinates of the image after a dilation three times the original size.

12. Use matrices to represent the vertices of graph f and graph g.


13. What are the coordinates of the image after a rotation of 90?

14. Use an augmented matrix to solve .

(2, 5, 1)

(4, 6, 1)

(4, 5, 6)

none of these

15. What are the dimensions of ?

4 5

7 3

5 4

6 4


16. Find + .

17. Assign a number to each letter of the alphabet and a blank space as shown by the table below:

Matrix A was used to encode the cryptogram below.

A = 14 _ 5 18 18 _ 16 _ 11 4 _ 3 2 1 3 _ 17 9 3 17 20 _ 12 16 Use A1 to decode the cryptogram.

LEAPS AND BOUNDS

GRIND TO A HALT

RUN LIKE THE WIND

IN BROAD DAYLIGHT

18. Use matrices A, B, and C. Find B C if you can.

A = , B = , C =

not possible

19. Determine whether has an inverse. If an inverse matrix exists, find it.


does not exist

20. Determine whether J = and K = are multiplicative inverses.

yes

no

Matrices

1. Find .

CORRECT:

2. What is the determinant of ?

CORRECT: 10

3. Identify a23 in .

CORRECT: 21

4. Find BA, if A = , and B = , if it is defined. If it is not defined, choose undefined.

CORRECT: undefined 5. Which product is undefined?

A = , B = , C = , D =

CORRECT: BA


6. Find the inverse of , if it exists.

CORRECT:

7. Solve X = .

CORRECT: 8. Write the coordinates of the image after a translation 3 units right and 2 units down.

CORRECT:

9. Find 4 .

CORRECT:

10. What are the dimensions of ? Identify a12.

CORRECT: 3 2, 4


11. Write the coordinates of the image after a dilation three times the original size.

CORRECT: 12. Use matrices to represent the vertices of graph f and graph g.

CORRECT: 13. What are the coordinates of the image after a rotation of 90?

CORRECT:


14. Use an augmented matrix to solve .

CORRECT: (2, 5, 1)

15. What are the dimensions of ?

CORRECT: 5 4

16. Find + .

CORRECT: 17. Assign a number to each letter of the alphabet and a blank space as shown by the table below:

Matrix A was used to encode the cryptogram below.

A = 14 _ 5 18 18 _ 16 _ 11 4 _ 3 2 1 3 _ 17 9 3 17 20 _ 12 16 Use A1 to decode the cryptogram.

CORRECT: IN BROAD DAYLIGHT 18. Use matrices A, B, and C. Find B C if you can.

A = , B = , C =

CORRECT:

20. Determine whether J = and K = are multiplicative inverses.

CORRECT: yes


Quadratic Equations and Functions

1. Solve x2 + 6x 2 = 0 by completing the square.

3

3

3

11

2. A rocket is launched from atop a 63-foot cliff with an initial velocity of 95 feet per second. The height of the rocket t seconds after launch is given by the equation h = 16t2 + 95t + 63. Graph the equation to find out how long after the rocket is launched it will hit the ground. Estimate your

answer to the nearest hundredth of a second.

6.54 sec

2.97 sec

2.97 sec

5.94 sec


3. Factor x2 15x + 54.

(x 3)(x 18)

(x + 1)(x + 54)

(x + 9)(x + 6)

(x 9)(x 6)

4. Is y = (x + 3)(x 6) 2x2 linear or quadratic? Identify the quadratic, linear, and constant terms.

linear function linear term: 3x

constant term: 18

quadratic function quadratic term: 2x2

linear term: 3x

constant term: 18

quadratic function quadratic term: x2

linear term: 3x

constant term:18

linear function linear term: 9x

constant term:18

5. Solve 11x + 9 + 5x2 = 0 using the Quadratic Formula.

,

6. Identify the vertex and y-intercept of the graph of f(x) = 4(x + 3)2 + 7.

(3, 7), y-intercept 5





7. A rocket is launched from atop a 54-foot cliff with an initial velocity of 119 feet per second. The height of the rocket t seconds after launch is given by the equation h = 16t2 + 119t + 54. Graph the equation to find out how long after the rocket is launched it will hit the ground. Estimate your answer to the nearest hundredth of a second.

3.72 sec

3.72 sec

7.87 sec

7.44 sec

8. Find the absolute value of |1 + 3i|.

2

10

8



,

10. Solve x2 5x 1 = 0 using the Quadratic Formula. Find the exact solutions. Then approximate

any radical solutions. Round to the nearest hundredth.

; 0.19, 5.19

; 4.79, 0.21

; 0.21, 4.79

; 5.19, 0.19

11. Find a quadratic model for the values in the table.

f(x) = 7x2 9x + 3

f(x) = 3x2 9x + 7

f(x) = 9x2 + 3x + 7

f(x) = 3x2 + 7x 9 12. Write the equation of the parabola with vertex (9, 2) and point (12, 16) in vertex form.

y = (x + 9)2 + 2

y = (x 12)2 16

y = (x 9)2 2

y = 3(x + 12)2 + 16 13. Factor x2 6x + 8.

(x + 4)(x + 2)

(x 4)(x 2)

(x 2)(x 6)

(x + 2)(x + 6)


14. Identify the vertex and the axis of symmetry of the parabola.

vertex: ( , ); axis of symmetry: x =

vertex: ( , ); axis of symmetry: y =

vertex: ( , ); axis of symmetry: x =

vertex: ( , ); axis of symmetry: y = 15. Solve 6x2 = 12. If necessary, round your answer to the nearest hundredth.

6 8.49

1.41

2 3.46

none of these

16. Find the quadratic function y = ax2 + c with a graph that includes (0, ) and (3, ).

y = 4x2

y = 4x2 +

y = 4x2

none of these

17. Solve 5x2 11x + 2 = 0 using the Quadratic Formula.

2,

2,

,

,


18. Factor t2 10t + 25.

(t 25)(t + 1)

(t 5)(t + 5)

(t + 5)2

(t 5)2

19. Rewrite y = x2 + 5x 1 in vertex form.

y = (x )2 +

y = (x )2

y = (x + )2

y = (x )2 + 20. Graph the quadratic function with vertex (3, 1) and y-intercept 2.


Quadratic Equations and Functions

1. Solve x2 + 6x 2 = 0 by completing the square.

CORRECT: 3 2. A rocket is launched from atop a 63-foot cliff with an initial velocity of 95 feet per second. The height of the rocket t seconds after launch is given by the equation h = 16t2 + 95t + 63. Graph the equation to find out how long after the rocket is launched it will hit the ground. Estimate your

answer to the nearest hundredth of a second.

CORRECT: 6.54 sec 3. Factor x2 15x + 54.

CORRECT: (x 9)(x 6) 4. Is y = (x + 3)(x 6) 2x2 linear or quadratic? Identify the quadratic, linear, and constant terms.

CORRECT: quadratic function quadratic term: x2

linear term: 3x

constant term:18


CORRECT: 6. Identify the vertex and y-intercept of the graph of f(x) = 4(x + 3)2 + 7.

CORRECT: (3, 7), y-intercept 29 7. A rocket is launched from atop a 54-foot cliff with an initial velocity of 119 feet per second. The height of the rocket t seconds after launch is given by the equation h = 16t2 + 119t + 54. Graph the equation to find out how long after the rocket is launched it will hit the ground. Estimate your answer to the nearest hundredth of a second.


CORRECT: 7.87 sec 8. Find the absolute value of |1 + 3i|.

CORRECT: 9. Solve 9x + 6 + 5x2 = 0 using the Quadratic Formula.

CORRECT: 10. Solve x2 5x 1 = 0 using the Quadratic Formula. Find the exact solutions. Then approximate

any radical solutions. Round to the nearest hundredth.

CORRECT: ; 0.19, 5.19 11. Find a quadratic model for the values in the table.

CORRECT: f(x) = 3x2 + 7x 9 12. Write the equation of the parabola with vertex (9, 2) and point (12, 16) in vertex form.

CORRECT: y = (x 9)2 2 13. Factor x2 6x + 8.

CORRECT: (x 4)(x 2) 14. Identify the vertex and the axis of symmetry of the parabola.

CORRECT: vertex: ( , ); axis of symmetry: x = 15. Solve 6x2 = 12. If necessary, round your answer to the nearest hundredth.

CORRECT: 1.41


16. Find the quadratic function y = ax2 + c with a graph that includes (0, ) and (3, ).

CORRECT: y = 4x2 + 17. Solve 5x2 11x + 2 = 0 using the Quadratic Formula.

CORRECT: 2, 18. Factor t2 10t + 25.

CORRECT: (t 5)2 19. Rewrite y = x2 + 5x 1 in vertex form.

CORRECT: y = (x )2 + 20. Graph the quadratic function with vertex (3, 1) and y-intercept 2.

CORRECT:

Polynomials and Polynomial Functions

1. Which polynomial function does NOT have 0 as a root?

f(x) = x4 1

h(x) = x4 + 3x2 x

k(x) = x2 x3

g(x) = x2 3x 2. Morris Mouse can find his way through the maze easily from entrance A to exit B. However, he receives food only if he finds the exit without going west or south. (North is towards the top of the diagram.) How many different paths can he take through the maze to receive food? (Note: Different


paths have at least one distinct section. See the diagram for an example.)

is different from

70 paths

140 paths

10 paths

16 paths

3. Use synthetic division and the Remainder Theorem to find P(3) if P(x) = x4 + 19x3 + 108x2 + 236x + 176.

9

8

7

6

4. The volume of a certain rectangular prism is given by 15x3 + 76x2 92x 48. Give the

dimensions of the prism.

(5x + 2)(x + 6)(3x + 4)

(5x + 2)(x + 6)(3x 4)

(5x 2)(x + 6)(3x 4)

(5x + 2)(x 6)(3x 4)

5. Write a polynomial in standard form with zeros 1, 2, and 5.

x3 2x2 + 13x 10 = 0

x3 4x2 + 17x + 10 = 0

x3 + 2x2 13x + 10 = 0

x3 + 6x2 3x + 10 = 0


6. A polynomial equation with rational coefficients has roots 1 + and 3 . Find two additional

roots.

1 , 3 +

1 , 3 +

1 + , 3 +

1 , 3 +

7. Divide (2x3 + 3x2 4x 4) (x + 2) using long division.

2x2 x 1

2x2 + x + 2

2x2 x + 2

2x2 + x 2

8. Solve 8x3 + 27 = 0.

x = ,

x = ,

x = ,

x = ,

9. Evaluate .

99

110

55

77

10. Evaluate .

60,480

504

84


11. Evaluate 9!.

9

362,880

3,628,800

99

12. Divide (3x3 + 4x2 5x 2) (x + 2) using long division.

3x2 + 2x 1

3x2 + 2x + 1

3x2 2x 1

3x2 2x + 1

13. Use the Binomial Theorem to expand (v u)6.

v6 8v5u + 17v4u2 22v3u3 + 17v2u4 8vu5 + u6

v6 6v5u + 15v4u2 20v3u3 + 15v2u4 6vu5 + u6

v6 u6

v6 6v5u 30v4u2 + 120v3u3 + 360v2u4 720vu5 720u6

14. Find the factored form of 4x3 8x2 12x.

4x(x2 2x 3)

4(x2 + 1)(x 3)

4x(x + 1)(x 3)

x(4x + 1)(x 3)

15. Use Pascal's Triangle to expand (3a b)4.

81a4 12a3b + 18a2b2 12ab3 + b4

81a4 108a3b + 54a2b2 12ab3 + b4

81a4 + 108a3b + 54a2b2 + 12ab3 + b4

81a4 + 12a3b + 18a2b2 + 12ab3 + b4 16. Assume red and green are equally likely occurrences. What is probability that you will get one

green light in a row of five lights?


17. In how many ways can 4 singers be selected from 8 who came to an audition?

70 ways

4 ways

32 ways

1344 ways

18. Use Pascal's Triangle to expand (2x + y)4.

16x4 32x3y + 24x2y2 8xy3 + y4

16x4 + 32x3y + 24x2y2 + 8xy3 + y4

16x4 + 64x3y + 48x2y2 + 16xy3 + y4

16x4 64x3y + 48x2y2 16xy3 + y4 19. Use the Rational Root Theorem to determine all possible rational roots of 5x3 + 4x2 + 4x + 10 =

0. Do NOT find the actual roots.

2, 5, 10, , ,

1, 2, 5, 10, ,

0, 1, 2, 5, ,

2, 5, 10, 50, ,

20. Use synthetic division and the factor x + 3 to completely factor y = 4x3 + 5x2 23x 6.

y = (x + 3)(x + 2)(4x + 1)

y = (x + 3)(x 2)(4x 1)

y = (x + 3)(x + 2)(4x 1)

y = (x + 3)(x 2)(4x + 1)

Polynomials and Polynomial Functions

1. Which polynomial function does NOT have 0 as a root?

CORRECT: f(x) = x4 1 2. Morris Mouse can find his way through the maze easily from entrance A to exit B. However, he

receives food only if he finds the exit without going west or south. (North is towards the top of the diagram.) How many different paths can he take through the maze to receive food? (Note: Different


paths have at least one distinct section. See the diagram for an example.)

is different from

CORRECT: 70 paths 3. Use synthetic division and the Remainder Theorem to find P(3) if P(x) = x4 + 19x3 + 108x2 + 236x + 176.

CORRECT: 8 4. The volume of a certain rectangular prism is given by 15x3 + 76x2 92x 48. Give the

dimensions of the prism.

CORRECT: (5x + 2)(x + 6)(3x 4) 5. Write a polynomial in standard form with zeros 1, 2, and 5.

CORRECT: x3 + 2x2 13x + 10 = 0 6. A polynomial equation with rational coefficients has roots 1 + and 3 . Find two additional roots.

CORRECT: 1 , 3 + 7. Divide (2x3 + 3x2 4x 4) (x + 2) using long division.

CORRECT: 2x2 x 1 8. Solve 8x3 + 27 = 0.

CORRECT: x = ,

9. Evaluate .

CORRECT: 55

10. Evaluate .

CORRECT: 504 11. Evaluate 9!.

CORRECT: 362,880 12. Divide (3x3 + 4x2 5x 2) (x + 2) using long division.

CORRECT: 3x2 2x 1 13. Use the Binomial Theorem to expand (v u)6.

CORRECT: v6 6v5u + 15v4u2 20v3u3 + 15v2u4 6vu5 + u6


14. Find the factored form of 4x3 8x2 12x.

CORRECT: 4x(x + 1)(x 3) 15. Use Pascal's Triangle to expand (3a b)4.

CORRECT: 81a4 108a3b + 54a2b2 12ab3 + b4 16. Assume red and green are equally likely occurrences. What is probability that you will get one

green light in a row of five lights?

CORRECT: 17. In how many ways can 4 singers be selected from 8 who came to an audition?

CORRECT: 70 ways 18. Use Pascal's Triangle to expand (2x + y)4.

CORRECT: 16x4 + 32x3y + 24x2y2 + 8xy3 + y4 19. Use the Rational Root Theorem to determine all possible rational roots of 5x3 + 4x2 + 4x + 10 =

0. Do NOT find the actual roots.

CORRECT: 1, 2, 5, 10, , 20. Use synthetic division and the factor x + 3 to completely factor y = 4x3 + 5x2 23x 6.

CORRECT: y = (x + 3)(x 2)(4x + 1)

Radical Functions and Rational Exponents

1. Let f(x) = x2 + 3 and g(x) = . Find (f g) (2).

9

7

14

2. Graph y = 5 + 5. Find the domain and range of the function.

domain: x 0, range: y 5





3. Rationalize the denominator of . Assume that all variables are positive.

2

4. Multiply (3 10 )(3 + 10 ). Assume that all the variables are positive.

9x 100yz

9x2 100y2z2

9x 60 100yz

9x2 + 60 xyz 100y2z2

5. Rationalize the denominator of . Simplify the answer.


6. Rewrite y = + 2 to make it easy to graph using a translation. Describe the graph. Find the domain and the range of the function.

y = 5 + 2; it is the graph of y = 5 translated unit left and 2 units up.

domain: x , range: y 2

y = 5 ( + ); it is the graph of y = 5 translated unit left and units up.


y = 5 + 2; it is the graph of y = 5 translated unit right and 2 units up.


y = 5( ); it is the graph of y = 5 translated unit left and units down.


7. Let f(x) = x2 + 7 and g(x) = . Find (f g) (4).

8. Find the inverse of y = 3x2 + 4. Is the inverse a function?

y = , no

y = , no

y = , yes

y = , yes


9. Graph y = . Find the domain and range of the function.




domain: x 0, range: y 2 10. Solve (x + 4) = 16.

4 2

64

60

68



12. Multiply and simplify . Assume that all variables are positive.

6x2

6x2

6x2

6x2

13. Simplify x1(x )(x ). Assume that all variables are positive.

14. Simplify . Use absolute value symbols when needed.

3x5y4

3|x5|y4

3|x5y4|

|3x5|y4


3


16. Multiply 5 6 . Simplify if possible.

3

180

9

360

17. Simplify 7 2 2 .

3

3

4 2

4

18. Let f(x) = 3x + 7 and g(x) = 4x 2. Find (f g) (6).

46

72

71

47

19. For the function f(x) = (4 2x)2, find f1 and the domain and range of f andf1. Determine whether f1 is a function.

f1(x) = 2 , domain of f: all real numbers, range of f: f(x) 16, domain of f1: x 16, range of f1: all real numbers; function

f1(x) = 2 , domain of f: all real numbers, range of f: f(x) 0, domain off1: x 0, range of f1: all real numbers; not a function

f1(x) = , domain of f: all real numbers, range of f: f(x) 4, domain off1: x 4, range of f1: f(x) 0; function

f1(x) = , domain of f: all real numbers, range of f: f(x) 16, domain of f1: x 16, range of f1: all real numbers; not a function



Radical Functions and Rational Exponents

1. Multiply (7 + )2.

CORRECT: 51 + 14

2. Simplify . Assume that all variables are positive.

CORRECT: x y 3. Simplify 2 2 4 .

CORRECT: 14 12 4. Simplify . Assume that all variables are positive.

CORRECT: 4a2b 5. Solve (x + 4) = 16.

CORRECT: 60

6. The formula for the volume of a sphere isV = pir3. Find the radius, to the nearest hundredth, of a sphere with a volume of 18 in.3.

CORRECT: 1.63 in. 7. Simplify x1(x )(x ). Assume that all variables are positive.

CORRECT: 8. Solve = x 4. Check for extraneous solutions.

CORRECT: 6

9. Rationalize the denominator of . Simplify the answer.

CORRECT: 10. Find all the real cube roots of 0.000064.

CORRECT: 0.04


11. Which function matches the graph?

CORRECT: y = 5 12. Multiply and simplify . Assume that all variables are positive.

CORRECT: m2 13. Identify the graph of y = 2x2 5 and its inverse.

CORRECT: 14. Find the real-number root of .

CORRECT: 0.5 15. Simplify 6 7 .

CORRECT: 4

16. Let f(x) = 3x2 + 10x 8 and g(x) = x + 4. Find .

CORRECT: 9x + 6, all real numbers x 4 17. Graph y = 5 + 1. Find the domain and range of the function.

CORRECT: domain: x 0, range: y 1


18. Simplify . Use absolute value symbols when needed.

CORRECT: 4|k|5 19. Which function matches the graph?

CORRECT: y = + 4

20. Find the real-number root of .

CORRECT:


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Algebra 1

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