writing equations - homewood-flossmoor high school · 2013. 6. 5. · name date periodworkbook...

Name Date Period Workbook Activity

Chapter 1, Lesson 11

Writing Equations

One-half of a number plus 3 is 13.

�12

�x � 3 � 13

Directions Write an equation for each statement. Let x be the number.

1. 6 times a number is 18. ___________________________

2. One-fifth of a number is 25. ___________________________

3. 8 plus some number is 22. ___________________________

4. Two-thirds of a number is 12. ___________________________

5. 4 times a number plus 4 is twice the number. ___________________________

6. A number plus one-fourth of the number is 150. ___________________________

7. 3 times a number subtracted from 45 is 30. ___________________________

8. 14 less than one-half of a number is 2 more than the number. ___________________________

Directions Circle the equation that solves the problem.

9. Sandra rode her bike 54 miles in one day. She rode 6 times thenumber of miles Caleb rode his bike. How many miles did Caleb ridehis bike? Let b represent the number of miles Caleb rode his bike.

A b � 54 � 6

B 54b � 6

C 6b � 54

10. Jordan went to Europe for vacation. He spent �23� of his time in Spain.If he was in Spain for 14 days, how long was he in Europe? Let v represent the number of days he was in Europe.

A �23�v � 14

B �23�(14) � v

C v � 14 � �23�

©AGS Publishing. Permission is granted to reproduce for classroom use only. Algebra 2

EXAMPLE



Axioms of Equality (Rules for Equations)

Directions Draw a line to match the axiom of equality on the left withthe statement on the right.

1. Reflexive Law

2. Symmetric Law

3. Transitive Law

4. Substitution Principle

If 3x � 15 and �12�y � 15, then 3x � �12

�y.

This statement illustrates the transitive law.

Directions Write the axiom of equality that is illustrated.

5. 14 � 2x and 2x � 14 ___________________________

6. 23y � 6 � 23y � 6 ___________________________

7. x � y and 4x � 17; 4y � 17 ___________________________

8. If 9m � 12 and n � 12, then 9m � n ___________________________

9. x � y and �13�y � 18; then �13

�x � 18 ___________________________

10. x � 8 � 12 and 12 � x � 8 ___________________________

11. 18 � �23�x � 15 and x � 3y; then 18 � �23

�(3y) � 15 ___________________________

Directions Complete each statement to illustrate the axiom ofequality given.

12. Substitution Principle: If a � b and b � 2 � 6, then a � ______ � 6

13. Reflexive Law: 8y � 3 � ______ � 3

14. Symmetric Law: If 3x � 7 � 15, then 15 � ______

15. Transitive Law: If 5x � 3 � 7 and 4x � 7, then 5x � 3 � ______


A If a � b, then b � a.

B Things that are equal to the same thing areequal to each other.

C Equals may be substituted for equals. If a � b,then b can be substituted for a in anymathematical statement without changing itstruth or falsehood.

D Anything is equal to itself.

EXAMPLE



Solutions by Addition or Subtraction

Write the missing step in solving the equation x � 1 � 3.

x � 1 � 3 x � 1 � 3

? � ? �1 � �1

x � 0 � 2 x � 0 � 2

Directions Write the missing step in solving each equation.

Solve for x: x � 14 � 3 Check: Let x � 17; x � 14 � 3 �� 14 � �14 17 � 14 � 3 � 3 � 3

x � 17 True.

Directions Solve each equation for x. Use the substitution principle tocheck your answers.


EXAMPLE

1. x � 7 � 12

______ � ______

x � 0 � 19

2. x � 9 � 1

______ � ______

x � 0 � �8

3. x � 11 � �33

______ � ______

x � 0 � �44

4. 8.5 � x � 12

______ � ______

0 � x � 3.5

5. x � 1�12� � 14�12

�

______ � ______

x � 0 � 13

6. x � �23� � 5

______ � ______

x � 0 � 5�23

�

7. x � �17� � ��27

�

______ � ______

x � 0 � ��37

�

8. 8�23� � x � 5�13

�

______ � ______

0 � x � �3�13

�

9. x � 10 � 27 ____________________

10. x � 12 � 7 ____________________

11. x � 3�12� � �8 ____________________

12. 19 � x � 9 ____________________

13. �12� � x � �2 ____________________

14. 16 � x � 8 ____________________

15. x � �13� � 9 ____________________

16. �8.7 � x � 12 ____________________

17. x � 0.8 � �2.3 ____________________

18. 5 � x � 20�14� ____________________

19. x � 16.6 � �3.4 ____________________

20. �9�23� � x � �1�13

� ____________________

EXAMPLE



Solutions by Multiplication or Division

Write how to use the rule for multiplication or division to solve 3x � 18.

Multiply by �13� or divide by 3.

Directions Write how you can use the rule for multiplication or divisionto solve the equation.

1. 6x � 12 multiply by ______ or divide by ______

2. �13�x � 15 multiply by ______ or divide by ______

3. �7x � 21 multiply by ______ or divide by ______

4. �23�x � 15 multiply by ______ or divide by ______

5. ��12�x � 4 multiply by ______ or divide by ______

Solve for x: �12�x � 14 Check: Let x � 28; �12

�x � 14 �

(2)(�12�x) � (2)(14) �12

�(28) � 14 � 14 � 14

x � 28 True.

Directions Solve for x. Use the substitution principle to check your answers.

Directions Solve each problem.

16. 8 times what number equals 24? __________________________________



19. �13� of what number is 9? __________________________________

20. �25� of what number is 1? __________________________________


6. 9x � 36 _____________________

7. 10x � 50 _____________________

8. 18x � 6 _____________________

9. �7x � 49 _____________________

10. �8x � �4 _____________________

11. �14�x � 2 _____________________

12. ��32�x � 12 _____________________

13. �110�x � 45 _____________________

14. �1,0100�x � 8.35 _____________________

15. ��14�x � �12

� _____________________

EXAMPLE

EXAMPLE



Multistep Solutions

Directions One step is missing in the solution to each equation.Using a complete sentence, write the missing step.

1. 8x � 18 � 46

Step 1 Add 18 to both sides of the equation.

Step 2 _________________________________________________________________

2. ��14�x � 16 � �4

Step 1 Subtract 16 from both sides of the equation.

Step 2 _________________________________________________________________

3. �15�x � 12 � 18

Step 1 _________________________________________________________________

Step 2 Multiply both sides of the equation by 5 (or divide by �15

�).

4. �23�x � 6 � 24

Step 1 Subtract 6 from both sides of the equation.

Step 2 _________________________________________________________________

Solve for x: 9x � 16 � 43 Check: Let x � 3; 9x � 16 � 43 � 9(3) � 16 � 43 �9x � 16 � 43 27 � 16 � 43 � 43 � 43

9x � 16 � 16 � 43 � 16 True.9x � 27

(�19�)(9x) � (27)(�19

�)x � 3

Directions Solve each equation. Use the substitution principle to checkyour answers.


5. 5x � 3 � 18 __________________

6. 30 � 5x � 0 __________________

7. 17 � 3x � 11 __________________

8. 22x � 5 � 93 __________________

9. 7 � 4x � 11 __________________

10. 3 � 6x � 21 __________________

11. 4x � 10 � 26 __________________

12. �3x � 7 � �14 __________________

13. �12�x � 12 � 16 __________________

14. 35 � �23�x � 13 __________________

15. ��110�x � 8 � 46 __________________

16. �45�x � 4 � 8 __________________

17. ��27�x � 6 � �4 __________________

18. �9x � 9 � �9 __________________

19. ��56�x � 3 � 27 __________________

20. �53�x � 18 � �43 __________________

EXAMPLE

Axioms of Inequality and Real Number Line

Directions On the line beside each inequality, write the letter of thegraph from the right column that matches the inequality.

1. ____________ 4 � x � 0 A

2. ____________ x � 2 � 0 B

3. ____________ 3x � 6 � 0 C

4. ____________ 18 � 6x � 0 D

5. ____________ 4x � 4 � 0 E

6. ____________ �2 � x � 0 F

7. ____________ 100 � 20x � 0 G

Solve for x. Graph the solution.8 � 2x � 0

8 � 2x � 8 � 0 � 8�2x � �8

(��12�)(�2x) � (�8)(��12

�)x � 4

Directions Solve each inequality for x. Graph each solution on the number line provided.

8. x � 6 � 0

9. 12x � 48 � 0

10. 9x � 18 � 0

11. �12�x � 8 � 0

12. �24 � 6x � 0

13. 25 � 100x � 0

14. 45x � 90 � 0

15. ��43�x � 16 � 0

2 3 4 5 6 7 8 9

�4 �3�2 �1 0 1 2 3

2 3 4 5 6 7 8 9

�6 �5�4 �3�2 �1 0 1

0 1 2 3 4 5 6 7




�4 �3�2 �1 0 1 2 3

0 1 2 3 4 5 6 7

�6�5 �4 �3 �2 �1 0 1

EXAMPLE



Comparing Pairs of Numbers

Directions For each ordered pair of numbers, write whether y � x, y � x,or y � x.

(3, 5) lies above the equals line.

(�2, �2) lies on the equals line.

(�1, �2) lies below the equals line.

Directions Write above, below, or on to tell where each point lies in relation to the equals line on the coordinate plane.

Directions Write a value for the missing coordinate in each pair to makeeach statement true.

22. (�3, ___) lies on the equals line.

23. (___, �6) lies above the equals line.

24. (0, ___) lies below the equals line.

25. (___, �18) lies below the equals line.


10. (�2, 3) ____________

11. (7, �4) ____________

12. (0, 4) ____________

13. (8, �8) ____________

14. (�3, �3) ____________

15. (0, 0) ____________

16. (�4, 2) ____________

17. (�14, �12) ____________

18. (7.2, �6.9) ____________

19. (�42, �50) ____________

20. (102, 102) ____________

21. (��12�, ��34

�) ____________

1. (4, �1) ____________

2. (�5, 6) ____________

3. (�4, �4) ____________

4. (�2, 3) ____________

5. (�8, �7) ____________

6. (19, �19) ____________

7. (�12�, ��14

�) ____________

8. (3�12�, �4) ____________

9. (�3, 2.5) ____________

EXAMPLES

x –4 –3 –2 –1 1 2 3 4

5

4

3

2

1

–1

–2

–3

–4

y(3, 5)

(–2, –2) (–1, –2)• •

•



Intervals on the Real Number Line

Directions On the line beside each inequality, write the letter of thegraphed interval from the right column that matches theinequality.

1. ____________ x � 1 or x � 3 A

2. ____________ �3 x 3 B

3. ____________ x � 2 or x � 2 C

4. ____________ x �3 or x 1 D

5. ____________ 0 x 4 E

6. ____________ x � 4 or x � 0 F

7. ____________ �3 � x � 1 G

Write an inequality for the interval shown.

3 x �4

Directions Write an inequality for each interval.

8. ____________________________

9. ____________________________

10. ____________________________

11. ____________________________

12. ____________________________

13. ____________________________

14. ____________________________

15. ____________________________


2 3 4 5 6 7 8 9

�4 �3�2 �1 0 1 2 3

�2 �1 0 1 2 3 4 5

�5�4 �3�2 �1 0 1 2 3 4 5

�4 �3 �2 �1 0 1 2 3

�2 �1 0 1 2 3 4 5

�4 �3 �2�1 0 1 2 3

�2 �1 0 1 2 3 4 5

�2 �1 0 1 2 3 4 5

�4 �3�2 �1 0 1 2 3

�4 �3�2 �1 0 1 2 3

�2 �1 0 1 2 3 4 5

�10�9�8�7�6 �5�4 �3�2

�5 �4�3�2�1 0 1 2 3 4 5

�4 �3�2�1 0 1 2 3

�10�9�8�7�6 �5�4 �3 �2

EXAMPLE



Solutions of Absolute Value Equations

For |x| � 15, x � 15 or x � �15.

Directions Write all the values for x that make each statement true.

For |x � 6| � 8, x � 6 � 8 or x � 6 � �8.

Directions Write the two equations that you need to solve to find thesolution of each absolute value equation.

6. |x � 3| � 6 _____________________ or _____________________

7. |x � 9| � 5 _____________________ or _____________________

8. |x � �12�| � 4 _____________________ or _____________________

9. |x � 3�14�| � 7�34

� _____________________ or _____________________

10. |x � 1.3| � 8.5 _____________________ or _____________________

|x| � 8 |x � 2| � 10 |4x � 1| � 15x � 8 or x � �8 x � 2 � 10 or x � 2 � �10 4x � 1 � 15 or 4x � 1 � �15

x � 2 � 2 � 10 � 2 or 4x � 1 � 1 � 15 � 1 or x � 2 � 2 � �10 � 2 4x � 1 � 1 � �15 � 1x � 8 or x � �12 4x � 16 or 4x � �14

(�14�)(4x) � (16)(�14

�) or (�14�)(4x) � (�14)(�14

�)

x � �146� � 4 or x � ��4

14� � �3�12�

Directions Solve for x. Use the substitution principle to check your answers.


1. |x| � 9 x � _____ or x � _____

2. |x| � 7 x � _____ or x � _____

3. |x| � 42 x � _____ or x � _____

4. |x| � 2�12� x � _____ or x � _____

5. |x| � �14� x � _____ or x � _____

11. |x| � 25 ____________________

12. |x � 6| � 10 ____________________

13. |x � 17| � 25 ____________________

14. |2x � 3| � 5 ____________________

15. |7x � 2| � 12 ____________________

16. |12 � 4x| � 16 ____________________

17. |3x � 10| � 17 ____________________

18. |65 � 5x| � 10 ____________________

19. |17x � 1| � 0 ____________________

20. |�23�x � 4| � 18 ____________________

EXAMPLE

EXAMPLE

EXAMPLES



Solutions of Absolute Value Inequalities

Directions On the line beside each absolute value inequality, write theletter of the graph from the right column that matches theinequality.

1. ____________ |x � 2| 5 A

2. ____________ |x � 3| � 1 B

3. ____________ |2x � 1| 3 C

4. ____________ |3x � 6| � 0 D

5. ____________ |4x � 4| � 12 E

6. ____________ |x � 1| 1 F

7. ____________ |10x � 5| � 15 G

8. ____________ |4x � 2| � 6 H

Solve for x. Graph.|5x � 10| 10

5x � 10 10 or 5x � 10 �105x � 10 � 10 10 � 10 or 5x � 10 � 10 �10 � 10

5x 0 or 5x �20

(�15�)(5x) (�15

�)(0) or (�15�)(5x) (�15

�)(�20)

x 0 or x �4

Directions Solve each inequality for x. Graph each solution on the number line provided.

9. |x| 5

10. |3x| � 9

11. |12x| 36

12. |x � 1| � 4

13. |x � 5| 10

14. |4x � 8| � 12

15. |�12�x � 4| � 2


�4 �3�2 �1 0 1 2 3

�2�1 0 1 2 3 4 5

�3�2 �1 0 1 2 3 4 5

�10�8�6�4 �2 0 2 4 6

�6 �5�4 �3�2 �1 0 1 2

�6�4�2 0 2 4 6 8

�4 �3�2 �1 0 1 2 3

�4�3 �2 �1 0 1 2 3

EXAMPLE�5 �4�3 �2 �1 0 1 2 3



Geometry Connection: Relating Lines

�1 x 2 x 1 3 x 3

Directions Draw a geometric picture that fits with each algebra statement. Tell whether the picture is a line, ray, or segment.

Directions Write whether the graph of the solution set of each equationor inequality below is a point, two points, a line, a ray, tworays, or a segment.


11. x 4 ______________________

12. x �45 ______________________

13. |x| 1 ______________________

14. x ��43� ______________________

15. |x| 0 ______________________

16. |x| 8 ______________________

17. x � 2 � 7 ______________________

18. |x � 2| � 7 ______________________

19. x � 2 5 ______________________

20. |x| � 12 ______________________

21. |x| 5 ______________________

22. |x � 3| � 7 ______________________

23. |x � 3| 7 ______________________

24. 3x � 5 � 13 ______________________

25. 3x � 5 13 ______________________

26. |3x � 5| 13 ______________________

27. |3x � 5| 13 ______________________

28. x � 3 � 3 ______________________

29. 6 � 2x � 10 ______________________

30. 6 � 2x 10 ______________________

�2�1 0 1 2 0 1 2 3 4 1 2 3 4 5

segment ray lineEXAMPLES

1. x 4 ________________

2. x 3 or x �3 ________________

3. x 0 or x 0 ________________

4. �3 x 20 ________________

5. x 57 ________________

6. 1 x 100 ________________

7. x �0.783 or x 0.783________________

8. x 100 or x 100 ________________

9. x �5.6 ________________

10. �99 x 10 ________________



Functions as Ordered Pairs

Is this set of ordered pairs a function?(5, 4), (7, 2), (9, 0), (11, �2)The set of ordered pairs is a function because no x-coordinates have been repeated.

Directions Tell whether the sets of ordered pairs are functions or not.Write yes or no and explain your answer.

Directions If a vertical line passes through two or more points of a graph,the graph does not represent a function. Use this vertical linetest to determine if the graph is a function or not.Write yes or no.

Write the domain and range of this function.(7, �2), (1, 4), (3, 6), (�4, �1)The domain is 7, 1, 3, �4. The range is �2, 4, 6, �1.

Directions Write the domain and range for each function below.


EXAMPLE

1. (1, 0), (4, 2), (7, 4), (10, 6)

____________________________________

2. (5, �2), (5, �1), (5, 0), (5, 1)

____________________________________

3. (�3, 3), (�2, 2), (�1, 1), (0, 0)

____________________________________

4. (9, �2), (8, 1), (7, 4), (6, 7)

____________________________________

5. (0, 0), (�1, 2), (1, 0), (1, 2)

____________________________________

EXAMPLE

8. (1, �2), (0, 2), (�1, 6), (�2, 10) ________

9. (5, 0), (3, �2), (1, �4), (�1, �6) ________

10. (0, 4), (2, �1), (4, �6), (�2, 8) ________

6. 7.

x –4 –3 –2 –1 1 2 3 4

5

4

3

2

1

–1

–2

–3

–4

y

•

•

•x

–8 –6 –4 –2 2 4 6 8

8

6

4

2

–2

–4

–6

–8

y

• •••



Functions as a Rule

Calculate f(x) for the given domain values.f(x) � 3x; x � 1, 3, 8, 10, 100f(x) � 3, 9, 24, 30, 300 for the given domain values.

Directions Calculate f(x) for the given domain values.

Choose any number; then multiply it by 7.

f(x) � 7x is a rule in function notation for the example above. The reason that it is a function is that each x has one and only one 7x.

Directions Write a rule using function notation, f(x) � _____.Then give a reason why it is a function.

11. Choose any number; then divide it by 6. _____________________________

12. Choose any number; then multiply it by 4. _____________________________

13. Choose any number; multiply it by 3, then add 15. _____________________________

14. Choose any number; then subtract 9. _____________________________

15. Choose any number; then divide it by �2. _____________________________

16. Choose any number; then multiply it by �5. _____________________________

17. Choose any number; multiply it by �8, then subtract 7. _____________________________

18. Choose any number; divide it by 3, then add 13. _____________________________

19. Choose any number; multiply it by 4, then subtract 52. _____________________________

Directions Solve the problem.

20. Each month Daisy shoots eight rolls of film. Write a rule that showshow many rolls of film she shoots for a given number of months.Write the rule in function notation.


EXAMPLE

1. f(x) � 5x; x � 4, 6, 8, 10, 20

2. f(x) � �3x; x � 0, �1, �2, �3, �4

3. f(x) � �16�x; x � 6, 12, �12, �42, 60

4. f(x) � 5x � 2; x � 0, 1, 2, 3, 4

5. f(x) � 7x � 11; x � 3, 6, 9, 12, 15

6. f(x) � �12�x � 5; x � 0, 4, 10, 50, �100

7. f(x) � 4x � 8; x � 1, 11, 21, 31, 101

8. f(x) � �2x � 14; x � �1, �5, �10, �15, 12

9. f(x) � �13�x � 22; x � 9, 6, 3, 0, �3

10. f(x) � �78�x � 12; x � 16, 24, 48, �8, �64

EXAMPLE



Zeros of a Function

f(x) � 3x � 6 Find the zeros of f(x).

Let f(x) � 0 and solve for x.0 � 3x � 66 � 3x2 � x

Check: f(2) � 3(2) � 6f(2) � 6 � 6f(2) � 0

Directions Find the zeros of f(x).


EXAMPLE

1. f(x) � �2x � 12 __________________

2. f(x) � �25�x � 10 __________________

3. f(x) � 4x � 4 __________________

4. f(x) � �13�x � 9 __________________

5. f(x) � x � 8 __________________

6. f(x) � x2 � 64 __________________

7. f(x) � �14�x � 3 __________________

8. f(x) � 5x � 10 __________________

9. f(x) � �x � 8 __________________

10. f(x) � 6x � 42 __________________

11. f(x) � 9x � 9 __________________

12. f(x) � 7x � 3 __________________

13. f(x) � �38�x � 1 __________________

14. f(x) � x2 � 81 __________________

15. f(x) � 8x � 4 __________________

16. f(x) � �27�x � 4 __________________

17. f(x) � x3 � 125 __________________

18. f(x) � �34�x � 12 __________________

19. f(x) � 10x � 25 __________________

20. f(x) � x3 � 27 __________________

21. f(x) � 2x � 10 __________________

22. f(x) � �110�x � 100 __________________

23. f(x) � 7x � 91 __________________

24. f(x) � 6x �15 __________________

25. f(x) � �1201�x � 1 __________________

26. f(x) � 30x � 450 __________________

27. f(x) � x4 � 81 __________________

28. f(x) � �16�x � 2 __________________

29. f(x) � 15x � 75 __________________

30. f(x) � x5 � 32 __________________



Graphs of Linear Functions

Graph f(x) � 3x � 5.

Step 1 Let x � 0.f(0) � 3(0) � 5 � 5 � (0, 5) is point A.y � 5 is the y-intercept.

Step 2 Let x � �1.f(�1) � 3(�1) � 5 � 2 � (�1, 2) is point B.

Step 3 Graph the two points; then draw the line y � f(x) � 3x � 5.

Directions Graph each linear function and label the y-intercept.(Use graph paper. Label the x- and y-axes first.)


EXAMPLE

x –8 –6 –4 –2 2 4 6 8

8

6

4

2

–2

–4

–6

–8

y

••

A

B

1. f(x) � 2x

2. f(x) � 3x � 2

3. f(x) � �4x

4. f(x) � 2x � 4

5. f(x) � 5x � 1

6. f(x) � �14�x

7. f(x) � �3x � 8

8. f(x) � 2x � 7

9. f(x) � �38�x � 2

10. f(x) � 5x � 2

11. f(x) � �27�x

12. f(x) � 4x � 5

13. f(x) � ��15�x � 3

14. f(x) � x � 10

15. f(x) � �14�x � 6

16. f(x) � 6x � 6

17. f(x) � �170�x

18. f(x) � 10x � 8

19. f(x) � �15�x � 2

20. f(x) � 8x � 8



The Slope of a Line, Parallel Lines

Calculate the slope of f(x) � 3x � 4.

Step 1 Find two points.

f(1) � 3(1) � 4 � 7 � (1, 7) is point 1.f(0) � 4(0) � 4 � 4 � (0, 4) is point 2.

Directions Calculate the slope of each line. Remember, m � �((

x

y1

1

�

�

y

x2

2

)

)�.

Given f(x) � 5x and g(x) � 5x � 4, show that the lines are parallel by showing that their slopes are equal.

f(1) � 5(1) � 5 � (1, 5) is point 1. g(1) � 5(1) � 4 � 1 � (1, 1) is point 1.f(0) � 5(0) � 0 � (0, 0) is point 2. g(0) � 5(0) � 4 � �4 � (0, �4) is point 2.

m � �((51

��

00))

� � �51

� m � �((11 �

�40))

� � �51

�

m � 5 m � 5

Directions Show that the lines are parallel by showing that their slopesare equal.

Directions Solve the problem.

25. A hill has a height of 450 feet. The horizontal distance covered between thebottom of the hill and the top is 1,800 feet. Find the slope of the hill.


EXAMPLE

1. f(x) � x � 5 __________________

2. f(x) � 4x � 2 __________________

3. f(x) � �3x __________________

4. f(x) � 5x __________________

5. f(x) � �2x � 7 __________________

6. f(x) � �12�x __________________

7. f(x) � �37�x � 5 __________________

8. f(x) � �7x � 2 __________________

9. f(x) � ��29�x � 1 __________________

10. f(x) � x � 6 __________________

11. f(x) � �25�x � 1 __________________

12. f(x) � 2�12�x � 6 __________________

13. f(x) � �4x � 9 __________________

14. f(x) � ��115�x � 3 __________________

15. f(x) � 10x � 1 __________________

16. f(x) � �15x � 25 __________________

17. f(x) � �125�x � 8 __________________

18. f(x) � �4x � 11 __________________

19. f(x) � ��181�x � 5 __________________

20. f(x) � 18x � 1 __________________

21. f(x) � 2x � 5 and g(x) � 2x ______

22. f(x) � �6x and g(x) � �6x � 7 ______

23. f(x) � �13�x � 4 and g(x) � �13

�x � 4 ______

24. f(x) � �x � 100 and g(x) � �x � 8 ______

EXAMPLE

Step 2 Calculate m � �((xy

1

1

�

�

xy2

2

))

�.

m � �((71

��

40

))

� � �31

� � 3

m � 3



The Formula f(x) � y � mx � b

5x � y � 2 Change to y � mx � b. Give m and b.Solution: Subtract 5x from both sides. y � �5x � 2

m � �5, y-intercept � 2

Directions Change the given equation to the form y � mx � b.Give the value of m and b.


EXAMPLE

1. 2x � 4y � 8 __________________

2. �2x � y � 1 __________________

3. �4x � 4y � 4 __________________

4. �x � 3y � 9 __________________

5. 3x � y � �7 __________________

6. �2x � 2y � 2 __________________

7. x � 4y � 2 __________________

8. �3x � 6y � 12 __________________

9. 4x � 8 � y __________________

10. �6x � 10 � y __________________

11. �6x � 3y � 9 __________________

12. ��13�x � 6y � 2 __________________

13. �25�x � �15

�y � 5 __________________

14. �x � �13�y � 4 __________________

15. �3x � �15�y � �4 __________________

16. 2x � �15�y � 0 __________________

17. x � �110�y � 1 __________________

18. �15�x � 2y � 8 __________________

19. �10x � 8 � 5y � 2 __________________

20. �13�x � 9 � �13

�y � 6 __________________

21. �x � �34�y � �2 __________________

22. �6x � 9y � 3 __________________

23. x � �18�y � �4 __________________

24. �3x � y � 6 __________________

25. �12�x � 2y � 8 � 2y __________________

26. x � �16�y � 6 __________________

27. ��110�x � y � 10 __________________

28. �12x � 4y � 2y � 3 __________________

29. �x � y � 0 __________________

30. �x � y � 2 __________________



Reading Line Graphs: Slopes of Lines

The slope of this line is positive because it ascends to the right.The y-intercept is 8 because the line crosses the y-axis at (0, 8).The zero or root is �2 because the line crosses the x-axis at (�2, 0).

Directions Give the slope (positive, zero, or negative), the y-intercept,and the zero or root for each graph.


EXAMPLE

1. _______________

2. _______________

3. _______________

4. _______________

5. _______________

6. _______________

7. _______________

8. _______________

9. _______________

10. _______________

x –4 –3 –2 –1 1 2 3 4

4

3

2

1

–1

–2

–3

–4

y

x –4 –3 –2 –1 1 2 3 4

4

3

2

1

–1

–2

–3

–4

y

x –4 –3 –2 –1 1 2 3 4

4

3

2

1

–1

–2

–3

–4

y

x –20 –15 –10 –5 5 10 15 20

20

15

10

5

–5

–10

–15

–20

y

x –4 –3 –2 –1 1 2 3 4

4

3

2

1

–1

–2

–3

–4

y

x –8 –6 –4 –2 2 4 6 8

8

6

4

2

–2

–4

–6

–8

y

x –4 –3 –2 –1 1 2 3 4

4

3

2

1

–1

–2

–3

–4

y

x –20 –15 –10 –5 5 10 15 20

20

15

10

5

–5

–10

–15

–20

y

x –4 –3 –2 –1 1 2 3 4

4

3

2

1

–1

–2

–3

–4

y

x –20 –15 –10 –5 5 10 15 20

20

15

10

5

–5

–10

–15

–20

y

x –8 –6 –4 –2 2 4 6 8

8

6

4

2

–2

–4

–6

–8

y



Writing Equations of Lines

Write the equation of the line with m � 2 and y-intercept � 7.y � 2x � 7

Directions Given m and b, write the equation of the line.

Write the equation of a line passing through (0, 4) and (1, 1).

Step 1 Calculate m. Let (x1, y1) � (0, 4) and (x2, y2) � (1, 1).

m � �((yx

1

1

�

�

yx

2

2

))

� � �((40

��

11))

� � ��31� � �3

m � �3 so y � �3x � b

Step 2 Substitute one point in y � �3x � b and solve for b.(0, 4) � x � 0, y � 4 4 � �3(0) � b4 � 0 � b4 � b

Step 3 Write the equation: y � �3x � 4.

Directions Write the equation of the line passing through the two points.


EXAMPLE

1. m � �3; b � 4 __________________

2. m � 1; b � 3 __________________

3. m � 5; b � �6 __________________

4. m � 8; b � ��12� __________________

5. m � 2; b � �2 __________________

6. m � �15�; b � �5 __________________

7. m � 4; b � 4 __________________

8. m � �1; b � 0 __________________

9. m � 0; b � 4 __________________

10. m � �37�; b � �1 __________________

11. m � 5; b � ��56� __________________

12. m � ��13�; b � 4 __________________

13. m � �2; b � 0 __________________

14. m � 0; b � �5 __________________

15. m � �45�; b � 1�12

� __________________

16. m � 0; b � 1 __________________

17. m � 5�14�; b � �4 __________________

18. m � �110�; b � 0 __________________

19. m � 11; b � 11 __________________

20. m � �4; b � 14 __________________

21. (1, 1) and (2, 6) ________________

22. (1, 4) and (0, 5) ________________

23. (0, 3) and (�1, 4) ________________

24. (2, 4) and (1, 5) ________________

25. (3, 6) and (1, 7) ________________

26. (5, 0) and (4, �1) ________________

27. (�4, 1) and (0, 2) ________________

28. (�10, 2) and (�11, 3) ________________

29. (6, 2) and (2, 6) ________________

30. (1, 1) and (7, 6) ________________

EXAMPLE



Graphs of y � mx � b, y � mx � b

y � 3x � 2 y 3x � 2 y 3x � 2.

Directions Write the inequality for the shaded region.

Directions Sketch each of the following inequalities in the coordinate plane.


EXAMPLE

1. _____________________

2. _____________________

3. _____________________

4. _____________________

5. _____________________

6. _____________________

7. x 4 8. y � 3 9. y � �4 10. y 4x

x –4 –3 –2 –1 1 2 3 4

4

3

2

1

–1

–2

–3

–4

y

x –4 –3 –2 –1 1 2 3 4

4

3

2

1

–1

–2

–3

–4

y

x –4 –3 –2 –1 1 2 3 4

4

3

2

1

–1

–2

–3

–4

y

x –4 –3 –2 –1 1 2 3 4

4

3

2

1

–1

–2

–3

–4

y

x –5 –4 –3 –2 –1 1 2 3 4 5

5

4

3

2

1

–1

–2

–3

–4

–5

y

x –5 –4 –3 –2 –1 1 2 3 4 5

5

4

3

2

1

–1

–2

–3

–4

–5

y

x –5 –4 –3 –2 –1 1 2 3 4 5

5

4

3

2

1

–1

–2

–3

–4

–5

y

x –5 –4 –3 –2 –1 1 2 3 4 5

5

4

3

2

1

–1

–2

–3

–4

–5

y

x –5 –4 –3 –2 –1 1 2 3 4 5

5

4

3

2

1

–1

–2

–3

–4

–5

y

x –5 –4 –3 –2 –1 1 2 3 4 5

5

4

3

2

1

–1

–2

–3

–4

–5

y

x –5 –4 –3 –2 –1 1 2 3 4 5

5

4

3

2

1

–1

–2

–3

–4

–5

y

x –10 –8 –6 –4 –2 2 4 6 8 10

10

8

6

4

2

–2

–4

–6

–8

–10

y

x –5 –4 –3 –2 –1 1 2 3 4 5

5

4

3

2

1

–1

–2

–3

–4

–5

y



Geometry Connection: Lines

In algebra’s coordinate plane, the Lines that have the same slope such asx- and y-axes are perpendicular. y � x � 1 and y � x � 1 are parallel.

Directions Write a reason from geometry why each statement is true.


EXAMPLE

1. The line x � 7 is parallel to the y-axis. Why?

2. The line y � �3 is parallel to the x-axis. Why?

3. The vertical lines are parallel to the y-axis.Why?

4. The horizontal lines are parallel to the x-axis.Why?

5. y � 3x � 5 and y � 3x � 2 are parallel. Why?

x

y

x

y

b b••

•• –1 1

1

–1

y = x + 1

y = x – 1

x –8 –6 –4 –2 2 4 6 8

8

6

4

2

–2

–4

–6

–8

y

x = 7

x –4 –3 –2 –1 1 2 3 4

4

3

2

1

–1

–2

–3

–4

y

y = –3

x –4 –3 –2 –1 1 2 3 4

4

3

2

1

–1

–2

–3

–4

y

x –5 –4 –3 –2 –1 1 2 3 4 5

5

4

3

2

1

–1

–2

–3

–4

–5

y

y = 3x + 5

y = 3x – 2

b b

x –4 –3 –2 –1 1 2 3 4

4

3

2

1

–1

–2

–3

–4

y



The Distributive Law—Multiplication

6(x � y) � 6x � 6y

Directions Multiply, using the distributive law.

(2 � 7)(y � x) � 2y � 2x � 7y � 7x� 9y � 9x

Directions Multiply, using the distributive law twice. Simplify by addinglike terms.

(x � 3)(x � y � 8) �x2 � xy � 8x � 3x � 3y � 24 �x2 � xy � 11x � 3y � 24

Directions Multiply.


EXAMPLE

1. 3(8 � 2) ____________________

2. 6(x � y) ____________________

3. a(b � c) ____________________

4. x(a � b � c) ____________________

5. x(3x � 9) ____________________

6. y(x � y3) ____________________

7. x(a � b � c) ____________________

8. x2(x3 � y3) ____________________

9. x4(x � z � y) ____________________

10. x3(5x3 � x2) ____________________

11. (6 � 4)(a � b) ____________________

12. (a � 2)(a � 4) ____________________

13. (x � y)(a � b) ____________________

14. (x � 3)(x � 5) ____________________

15. (y � 4)(y � 4) ____________________

16. (2a � 4)(a � 5) ____________________

17. (x � y)(y � x) ____________________

18. (a � 2b)(4a � b) ____________________

19. (a � b)(a � b) ____________________

20. (x � y)(3x � 3y) ____________________

21. (x � 5)(x � y � 4) ________________

22. (x � y)(6x � y � z) ________________

23. (x � y)(3x2 � 4y � 7) ________________

24. (x � 4)(4x � y � z) ________________

25. (a � b)(3a � 6b � ab) ________________

26. (a � b)(a3 � b2 � 1) ________________

27. (a � b)(a � 2b � 4ab) ________________

28. (x � 3)(3x � y � 8) ________________

29. (x � 4y)(x � y � xy) ________________

30. (x � y)(x � y � 10) ________________

EXAMPLE

EXAMPLE



The Distributive Law—Factoring

Examples rb � rc � r(b � c)

3yx2 � 6yx � 9y2 � 3y(x2 � 2x � 3y)

Directions Factor the expressions by finding the common factor(s) first.

Factor x2 � 6x � 9.

Step 1 x2 � 6x � 9 � (x � ___)(x � ___)

Step 2 The factors of 9 are 3 and 3; 1 and 9; �3 and �3; and �1 and �9. So the possible factors for x2 � 6x � 9 include (x � 3)(x � 3); (x � 1)(x � 9); (x � 3)(x � 3); and (x � 1)(x � 9).

Step 3 Substitute each set of factors in the product and check.x2 � 6x � 9 � (x � 1)(x � 9)

� x(x � 9) � 1(x � 9)� x2 � 9x � x � 9� x2 � 10x � 9 Incorrect.

x2 � 6x � 9 � (x � 3)(x � 3)� x(x � 3) � 3(x � 3)� x2 � 3x � 3x � 9� x2 � 6x � 9 Correct.

Directions Factor, using the model (x � ___)(x � ___).Check by multiplying.


EXAMPLES

1. kl � kj ____________________

2. 9x � 6y ____________________

3. x2 � xy � x ____________________

4. xb � xc � xd ____________________

5. 2x2 � 6xy � 4x ____________________

6. ab � ac � a3 ____________________

7. axy � xy2 ____________________

8. 5xy � 10xya ____________________

9. 4x2y � 12xy � 10y2 ____________________

10. g2 � g3 ____________________

11. x2 � 7x � 6 ____________________

12. x2 � x � 6 ____________________

13. x2 � 8x � 15 ____________________

14. x2 � 2x � 15 ____________________

15. x2 � 2x � 8 ____________________

16. x2 � 3x � 18 ____________________

17. x2 � 25 ____________________

18. x2 � 6x � 5 ____________________

19. x2 � 6x � 7 ____________________

20. x2 � 10x � 25 ____________________

EXAMPLE



Solutions to ax2 � bx � 0

Example Solve for x and check: 2x2 � 8x � 0.

Step 1 Factor: 2x2 � 8x � 0 � 2x(x � 4) � 0

Step 2 Set each factor equal to 0 and solve for x:2x � 0 or x � 4 � 0x � 0 or x � �4

Check: x � 0, 2x2 � 8x � 0 � 2(0)2 � 8(0) � 0 � 0 � 0x � �4, 2x2 � 8x � 0 � 2(�4)2 � 8(�4) � 32 � 32 � 0

Directions Solve for x and check.


EXAMPLE

1. x2 � 12x � 0 __________________

2. x2 � 3x � 0 __________________

3. x2 � 10x � 0 __________________

4. x2 � 25x � 0 __________________

5. x2 � 13x � 0 __________________

6. x2 � 7x � 0 __________________

7. x2 � 19x � 0 __________________

8. x2 � 23x � 0 __________________

9. x2 � 36x � 0 __________________

10. x2 � 45x � 0 __________________

11. 2x2 � 8x � 0 __________________

12. 3x2 � 15x � 0 __________________

13. 4x2 � 4x � 0 __________________

14. 10x2 � 25x � 0 __________________

15. 8x2 � 16x � 0 __________________

16. 6x2 � 21x � 0 __________________

17. 2x2 � 40x � 0 __________________

18. 3x2 � 30x � 0 __________________

19. 4x2 � 36x � 0 __________________

20. 5x2 � 45x � 0 __________________

21. 2x2 � 48x � 0 __________________

22. 3x2 � 48x � 0 __________________

23. 4x2 � 52x � 0 __________________

24. 5x2 � 75x � 0 __________________

25. 6x2 � 90x � 0 __________________

26. 12x2 � 6x � 0 __________________

27. 20x2 � 4x � 0 __________________

28. 15x2 � 3x � 0 __________________

29. 24x2 � 6x � 0 __________________

30. 35x2 � 7x � 0 __________________



Solutions to x2 � bx � c � 0 by Factoring

Example Solve for x by factoring x2 � 7x � 10 � 0. Then check.

Step 1 Factor: x2 � 7x � 10 � 0(x � __)(x � __) � 0 Think: Factors of 10 are 2, 5, 1, 10.(x � 2)(x � 5) � 0

Step 2 Set each factor equal to 0: x � 2 � 0 or x � 5 � 0Solve for x: x � �2 or x � �5

Check: x � �2, x2 � 7x � 10 � 0 � (�2)2 � 7(�2) � 10 � 4 � 14 � 10 � 0x � �5, x2 � 7x � 10 � 0 � (�5)2 � 7(�5) � 10 � 25 � 35 � 10 � 0

Directions Solve for x by factoring. Check your answers.


EXAMPLE

1. x2 � 2x � 8 � 0 __________________

2. x2 � 2x � 15 � 0 __________________

3. x2 � 6x � 9 � 0 __________________

4. x2 � 3x � 18 � 0 __________________

5. x2 � 4x � 21 � 0 __________________

6. x2 � 10x � 25 � 0 __________________

7. x2 � 9x � 14 � 0 __________________

8. x2 � 3x � 10 � 0 __________________

9. x2 � 5x � 6 � 0 __________________

10. x2 � 6x � 27 � 0 __________________

11. x2 � 11x � 26 � 0 __________________

12. x2 � 12x � 35 � 0 __________________

13. x2 � 14x � 45 � 0 __________________

14. x2 � 2x � 80 � 0 __________________

15. x2 � 20x � 100 � 0 __________________

16. x2 � 6x � 55 � 0 __________________

17. x2 � 8x � 33 � 0 __________________

18. x2 � 8x � 65 � 0 __________________

19. x2 � 13x � 36 � 0 __________________

20. x2 � 14x � 40 � 0 __________________

21. x2 � 30x � 29 � 0 __________________

22. x2 � 9x � 52 � 0 __________________

23. x2 � 16x � 64 � 0 __________________

24. x2 � 19x � 84 � 0 __________________

25. x2 � 20x � 69 � 0 __________________

26. x2 � 3x � 70 � 0 __________________

27. x2 � 17x � 30 � 0 __________________

28. x2 � x � 56 � 0 __________________

29. x2 � x � 72 � 0 __________________

30. x2 � 3x � 108 � 0 __________________



Solutions to ax2 � bx � c � 0 by Factoring

Example Solve for x by factoring 2x2 � 6x � 4 � 0. Then check.

Step 1 Factor: 2x2 � 6x � 4 � 0(__x � __)(__x � __) � 0 Factors of 2: 2 and 1Factors of 4: 2, 2, 1, and 4Some trial factors:(2x � 1)(x � 4) � 0 (2x � 2)(x � 2)� 2x(x � 4) � 1(x � 4) � 2x(x � 2) � 2(x � 2)� 2x2 � 8x � x � 4 � 2x2 � 4x � 2x � 4� 2x2 � 9x � 4 � 2x2 � 6x � 4No YesThe factors of 2x2 � 6x � 4 � 0 are (2x � 2) and (x � 2).

Step 2 Set each factor equal to 0: 2x � 2 � 0 or x � 2 � 0Solve for x: 2x � �2 or x � �2 x � �1 or x � �2

Check: Let x � �1 and Let x � �2 and2x2 � 6x � 4 � 0 2x2 � 6x � 4 � 02(�1)2 � 6(�1) � 4 � 0 2(�2)2 � 6(�2) � 4 � 02(1) � 6 � 4 � 0 2(4) � 12 � 4 � 02 � 6 � 4 � 0 8 � 12 � 4 � 00 � 0 0 � 0

Directions Solve each equation by factoring. Check your answers.


EXAMPLE

1. 2x2 � 7x � 6 � 0 __________________

2. 3x2 � 8x � 4 � 0 __________________

3. 4x2 � 17x � 4 � 0 __________________

4. 6x2 � 12x � 6 � 0 __________________

5. 2x2 � 2x � 4 � 0 __________________

6. 6x2 � 11x � 10 � 0 __________________

7. 2x2 � 2x � 12 � 0 __________________

8. 8x2 � 10x � 3 � 0 __________________

9. 4x2 � 25 � 0 __________________

10. 4x2 � 9x � 5 � 0 __________________

11. 2x2 � 3x � 2 � 0 __________________

12. 12x2 � 9x � 3 � 0 __________________

13. 9x2 � 16 � 0 __________________

14. 4x2 � 4x � 8 � 0 __________________

15. 6x2 � 32x � 10 � 0 __________________

16. 8x2 � 18x � 7 � 0 __________________

17. 6x2 � 3x � 9 � 0 __________________

18. 6x2 � x � 1 � 0 __________________

19. 4x2 � 14x � 8 � 0 __________________

20. 6x2 � 13x � 15 � 0 __________________

21. 6x2 � 22x � 20 � 0 __________________

22. 6x2 � 37x � 6 � 0 __________________

23. 5x2 � 26x � 5 � 0 __________________

24. 4x2 � 13x � 3 � 0 __________________

25. 2x2 � 3x � 9 � 0 __________________

26. 6x2 � 5x � 21 � 0 __________________

27. 6x2 � 3x � 18 � 0 __________________

28. 10x2 � 99x � 10 � 0 __________________

29. 12x2 � 25x � 2 � 0 __________________

30. 15x2 � 14x � 3 � 0 __________________



Trinomials—Completing the Square

Example monomial one term a, b, cd, e2, and so on binomial two terms x � 7, xy � 2, (x2 � 6)trinomial three terms x2 � 4x � 3, 2x2 � 5x � 2polynomial many terms 3x2 � 7x � 6y � 2z2 � 5

Directions Identify each expression. Write monomial, binominal,trinomial, or polynomial.

Complete the square, given x2 � 10x � ___. Check.

Solution: Find �12� of 10 � 5. Square 5 and add to given expression.

x2 � 10x � 25, perfect square trinomial

x2 � 10x � 25

(x � 5)2

Check: (x � 5)2 � (x � 5)(x � 5) � x(x � 5) � 5(x � 5) � x2 � 5x � 5x � 25 � x2 � 10x � 25

Directions Complete the square. Check by factoring and multiplying.


EXAMPLE

1. 10x2 __________________

2. x2 � 4 __________________

3. 5x � 4y __________________

4. 5x2 � 6x � 3y � 8 __________________

5. 6x2 � 2x � 9 __________________

6. 2x2 � 5x � 2 __________________

7. 52 __________________

8. 5x2 � 3x __________________

9. 6x2 � 13x � 2 __________________

10. a � b � c � d __________________

11. x2 � 40x ____________________

12. x2 � 30x ____________________

13. x2 � 12x ____________________

14. x2 � 18x ____________________

15. x2 � 26x ____________________

16. x2 � 26x ____________________

17. x2 � 40x ____________________

18. x2 � 22x ____________________

19. x2 � 30x ____________________

20. x2 � 2x ____________________

EXAMPLE



Solutions by Completing the Square

Example Solve x2 � 12x � 13 � 0 by completing the square.

Solution:x2 � 12x � 13 � 0

x2 � 12x � 13 Change to x � bx � constant.x2 � 12x � 36 � 13 � 36 Complete the square. Add [�12�(b)]

2 to both sides.

(x � 6)2 � 49 Factor.�(x � 6�) 2� � ��49� Take the square root of each side.

x � 6 � �7x � 6 � 7 or x � 6 � �7

x � 1 or x � �13

Check: Let x � 1x2 � 12x � 13 � 0 � 1 � 12 � 13 � 0 � 13 � 13 � 0 � 0 � 0Let x � �13x2 � 12x � 13 � 0 � 169 � 156 � 13 � 0 � 169 � 169 � 0 � 0 � 0

Directions Solve by completing the square. Check your answers.

Solve x2 � 4x � 2 � 0 by completing the square.

Solution:x2 � 4x � 2 � 0

x2 � 4x � 2 Change to x � bx � constant.x2 � 4x � 4 � 2 � 4 Complete the square. Add [�12�(b)]

2 to both sides.

(x � 2)2 � 6 Factor.�(x � 2�)2� � ��6� Take the square root of each side.

x � 2 � ��6�x � 2 � ��6� or x � 2 � ��6�

x � �2 � �6� or x � �2 � �6�

Directions Solve by completing the square. Check your answers. You mayleave expressions for square roots in your answers.


EXAMPLE

1. x2 � 12x � 11 � 0 __________________

2. x2 � 10x � 9 � 0 __________________

3. x2 � 8x � 20 � 0 __________________

4. x2 � 16x � 28 � 0 __________________

5. x2 � 18x � 19 � 0 __________________

6. x2 � 4x � 12 � 0 __________________

7. x2 � 14x � 9 � 0 __________________

8. x2 � 16x � 10 � 0 __________________

9. x2 � 18x � 11 � 0 __________________

10. x2 � 20x � 12 � 0 __________________

EXAMPLE



The Quadratic Formula

You can rewrite any quadratic equation that is not in standard form so that it is in standard form, ax2 � bx � c � 0.

10x2 � 15 � �19x; standard form is 10x2 � 19x � 15 � 0.

Directions Rewrite in standard form.

Solve 2x2 � 9x � 4 � 0 by using the quadratic formula x � .a � 2, b � 9, c � 4Substitute in the formula: x �

x �

x �

x �

The roots of the equation are x � ��94� 7� � �

�42� � ��

12

�

or x � ��94� 7� � �

�416� � �4

Check: Let x � ��12�; 2x

2 � 9x � 4 � 0 � 2(��12�)2 � 9(��12�) � 4 � 0 � �

24

� � �92

� � 4 � 0

� �12

� � �92

� � �82

� � 0 � 0 � 0

Let x � �4; 2x2 � 9x � 4 � 0 � 2(�4)2 � 9(�4) � 4 � 0 �32 � 36 � 4 � 0 � 36 � 36 � 0 � 0 � 0

Directions Solve, using the quadratic formula.

�9 � 7�

4

�9 � �49��

4

�9 � �81 ��32��

4

�9 � �92 � 4�(2)(4)��

2(2)

�b � �b2 � 4�ac��

2a


EXAMPLE

EXAMPLE

1. x2 � �6x � 4 __________________

2. 3x � 6 � 10x2 __________________

3. 2x2 � 2x � 27 __________________

4. 25x � 9 � 11x2 __________________

5. 15x2 � 4x � �18 __________________

6. 2x2 � 26x � 54 __________________

7. 4x2 � 70x � 15 __________________

8. 2x2 � 4 � 5x __________________

9. 5 � 3x2 � 16x __________________

10. 3x2 � x � 36 __________________

11. 6x2 � 19x � 3 � 0 __________________

12. x2 � 5x � 4 � 0 __________________

13. 7x2 � 11x � 6 � 0 __________________

14. 8x2 � 14x � 6 � 0 __________________

15. 2x2 � 2x � 4 � 0 __________________



Complex Roots

Example Are the roots of 2x2 � 3x � 7 � 0 real or complex?a � 2, b � �3, c � 7Radicand � b2 � 4ac� (�3)2 � 4(2)(7)� 9 � 56 � �47

Radicand � 0, so roots are complex.

Directions Evaluate the radicand b2 � 4ac. Then write if the roots of thegiven equation are real or complex.

Using Gauss’s definition, x � �i, and i � ��1�, substitute i for ��1�:

x �

Factor x �

Substitute i for ��1� x �

x � ��44i� � �i

The solutions are x � �i.

Directions Rewrite each number, using i for ��1�.

Directions Use any method to solve these equations. Write complex rootsusing i for ��1�.

13. x2 � 7x � 3 � 0 __________________

14. 3x2 � 5x � 11 � 0 __________________

15. 2x2 � 3x � 4 � 0 __________________

�i�16��

4

�(��1�)(�16�)��

4

��16��

4


EXAMPLE

1. 5x2 � 2x � 3 � 0 __________________

2. x2 � 10x � 16 � 0 __________________

3. 4x2 � 5x � 2 � 0 __________________

4. �x2 � 6x � 3 � 0 __________________

5. x2 � 15x � 20 � 0 __________________

6. 3x2 � 5x � 1 � 0 __________________

7. 4x2 � 8x � 12 � 0 __________________

8. x2 � x � 1 � 0 __________________

9. ��17�

10. ��23�

11. ��y�

12. ��81�

EXAMPLE



Geometry Connection: Areas

Example Write a formula to find the side of a squarewhose area is 81 cm2. Then solve for the side.Let x � s.

x2 � 81x � ��81�

Take square roots.x � �9

The root x � �9 does not makesense for this problem because lengthcannot be a negative number. �9 is called an extraneous root.Solution: s � 9 cm

Directions Find the sides of each square with the given area.

Write a formula to find the sides of a rectangle whosewidth is 7 cm less than its length, with an area260 cm2. Then solve for l and w.Let x � l and x � 7 � w

260 � x(x � 7)260 � x2 � 7x Solution: x � 20, so l � 20 cm.

0 � x2 � 7x � 260 (x � �13 is an extraneous root.)0 � (x � 20)(x � 13) w � x � 7, w � 20 � 7 � 13 cm

x � 20 � 0 or x � 13 � 0 Check: Area � (20 cm)(13 cm) �x � 20 or x � �13 260 cm2

Directions Find the length and width of the following rectangles.

7. Area � 88 yd2; its width is 3 more than its length. __________________

8. Area � 48 ft2; its width is 8 less than its length. __________________

Directions Find the base and height of the following parallelograms.

9. Area � 117 in.2; its height is 4 less than its base. __________________

10. Area � 105 cm2; its height is 8 more than its base. __________________


EXAMPLE

1. 10,000 m2 __________________

2. 1,600 ft2 __________________

3. 196 cm2 __________________

4. 900 in.2 __________________

5. 625 in.2 __________________

6. 2,500 yd2 __________________

EXAMPLE

x A � 81 cm2

x

x � 7

A � 260 cm2



f(x) � ax2 � bx � c, Quadratic Functions

Given f(x) � 3x2 � 4x � 1:

Find the values for f(x) for x � 0, 1, and �1.x � 0: f(0) � 3(0)2 � 4(0) � 1 � 1

independent (x, y) dependentx � 0 (0, 1) f(0) � 1

x � 1: f(1) � 3(1)2 � 4(1) � 1 � 8

independent (x, y) dependentx � 1 (1, 8) f(1) � 8

x � �1: f(�1) � 3(�1)2 � 4(�1) � 1 � 0

independent (x, y) dependentx � �1 (�1, 0) f(�1) � 0

Directions Find the values of f(x) for the given domain values. Completea table like the one at the right, listing x and f(x) values foreach function.


EXAMPLE

1. f(x) � 3x2 � 4x � 1

x � �2, �1, 0, 1, 2

2. f(x) � x2 � 5x � 4

x � �1, 0, 1, 2, 3

3. f(x) � 2x2 � 5x � 2

x � �2, �1, 0, 1, 2

4. f(x) � 4x2 � 4x � 3

x � �1, ��12

�, 0, �12

�, 1

5. f(x) � x2 � 6x � 5

x � �2, �1, 0, 1, 2

6. f(x) � x2 � x � 12

x � �2, �1, 0, 1, 2

7. f(x) � 4x2 � 10x � 6

x � �1, 0, 1, 2, 3

8. f(x) � 3x2 � 10x � 3

x � �2, �1, 0, 1, 2

9. f(x) � x2 � 10x � 8

x � �1, ��12

�, 0, �12

�, 1

10. f(x) � 6x2 � 12x � 6

x � �2, �1, 0, 1, 2

x y � f(x)



Graphing f(x) � ax2 and f(x) � �ax2

Graph f(x) � 3x2, domain � all real numbers.

Step 1 Let x � ±3, ±2, ±1, 0.Notice that all the range values are positive except 0 and the graph isabove the x-axis.

Step 2 Sketch curve.

Graph of f(x) � 3x2

Domain � all real numbersRange � 0 and all positive numbersThe curve is a parabola.

Directions Find seven points for each function. Then sketch the parabola.

1. f(x) � 5x2 ______________________________________

2. f(x) � �19�x2 ______________________________________

3. f(x) � �8x2 ______________________________________

4. f(x) � ��19�x2 ______________________________________

5. f(x) � �18�x2 ______________________________________

6. f(x) � �7x2 ______________________________________

7. f(x) � �4x2 ______________________________________

8. f(x) � ��112�x2 ______________________________________

9. f(x) � ��18�x2 ______________________________________

10. f(x) � 15x2 ______________________________________


EXAMPLEx f(x) � 3x2

�3 3(�3)2 � 27

3 3(3)2 � 27

�2 3(�2)2 � 12

2 3(2)2 � 12

�1 3(�1)2 � 3

1 3(1)2 � 3

0 3(0)2 � 0

30

25

20

15

10

5

–5

–10

y

x –20 –15 –10 –5 5 10 15 20

•

•

•

•

•

••



Graphing f(x) � ax2 � c

Sketch the graph of f(x) � 2x2 � 8.Tell if the roots are real or complex.

Solution:

Step 1 Think of f(x) � 2x2.

Step 2 Move curve down by 8 to get f(x) � 2x2 � 8.

Directions Sketch the following parabolas. Write whether the roots arereal or complex numbers.

Directions Sketch the parabolas and answer the questions about eachparabola.


EXAMPLE

1. f(x) � 5x2 � 3 __________________

2. f(x) � �x2 � 15 __________________

3. f(x) � �6x2 � 9 __________________

4. f(x) � 4x2 � 16 __________________

5. f(x) � 4x2 � 5 __________________

6. f(x) � 6x2 � 12 __________________

7. f(x) � 9x2 � 10 __________________

8. f(x) � �5x2 � 5 __________________

9. f(x) � �4x2 � 7 __________________

10. f(x) � 8x2 � 3 __________________

11. Sketch a narrow parabola that spills water andhas a turning point of (0, �4). Will thisparabola have real or complex roots?

____________________________________

12. Sketch a parabola that has a turning point of(0, �6) and complex roots. What else can youinfer about the parabola?

____________________________________

13. Sketch a wide parabola with real roots and aturning point of (0, �10). Is the turning pointa minimum or maximum value for f(x)?

____________________________________

14. Sketch a parabola with complex roots and aturning point of (0, 8). Is the turning point aminimum or maximum value for f(x)?

____________________________________

15. Sketch a narrow parabola that holds waterand has a turning point of (0, 6). Will thisparabola have real or complex roots?

____________________________________

x

8

7

6

5

4

3

2

1

y

–4 –3 –2 –1 1 2 3 4 •

• •

• •

f(x) = 2x2

–1

–2

–3

–4

–5

–6

–7

–8

y

x –4 –3 –2 –1 1 2 3 4

•

• •

• •

f(x) = 2x – 82



Graphing, Using Roots and the Turning Point

Sketch the graph of f(x) � x2 � x � 6.

Step 1 Find the roots by factoring or using the quadratic formula.

x �

x �

x �

x � ��12� 5� or x � ��12

� 5�

x � �42� � 2 or x � ��26� � �3

Directions Sketch the graphs, using the function and three points: tworoots and the turning point.

Given the roots x � �2 and x � 6, sketch the graph.

Step 1 Graph roots.

Step 2 Find midpoint between roots onx-axis: add x-values and divide by 2.

x � �� 22� 6� � �

42

� � 2

(x-value of turning point and axis of symmetry)

Directions Sketch the graphs, using the given roots to determine the function and the turning point.

�1 � �25��

2

�1 � �1 � 2�4��

2

�1 � �12 � 4�(1)(�6�)��

2(1)


EXAMPLE

1. f(x) � x2 � 7x � 8

2. f(x) � x2 � 5x � 6

3. f(x) � x2� 3x � 10

4. f(x) � x2 � 6x � 16

5. f(x) � x2 � 4x � 5

6. x � �3, x � 1

7. x � �5, x � 3

8. x � �2, x � 4

9. x � �8, x � �6

10. x � 3, x � 7

Step 3 Determine f(x). Calculate f(x) for x � 2 tofind the y-value of the turning point.Roots of x � �2 and x � 6 mean factorsare (x � 2) and (x � 6).f(x) � (x � 2)(x � 6)

� x2 � 4x � 12f(2) � (2)2 � 4(2) � 12

� 4 � 8 � 12� �16

The turning point is (2, �16).

EXAMPLE

Step 2 Find the x-value of the turning point.

x � ��2ab� � �2

�(11)

� � ��12

�

Step 3 Find the y-value of the turning point. Substitute the x-value into the equation and solve for y.

x � ��12�; f(��12

�) � (��12�)2

� (��12�) � 6

� �14

� � �12

� � 6 � �6�14�

The turning point is (��12�, �6�14

�).

–2

–4

–6

–8

–10

–12

–14

–16

y

x –8 –6 –4 –2 2 4 6 8

•

• •

(2, –16)

(–2, 0) (6, 0)



Reading Quadratic Graphs

Given f(x) � ax2 � bx � c, x �

Which graph represents the described f(x)?

f(x) has a zero radicand.Solution: If f(x) has a zero radicand, f(x)

has two equal roots; graph B represents the function.

f(x) has a negative radicand.Solution: If f(x) has a negative radicand,

f(x) has complex roots; graph C represents the function.

f(x) has a positive radicand.Solution: If f(x) has a positive radicand,

f(x) has real roots; graph F represents the function.

Directions Read the graphs and determine whether the parabola is afunction. Write function or not a function.

Directions Read the graph. Circle the function that represents the graph.

Directions Read the graph. Decide if the roots are real or complex.Write real or complex.

5. ______________________

�b � �b2 � 4�ac��

2a


EXAMPLES

1. __________________ 2. __________________

3. A f(x) � �6x2

B f(x) � 6x24. A f(x) � �3x2 � 8

B f(x) � �3x2 � 8

–2

–4

–6

–8

y

x –8 –6 –4 –2 2 4

•

• •

x –2 2 4 6 8 10

4

2

–2

–4

y

•

•

•

x

y

x

y

x

y

x

y

x

y

x

y

BA

DC

FE

x

y

x

y

x

y



Parabolas and Straight Lines

Find the common solutions to y � f(x) � x2 � 3 and y � 3x � 1.

Step 1 Equate y-values.Equate y � x2 � 3 and y � 3x � 1.x2 � 3 � 3x � 1x2 � 3x � 20 � x2 � 3x � 2

Step 2 Solve for x. Step 3 Substitute in either function.0 � x2 � 3x � 2 x � 2, y � 3x � 1 � y � 3(2) � 1 � 70 � (x � 2)(x � 1) Point A is (2, 7).x � 2 � 0 or x � 1 � 0 x � 1, y � 3(1) � 1 � 4x � 2 or x � 1 Point B is (1, 4).

Directions Find the common solutions. Give the coordinates of points A and B.

Use algebra to show that there are no common solutions toy � f(x) � x2 and y � x � 4.

Step 1 Equate y-values.Equate y � x2 and y � x � 4.x2 � x � 4x2 � x � 4 � 0

Step 2 Solve for x.

x2 � x � 4 � 0

x �

x �

Directions Use algebra to show that there are no common solutions.

1 � i�15��

2

1 � ��15��

2


EXAMPLE

4. y � f(x) � x2 � 3y � x � 8

5. y � f(x) � x2

y � �4

1. y � f(x) � 2x2

y � 8

_________________

2. y � f(x) � �13�x2

y � x � 6

_________________

3. y � f(x) � �4x2

y � 8x � 12

_________________

EXAMPLE

x

10

8

6

4

2

–2

–4

y

–8 –6 –4 –2 2 4 6 8

y = x – 4

y = f(x) = x 2

x

10

8

6

4

2

–2

–4

y

–8 –6 –4 –2 2 4 6 8

•

•A

B

y = 3x + 1

y = f(x) = x + 32

Step 3 x �

indicates the system of equations has complex roots. The graphs of the functions do not intersect.

1 � i�15��

2



The Straight Lines

Check if the system of linear equations has a common solution. If it does, find the common solution.

Step 1 Compare the slopes of each line. If they are not equal then they have a common solution.y � 3x � 7 m � 3y � x � 9 m � 1The slopes are not equal. Therefore, the system has a common solution.

Step 2 Elimination MethodEquate y from the equations.Rewrite y � 3x � 7 as y � 3x � 7Rewrite y � x � 9 as y � x � �9Subtract to eliminate y. y � 3x � 7

y � x � �9y � y � 3x � (�x) � 7 � (�9)

�2x � 16x � �8

Step 3 Use this value of x to find the corresponding value of y.x � �8, y � 3x � 7 � y � 3(�8) � 7 � �17 x � �8, y � x � 9 � y � (�8) � 9 � �17Common solution: (�8, �17)

Directions Determine whether the system has a common solution.Write yes or no.

Directions Find the common solution for each system of equations.


EXAMPLE

7. y � 4x � 2 _________________y � 2x � 3

8. 4y � 4x � 8 _________________y � 2x � 4

9. 2y � 4x � 12 _________________3x � y � 5

10. 5x � 6 � y _________________4x � 7 � y

1. y � 2x � 4 _________________y � 4x � 2

2. 3y � x � 6 _________________y � 3x

3. 2x � 8 � y _________________y � 2x � 6

4. 3x � 6y � 4 _________________5x � 3y � 6

5. y � 12 � 3x _________________y � 3x � 6

6. 9x � 3y � 6 _________________8y � 16x � 2



Word Problems and Linear Equations

The sum of two numbers is 26, and their difference is 8. What are the numbers?

Solution: Let x � one number, y � the other numberx � y � 26 Sum is 26.x � y � 8 Difference is 8.

Add the equations: x � y � 26x � y � 8

2x � 0 � 34x � 17

Substitute: x � y � 26 � 17 � y � 26 � y � 9

Common solution is (17, 9).Check: 17 � 9 � 26, 17 � 9 � 8 True.

Directions Solve.


EXAMPLE

1. The sum of two numbers is 34. The differenceof the two numbers is 12. What are thenumbers?

________________________________


________________________________


________________________________


________________________________


________________________________

6. You have 27 coins consisting of pennies andnickels. The coins total $0.83. How manycoins are pennies? How many are nickels?

________________________________

7. You have 31 coins consisting of nickels anddimes. The coins total $2.25. How many coinsare nickels? How many are dimes?

________________________________

8. You have 59 coins consisting of pennies andnickels. The coins total $2.87. How manycoins are pennies? How many are nickels?

________________________________

9. You have 67 coins consisting of dimes andnickels. The coins total $3.95. How manycoins are dimes? How many are nickels?

________________________________

10. You have 24 coins consisting of dimes andquarters. The coins total $3.15. How manycoins are dimes? How many are quarters?

________________________________



Geometry Connection: Axis of Symmetry

Matching points must be equidistant from the axis of symmetry.

l1 is an axis of symmetry. l2 is not an axis of symmetry.

Directions Which is an axis of symmetry for the given figure? Write l1 or l2.


EXAMPLES

1. __________________

2. __________________

3. __________________

4. __________________

5. __________________

l2

l1

l1

l2

l1

l2

l1

l2

l1

l2

l1

l2



Definitions, Addition

Add 5x3 � 9x2y � 7xy2 � 2x � 15 and 4x2y2 � x2y � 8xy2 � 5x �10.

Group like terms, then add.

5x3 � 9x2y � 7xy2 � 2x � 15

� 4x2y2 � x2y � 8xy2 � 5x � 10

5x3 � 4x2y2 � 10x2y � xy2 � 3x � 5

Directions Add the expressions.

1. x4y2 � 6x2y � 3xy2 and 3x3y2 � 8x2y � xy2 � 10x

__________________________________________

2. 3x3y2 � 8x2y � 12xy � 16x � 10 and 4x3y2 � 7x2y � 6xy � 4y2 � 4

__________________________________________

3. 9x4y3 � 3x3y2 � 7x2y � 6y3 � 6y2 � 18 and 12x3y2 � 14x2y � 7y2 � 11

__________________________________________

4. 3x4y2 � 6x3y2 � 7x2y2 � 10xy2 � 14y � 17 and �7x3y2 � 7x2y2 � 19xy2 � y � 4

__________________________________________

5. 10x4y3 � 7x3y3 � 16x2y � 8xy3 � 7xy � 2 and 4x3y3 � 8x2y � 8xy3 � 6xy � 12

__________________________________________

Directions: Find the difference.

6. Subtract 3x3y2 � 8x2y � xy2 � 10x from x4y2 � 6x2y � 3xy2.

__________________________________________

7. Subtract 4x3y2 � 7x2y � 6xy � 4y2 � 4 from 3x3y2 � 8x2y � 12xy � 16x � 10.

__________________________________________

8. Subtract 12x3y2 � 14x2y � 7y2 � 11 from 9x4y3 � 3x3y2 � 7x2y � 6y3 � 6y2 � 18.

__________________________________________

9. Subtract �7x3y2 � 7x2y2 � 19xy2 � y � 4 from 3x4y2 � 6x3y2 � 7x2y2 � 10xy2 � 14y � 17.

__________________________________________

10. Subtract 4x3y3 � 8x2y � 8xy3 � 6xy � 12 from 10x4y3 � 7x3y3 � 16x2y � 8xy3 � 7xy � 2.

__________________________________________


EXAMPLE



Products: (a � b)2, (a � b)2, (a � b)3, (a � b)3

Find (2x � 4y)2.Model method: (a � b)2 � a2 � 2ab � b2

Let a � 2x, b � �4y.(2x � 4y)2 � (2x)2 � 2(2x)(�4y) � (�4y)2

� 4x2 � 16xy � 16y2

Directions Write the expressions in expanded form.

Find (3x � y)3.

Factor method:(3x � y)3 � (3x � y)(3x � y)2

� (3x � y)(9x2 � 6xy � y2)� 3x(9x2 � 6xy � y2) � y(9x2 � 6xy � y2)� 27x3 � 18x2y � 3xy2 � 9x2y � 6xy2 � y3

� 27x3 � 27x2y � 9xy2 � y3

Model method: (a � b)3 � a3 � 3a2b � 3ab2 � b3

Let a � 3x, b � �y.(3x � y)3 � (3x)3 � 3(3x)2(�y) � 3(3x)(�y)2 � (�y)3

� 27x3 � 27x2y � 9xy2 � y3

Directions Write the expressions in expanded form.

11. (y � z)3 ___________________________

12. (3x � 4)3 ___________________________

13. (2x � y)3 ___________________________

14. (3x � 2y)3 ___________________________

15. (5x � 3y)3 ___________________________


EXAMPLE

1. (2x � 6)2 _____________________

2. (3x � 4)2 _____________________

3. (5x � 1)2 _____________________

4. (3x � 2y)2 _____________________

5. (4m � 4n)2 _____________________

6. (3a � 4b)2 _____________________

7. (6x � 6y)2 _____________________

8. (q � 5r)2 _____________________

9. (4t � 3v)2 _____________________

10. (8x � y)2 _____________________

EXAMPLE



Factoring a2 � b2, a3 � b3, and a3 � b3

Find the factors of 4x2 � 4y2.Solution: Use the model a2 � b2 � (a � b)(a � b).Let a � 2x, b � 2y; then 2x2 � 2y2 � (2x � 2y)(2x � 2y).

Find the factors of y3 � z3.Solution: Use the model a3 � b3 � (a � b)(a2 � ab � b2).Let a � y, b � z; then y3 � z3 � (y � z)(y2 � yz � z2).

Factor x3 � 8.Solution: Use the model a3 � b3 � (a � b)(a2 � ab � b2).Let a � x, b � 2; then x3 � 8 � (x � 2)(x2 � 2x � 4).

Directions Find the factors. Use a model.

1. m2 � n2 __________________________

2. 9x2 � y2 __________________________

3. 16x2 � 4y2 __________________________

4. 100x2 � 25y2 __________________________

5. 49m2 � 64n2 __________________________

6. 8x3 � y3 __________________________

7. p3 � r3 __________________________

8. x3 � 27y3 __________________________

9. 8x3 � 64y3 __________________________

10. 125s3 � 8t3 __________________________

11. t3 � w3 __________________________

12. 8x3 � 8y3 __________________________

13. 64a3 � 8 __________________________

14. 27 � b3 __________________________

15. 216a3 � 125b3 __________________________


EXAMPLES



Multiplication of Polynomials

Multiply the polynomials. Simplify and write the answer alphabeticallyand in descending order of the power of the terms.

(x � 3y)(x3 � 4x � 8)� x(x3 � 4x � 8) � 3y(x3 � 4x � 8)� x 4 � 4x2 � 8x � 3x3y � 12xy � 24y� x 4 � 3x3y � 4x2 � 8x � 12xy � 24y

(x � y)2(4x � y)2

� (x2 � 2xy � y2)(16x2 � 8xy � y2)� x2(16x2 � 8xy � y2) � 2xy(16x2 � 8xy � y2) � y2(16x2 � 8xy � y2)� 16x 4 � 8x3y � x2y2 � 32x3y � 16x2y2 � 2xy3 � 16x2y2 � 8xy3 � y 4

� 16x 4 � 24x3y � x2y2 � 6xy3 � y 4

Directions Multiply. Write the answer alphabetically in descending orderof the power of the terms.

1. (x2 � 3)(x3 � 2x) _______________________________

2. (2x � 4y)(x3 � x) _______________________________

3. (5x2 � 4)(x2 � 15) _______________________________

4. (x3 � 7x2 � 2)(x � y) _______________________________

5. (8x2 � y)(x � 2y � 7y2) _______________________________

6. (3x3 � y)(x2 � 2x � 4y � 1) _______________________________

7. (4x2 � 4)(6y2 � 2x � 3) _______________________________

8. (x � 3)(x2 � xy � y2) _______________________________

9. (3x � 2y)(x � 3y � 8y2) _______________________________

10. (x2 � 7)(x2 � 2xy2 � 3) _______________________________

11. (x � 4y � 8z)(4x � 9y � 7z) _______________________________

12. (x3 � 2x2y � 2y)(x2 � 3y � 2) _______________________________

13. (3x � y)(x2 � 3xy � 4) _______________________________

14. (7x � 2y � 8z)(2x � 4z) _______________________________

15. (2x2 � 4x)(x � 3y2 � 7y) _______________________________

16. (4x3 � 2y2 � 12)(2x � 3y2) _______________________________

17. (x2 � 7xy � 4y)(x2 � 5xy) _______________________________

18. (x � 3y)2(5x � 2y) _______________________________

19. (3x � 5)2(2x � y) _______________________________

20. (4x � y)2(x � y)2 _______________________________


EXAMPLES



Division of Polynomials; Rational Expressions

Find the quotient of (6x3 � 2x2).

6x3 � 2x2 � �62xx

3

2� � �(2)

((23))((xx))((xx))(x)

� � 3x

Find the quotient of (x3 � y3) � (x � y).

Factor the numerator; then look for common factors.

� x2 � xy � y2

Directions Divide.

1. 7x3 � x ________________________

2. (4x2 � x) � x ________________________

3. (15x4 � 3x3) � 3x ________________________

4. (18x5 � 6x2) � 2x2 ________________________

5. (x � y)5 � (x � y)3 ________________________

6. (6x � 4)6 � (6x � 4)3 ________________________

7. (x2 � 4xy � 4y2) � (x � 2y) ________________________

8. (4x2 � 20x � 25) � (2x � 5) ________________________

9. (9x2 � 24xy � 16y2) � (3x � 4y) ________________________

10. (x3 � y3) � (x � y) ________________________

11. (8x3 � 27) � (2x � 3) ________________________

12. (64x3 � y3) � (4x � y) ________________________

13. (16x2 � 4y2) � (4x � 2y) ________________________

14. (25 � 100y2) � (5 � 10y) ________________________

15. (8x3 � 8y3) � (4x2 � 4xy � 4y2) ________________________

(x � y)(x2 � xy � y2)��

(x � y)


EXAMPLES



Long Division of Polynom

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