66
COLLEGE ALGEBRA Assignment Problems Paul Dawkins
COLLEGE ALGEBRA Assignment Problems
Paul Dawkins
College Algebra
Table of Contents
Preface ............................................................................................................................................ ii Outline ........................................................................................................................................... iii Preliminaries .................................................................................................................................. 5
Introduction ................................................................................................................................................ 5 Integer Exponents ...................................................................................................................................... 5 Rational Exponents .................................................................................................................................... 7 Real Exponents .......................................................................................................................................... 9 Radicals ...................................................................................................................................................... 9 Polynomials ...............................................................................................................................................12 Factoring Polynomials ..............................................................................................................................13 Rational Expressions .................................................................................................................................16 Complex Numbers ....................................................................................................................................18
Solving Equations and Inequalities ............................................................................................ 19 Introduction ...............................................................................................................................................19 Solutions and Solution Sets .......................................................................................................................19 Linear Equations .......................................................................................................................................20 Application of Linear Equations ...............................................................................................................21 Equations With More Than One Variable .................................................................................................22 Quadratic Equations – Part I .....................................................................................................................23 Quadratic Equations – Part II ....................................................................................................................25 Solving Quadratic Equations : A Summary ..............................................................................................26 Application of Quadratic Equations ..........................................................................................................27 Equations Reducible to Quadratic Form ...................................................................................................27 Equations with Radicals ............................................................................................................................28 Linear Inequalities .....................................................................................................................................28 Polynomial Inequalities .............................................................................................................................29 Rational Inequalities .................................................................................................................................30 Absolute Value Equations .........................................................................................................................31 Absolute Value Inequalities ......................................................................................................................32
Graphing and Functions ............................................................................................................. 32 Introduction ...............................................................................................................................................33 Graphing ...................................................................................................................................................33 Lines ..........................................................................................................................................................34 Circles .......................................................................................................................................................36 The Definition of a Function .....................................................................................................................37 Graphing Functions ...................................................................................................................................40 Combining Functions ................................................................................................................................41 Inverse Functions ......................................................................................................................................42
Common Graphs ......................................................................................................................... 43 Introduction ...............................................................................................................................................43 Lines, Circles and Piecewise Functions ....................................................................................................43 Parabolas ...................................................................................................................................................44 Ellipses ......................................................................................................................................................45 Hyperbolas ................................................................................................................................................46 Miscellaneous Functions ...........................................................................................................................47 Transformations ........................................................................................................................................47 Symmetry ..................................................................................................................................................48 Rational Functions ....................................................................................................................................48
Polynomial Functions .................................................................................................................. 49 Introduction ...............................................................................................................................................49 Dividing Polynomials ...............................................................................................................................49 Zeroes/Roots of Polynomials ....................................................................................................................50
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College Algebra
Graphing Polynomials ...............................................................................................................................51 Finding Zeroes of Polynomials .................................................................................................................52 Partial Fractions ........................................................................................................................................52
Exponential and Logarithm Functions ...................................................................................... 54 Introduction ...............................................................................................................................................54 Exponential Functions ...............................................................................................................................54 Logarithm Functions .................................................................................................................................55 Solving Exponential Equations .................................................................................................................57 Solving Logarithm Equations ...................................................................................................................58 Applications ..............................................................................................................................................59
Systems of Equations ................................................................................................................... 60 Introduction ...............................................................................................................................................60 Linear Systems with Two Variables .........................................................................................................60 Linear Systems with Three Variables .......................................................................................................61 Augmented Matrices .................................................................................................................................62 More on the Augmented Matrix ................................................................................................................63 Non-Linear Systems ..................................................................................................................................64
Preface Here are a set of problems for my Algebra notes. These problems do not have any solutions available on this site. These are intended mostly for instructors who might want a set of problems to assign for turning in. I try to put up both practice problems (with solutions available) and these problems at the same time so that both will be available to anyone who wishes to use them.
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College Algebra
Outline Here is a list of sections for which problems have been written. Preliminaries
Integer Exponents Rational Exponents Real Exponents Radicals Polynomials Factoring Polynomials Rational Expressions Complex Numbers
Solving Equations and Inequalities Solutions and Solution Sets Linear Equations Applications of Linear Equations Equations With More Than One Variable Quadratic Equations, Part I Quadratic Equations, Part II Quadratic Equations : A Summary Applications of Quadratic Equations Equations Reducible to Quadratic Form Equations with Radicals Linear Inequalities Polynomial Inequalities Rational Inequalities Absolute Value Equations Absolute Value Inequalities
Graphing and Functions Graphing Lines Circles The Definition of a Function Graphing Functions Combining functions Inverse Functions
Common Graphs Lines, Circles and Piecewise Functions Parabolas Ellipses Hyperbolas Miscellaneous Functions Transformations
© 2007 Paul Dawkins iii http://tutorial.math.lamar.edu/terms.aspx
College Algebra
Symmetry Rational Functions
Polynomial Functions Dividing Polynomials Zeroes/Roots of Polynomials Graphing Polynomials Finding Zeroes of Polynomials Partial Fractions
Exponential and Logarithm Functions Exponential Functions Logarithm Functions Solving Exponential Equations Solving Logarithm Equations Applications
Systems of Equations Linear Systems with Two Variables Linear Systems with Three Variables Augmented Matrices More on the Augmented Matrix Nonlinear Systems
© 2007 Paul Dawkins iv http://tutorial.math.lamar.edu/terms.aspx
College Algebra
Preliminaries
Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. Integer Exponents Rational Exponents Real Exponents Radicals Polynomials Factoring Polynomials Rational Expressions Complex Numbers
Integer Exponents For problems 1 – 10 evaluate the given expression and write the answer as a single number with no exponents. 1. ( )222 5 4⋅ + − 2. 0 56 3− 3. 3 23 4 2 3⋅ + ⋅ 4. ( ) ( )4 41 2 3− + −
5. ( )20 2 27 4 3⋅ 6. ( )334 4− + − 7. 3 08 2 16−⋅ +
© 2007 Paul Dawkins 5 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
8. ( ) 11 12 3
−− −+
9. ( )32
2
3 26−
⋅ −
10. 2 3
4
4 53
−
−
⋅
For problems 11 – 18 simplify the given expression and write the answer with only positive exponents.
11. ( ) 12 43x y−− −
12. ( )( ) 332 42a b−−
13. 6 10
9 11
c bb c
−
−
14. ( ) 43 2
6 2 7
4a b ac a b
−
− −
15. ( )( )
12 4
3 10
6
2
v w
v w
− −
−
16. ( )
6021 3 8
9 1
8x y xy x
−
− −
17. 22 4 1
9 8 4
a b cb c a
−− −
− −
18. ( )
( )
336 7 2
21 4 10
p q p q
p q p
−−
− −
For problems 19 – 23 determine if the statement is true or false. If it is false explain why it is false and give a corrected version of the statement.
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College Algebra
19. 11 66
xx
−=
20. ( )73 10x x=
21. ( )23 4 12 8m n m n=
22. ( )( )432 24z z=
23. ( )3 3 3x y x y+ = +
Rational Exponents For problems 1 – 15 evaluate the given expression and write the answer as a single number with no exponents.
1. 1264
2. 1264−
3. 1216
4. 1416
5. ( )15243−
6. 12121
−
7. ( )1364 −−
8.
14625
256
9.
1327
8 −
© 2007 Paul Dawkins 7 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
10. 5249
11. 5664
−
12. ( )43729−
13.
32121
36
−
14.
2532
243 −
15.
3481
625
For problems 16 – 23 simplify the given expression and write the answer with only positive exponents.
16. ( )3
2 4 2p q− −
17.
313 2
2 34x x x−
18. 11 132 4a a a
−
19.
87 5 93 4m n
−−
20.
11 5
23
2 33 4
a b
b a
−
−
21.
31132
1 13 4
p q
p q
−
− −
© 2007 Paul Dawkins 8 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
22.
723 834
74
x y
x
−
23.
21 3
3 14
21 374 2
b c a
b a c
− −
−
For problems 24 & 25 determine if the statement is true or false. If it is false explain why it is false and give a corrected version of the statement.
24. 2332a a
−=
25. 1
n nx x− =
Real Exponents For problems 1 – 5 simplify the given expression and write the answer with only positive exponents.
1. ( )0.71.9 5.2a b−
2. ( ) 0.18.1 0.3 3.5x y z−− −
3. 0.21.1 2.2
3.3 4.4
m nn m
−
−
4. 22.6 0.4
10.1 1.6
p qp q
− −
− −
5. 6.22 3.4 0.7
2.1 1.9
a b cc a
−−
−
Radicals For problems 1 – 6 write the expression in exponential form.
© 2007 Paul Dawkins 9 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
1. 3n 2. 6 2y
3. 5 37x 4. 4 xyz 5. x y+
6. 3 3 3a b+ For problems 7 – 12 evaluate the radical. 7. 256 8. 4 256 9. 8 256 10. 5 1024− 11. 3 216− 12. 3 343 For problems 13 – 22 simplify each of the following. Assume that x, y and z are all positive.
13. 5z
14. 3 5z
15. 3 1716x
16. 116 128y
17. 3 17 4x y z
18. 3 20 54 x y x
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College Algebra
19. 7 134 729x y z
20. 2 5 23 34 10x y x y 21. 3 6 14x x x
22. 3 2 24 42 32xy x y For problems 23 – 26 multiply each of the following. Assume that x is positive. 23. ( )( )2 4 7x x+ −
24. ( )3 43 3 2x x x+
25. ( )( )2 2x y x y+ −
26. ( )2244 x x+
For problems 27 – 35 rationalize the denominator. Assume that x and y are both positive.
27. 9y
28. 37x
29. 4
1x
30. 5 2
123x
31. 2
4 x−
32. 9
3 2y +
33. 4
7 6 x−
© 2007 Paul Dawkins 11 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
34. 6
5 10x y−+
35. 4 x
x x+−
For problems 36 – 38 determine if the statement is true or false. If it is false explain why it is false.
36. 123 3x x=
37. 3 3 36 6x x+ = +
38. 24 x x= 39. For problems 13 – 35 above we always added the instruction to assume that the variables were positive. Why was this instruction added? How would the answers to the problems change if we did not have that instruction?
Polynomials For problems 1 – 18 perform the indicated operation and identify the degree of the result. 1. Add 5 310 2 1x x+ − to 4 3 28 16x x x− + 2. Add 27 13 4t t− + to 26 13 4t t− + − 3. Subtract 212 9 3z z− + − from 3 22 15 7z z z+ − + 4. Subtract 4 2100 19 7x x x− − from 3150 8 14x x+ − 5. Subtract 4 3 2 1w w w w+ + + + from 5w 6. ( )2 2 36 3 2y y y− + 7. ( )9 2 7 4x x x+ − 8. ( )( )7 5 4 10x x− − 9. ( )( )2 34 9 3t t t+ −
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College Algebra
10. ( )( )3 21 8 4 7y y y+ − + 11. ( )( )7 9 2 3x x− + 12. ( )( )2 2 21 1z z z− + 13. ( )( )22 4 6 7x x x− + + 14. ( )( )2 3 210 4 9 5 2w w w w− + + +
15. ( )2210 3x x+ 16. ( )( )21 5 4y y− + 17. Subtract ( )( )3 3x x− + from 2 7 10x x− +
18. Subtract ( )224 1x − from ( )239x x+ 19. If we multiply a polynomial with degree n and a polynomial of degree m what is the degree of the result? 20. If we add 2 polynomials of degree n and m with n m< what is the degree of the result? 21. If we subtract 2 polynomials of degree n and m with n m< what is the degree of the result? 22. If we add two polynomials, both of degree n, is it possible for the result to not be degree n? If it is not possible can you give an example of two polynomials, both of degree n, whose sum is not degree n? 23. If we subtract two polynomials, both of degree n, is it possible for the result to not be degree n? If it is not possible can you give an example of two polynomials, both of degree n, whose difference is not degree n?
Factoring Polynomials For problems 1 – 8 factor out the greatest common factor from each polynomial. 1. 3 8 106 10x x x− + 2. 6 5 825 15 30u u u− + 3. 6 4 10 2 22 3y z y z y z− +
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College Algebra
4. 10 7 8 9 6 127 14 35a b a b a b+ − 5. ( ) ( )( )3 9 7 2 9 7x x x+ − − + 6. ( ) ( )2 3 34 7 4z z z z z− + − 7. ( ) ( )4 938 2 7 2 2 7y y y y+ − +
8. ( )( ) ( ) ( )410 72 2 21 8 1 9 1 8 1w w w w w w+ − + + − For problems 9 – 13 factor each of the following by grouping. 9. 3 218 2 9x x x− + − 10. 4 3 26 3 14 7w w w w+ − − 11. 4 3 3 29 9y y y y+ + + 12. 4 3 621 56 12 32x x x x− − + 13. 3 4 5 66 3 2t t t t+ − − For problems 14 – 32 factor each of the following. 14. 2 10 9x x− + 15. 2 11 24t t+ + 16. 2 9 10z z− − 17. 2 3 28x x− − 18. 2 10 24x x+ − 19. 2 8 16w w− + 20. 2 6 9z z+ + 21. 2 144x − 22. 236 x− 23. 24 23 6z z− −
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College Algebra
24. 22 9 10y y− + 25. 212 31 7x x+ + 26. 26 35 36z z− + 27. 28 29 12t t+ − 28. 221 2w w− − 29. 236 49v − 30. 2100 20 1x x+ + 31. 225 40 16z z− + 32. 29 121y − For problems 33 – 38 factor each of the following. 33. 3 24 20 144x x x− − 34. 4 3 215 14t t t+ + 35. 8 6 46 3 3u u u− − 36. 8 45 24t t+ − 37. 4 22 5 12z z− − 38. 6 34 5x x+ − For problems 39 & 40 determine the possible values of a for which the polynomial will factor. 39. 2 16x ax+ − 40. 2 20x ax+ + For problems 41 – 44 use the knowledge of factoring that you’ve learned in this section to factor the following expressions. 41. 2 21 6x x−+ −
42. 22
12xx
− +
© 2007 Paul Dawkins 15 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
43. 42
49xx
−
44. 7 18x x− −
Rational Expressions For problems 1 – 6 reduce each of the following to lowest terms.
1. 3 2
2
106 40
x xx x
++ −
2. 2
2
18 722 11 6x x
x x+ ++ −
3. 2
2
3 2849
x xx
− −−
4. 2
2
6 13 53 26 35
x xx x
+ ++ +
5. 2
2
10 96 27
x xx x− + −− + +
6. 3 2
4 3 2
2012 36
x x xx x x
+ −− +
For problems 7 – 13 perform the indicated operation and reduce the answer to lowest terms.
7. 2 2
2 2
14 40 5 142 8 7 30
x x x xx x x x+ + + −+ − + −
8. 3 2 2
2 4 3
4 3 10 310 25
x x x x xx x x x
− − + −− + −
9. 2 2
2
5 24 125 4 1
x x x xx x x+ − + −
÷− + −
10. 2 3 4 3 2
2 2
6 3 94 6 16
x x x x xx x x+ − −
÷− + −
© 2007 Paul Dawkins 16 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
11. 2 2
2 2
3 23 14 6 13 64 3 2 1
x x x xx x x x+ + + +
÷+ + + +
12. 25 18 8
46
x xxx
− −−+
13. 3 2
2
24
6 173 4
xx x
x x
+++ −
For problems 14 – 22 perform the indicated operations.
14. 2 7 3
2 1 73 4 6x x x
− +
15. 2 1
9x x
x x−
−+
16. 1 61 7
xx x+
+− −
17. 2 2
9 74 4 4
xx x x
−− − +
18. 2
2 1 34 3 7 1 4 7
x x xx x x x
+ +− +
− − + −
19. 2 2
36 5 6
xx x x x
−− − −
20. 2 2
2 84 12 12 20
xx x x x
+− − + +
21. 2 2 2
3 9 25 10 25
xx x x x x
++ −
+ + +
22. ( ) ( )2 3
1 2 31 1 1x x x− −
+ + +
© 2007 Paul Dawkins 17 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
Complex Numbers Perform the indicated operation and write your answer in standard form. 1. ( )2 8 15i i+ − − 2. ( ) ( )12 9 2i i+ + + 3. ( )4 3 20i− − 4. ( ) ( )3 5 71
2 3 4 9i i− − + 5. ( ) ( ) ( )3 2 3 8 4 7i i i+ + − − − − 6. ( )2 9i i− + 7. ( )( )10 3 1 7i i+ − + 8. ( )26 2i+ 9. ( )( )2 14 2 14i i− + 10. ( )( )1 1
2 32 5i i− − + 11. ( )( )( )9 2 1 3 5 4i i i+ − +
12. 17
ii
+−
13. 2 49 3
ii
+− +
14. 6
4 7i
i− −
15. 12 2
9i
i−
16. 4 54 5
ii
+−
© 2007 Paul Dawkins 18 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
17. ( )
( )( )10 12
2 1 4i i
i i−
+ − +
Solving Equations and Inequalities
Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. Solutions and Solution Sets Linear Equations Applications of Linear Equations Equations With More Than One Variable Quadratic Equations, Part I Quadratic Equations, Part II Quadratic Equations : A Summary – No problems written yet. Applications of Quadratic Equations Equations Reducible to Quadratic Form Equations with Radicals Linear Inequalities Polynomial Inequalities Rational Inequalities Absolute Value Equations Absolute Value Inequalities
Solutions and Solution Sets For each of the following determine if the given number is a solution to the given equation or inequality. 1. Is 1u = − a solution to ( )24 40 10 2 1 6u u− = − − ? 2. Is 7t = a solution to ( ) ( )7 2 5 4 2t t+ = + + ?
© 2007 Paul Dawkins 19 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
3. Is 13
z = − a solution to ( )6 1 5 9z z− + = ?
4. Is 6x = − a solution to 2 10 24x x= − − ?
5. Is 14
t = a solution to ( )23 8 3 1t t t+ = − ?
6. Is 3w = − a solution to 2 22 10 7 8w w w− = − + ?
7. Is 12
x = a solution to 2
3 1 2x x− = ?
8. Is 2v = − a solution to 2 2 0
1v v
v+ −
=−
?
9. Is 1v = a solution to 2 2 0
1v v
v+ −
=−
?
10. Is 1x = − a solution to 2
3 1 6 72 3 4
x xx x x+ −
− =+ +
?
11. Is 4y = a solution to 2 34 5 2y y y− ≤ + ? 12. Is 0w = a solution to ( ) ( )3 7 2 1 10w w w− + + > ? 13. Is 7x = a solution to 3 4 24x x+ < + ?
Linear Equations Solve each of the following equations and check your answer. 1. ( ) ( )13 2 1 8 5 7u u u+ − = − + 2. ( ) ( )8 2 3 1 10 1z z z+ + = − + 3. ( ) ( )8 4 12 2 3 2 7 3t t t− − + = + − 4. ( ) ( )2 6 1 21 8 3 12x x x x x− + = − −
© 2007 Paul Dawkins 20 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
5. 3 1 7 21
5 15w w− +
+ =
6. 10 1 2 1
9 3 9y y −+ =
7. 2 5 12 3
4 3 3x x + − = −
8. 6 24 5
4xx+
=+
9. 2
3 2 7 47 5 14 2
vv v v v
−− =
+ + − −
10. 2
6 1 195 4 1t
t t t−
= −+ + +
11. 8 4 1023 2 3 2
z zz z−
= −− −
12. 2
4 1 8 4 32 6 8 4
w w ww w w w
− ++ =
− − + −
Application of Linear Equations 1. In a clearance bin everything has been reduced by 75%. One item is listed in the bin for $32.40. How much was the price of the item before it was put into the clearance bin? 2. A piece of electronics has been marked up 20% and is selling for $21.50. How much did the store pay for the item? 3. A widget is on sale for $715.80 and has been marked down by 11%. What was the original price of the widget? 4. Two cars start at the same point and move in the same direction. One car travels 5mph faster than the twice the speed of other car. After 10 hours the distance separating the two cars is 60 miles. What was the speed of each car? 5. Two people start out 100 meters apart from each other and start moving towards each other at the same time. One person is moving at half the speed of the other person and they meet after 25 seconds of travel time. What was the speed of each person?
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College Algebra
6. Two boats start at the same point. One boat starts traveling to the east at 45 mph and two hours later the second boat starts traveling to the east at 60 mph. At some point in time the faster boat will be 145 miles in front of the slower boat. How long has each boat been traveling when this happens? 7. One machine can complete a production run at a factory in 46 hours. Two machines can complete the production run in 25 hours if they work together. How long would it take the second machine to complete the production run if it had to do the job by itself? 8. One person can mow a field in 52 minutes and a second can mow the same field in 40 minutes. How long would it take the two of them to mow the field together? 9. One pump can fill a pool in 11 hours and a second pump can empty the same pool in 4 hours. While the pool is full both pumps are accidentally both turned on at the same time. How long will does it take to empty the pool? 10. How much pure acid should we add to a 32% acid solution to get 10 liters of a 60% acid solution. 11. We have 80 liters of a 2% saline solution. How much of a 10% saline solution should we add to this to increase the salinity to 4%? 12. We have 10 gallons of a 26% alcohol solution and we need 15 gallons of an 18% alcohol solution. What % alcohol solution should we add to the 26% solution to get the solution we want? 13. There is a field whose width is 6 meters less than its length. If both the length and width are doubled the perimeter will be 120 meters. What are the dimensions of the field? 14. A triangular piece of glass has been cut for a stained glass window. Two of the sides are the same length and the third side is 1 inch shorter than ½ the length of the other two sides. If the perimeter is 23 inches what are the lengths of the sides?
Equations With More Than One Variable 1. Solve ( )3 4 2A p r= − for p. 2. Solve ( )3 4 2A p r= − for r.
3. Solve 36 7
3c qT p p
c = + −
for p.
4. Solve 36 7
3c qT p p
c = + −
for c.
© 2007 Paul Dawkins 22 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
5. Solve 1 2 3n m q= − for n.
6. Solve 1 2 3n m q= − for q.
7. Solve ( )3 6 4 7A C A B C+ = − for C. 8. Solve ( )3 6 4 7A C A B C+ = − for A.
9. Solve 4 9
3xy −
= for x.
10. Solve 12
1y
x=
− for x.
11. Solve 7
10 9y
x=
+ for x.
12. Solve 8 59 7
xyx
−=
− for x.
13. Solve 2 111 4
xyx
+=
+ for x.
14. Solve 9 24
xyx
+=
− for x.
Quadratic Equations – Part I For problems 1 – 15 solve the quadratic equation by factoring. 1. 2 11 24 0z z− + = 2. 2 13 12 0w w+ + = 3. 2 32 12x x+ = 4. 2 6 27y y= + 5. 2 4 20 3 24u u u− − = + 6. 2 36 0z − =
© 2007 Paul Dawkins 23 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
7. 2144 25 0x − = 8. 27 19 6x x+ = 9. 24 15 6 4y y y+ + = 10. 26 11 15 12 5z z z− + = − 11. 2 220 3 5 5 1v v v v+ = + + 12. 2 4 16 4x x x− + = 13. 29 17 20 4 7y y y+ + = − 14. 27 9 0u u+ = 15. 214 3x x= For problems 16 – 18 use factoring to solve the equation. 16. 3 23 19 14 0v v v− − = 17. 6 5 420y y y+ = 18. 4 3 22 0z z z+ + = For problems 19 – 22 use factoring to solve the equation.
19. 2
2 1212 6
xx x x
−+ =
− + −
20. ( )2
4 51 42 2
ttt t t t
−+= +
+ +
21. 2 1 5 5
6 6w w ww w
− −= −
+ +
22. 2
2
2 19 34 39 10 9 1
y y y yy y y y− − + −
+ =− − + −
For problems 23 – 31 use the Square Root Property to solve the equation. 23. 2 144 0v − =
© 2007 Paul Dawkins 24 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
24. 281 25 0x − = 25. 24 1 0t + = 26. 27 3 0y − = 27. 214 2 0x+ = 28. ( )23 8 16 0t − − = 29. ( )211 6 0u + + = 30. ( )24 2 1 36 0x − − = 31. ( )24 121 0z− − =
Quadratic Equations – Part II For problems 1 – 6 complete the square. 1. 2 3w w+ 2. 2 10x x− 3. 2 14y y+ 4. 23 36u u− 5. 22 9t t− 6. 218x x− For problems 7 – 16 solve the quadratic equation by completing the square. 7. 2 3 10 0x x+ − = 8. 2 12 40 0z z− + = 9. 2 7 2 0t t− + = 10. 2 5 9 0u u+ + =
© 2007 Paul Dawkins 25 http://tutorial.math.lamar.edu/terms.aspx
College Algebra
11. 24 4 5 0x x− + = 12. 216 8 1 0w w+ + = 13. 24 24 29 0y y− + = 14. 281 54 10 0z z+ + = 15. 29 12 14 0t t− − = 16. 25 14 11 0v v− + = For problems 17 – 26 use the quadratic formula to solve the quadratic equation. 17. 2 14 245 0w w− + = 18. 23 20 31 0t t+ + = 19. 26 61 18 0x x+ + = 20. 2 4 23x x= − 21. 2 20 4 64y y y+ = − 22. 233 8z z= + 23. 2 22 49 32 2t t t+ = − 24. 240 25 10 11u u u+ = − 25. 2 210 10 4 3 10x x x x− = − + 26. 216 4 40 140 19z z z+ − = +
Solving Quadratic Equations : A Summary For problems 1 – 7 use the discriminant to determine the type of roots for the equation. Do not find any roots. 1. 225 120 619 0x x− + = 2. 2104 75 14 0x x− − = 3. 22 60 450 0x x+ + =
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College Algebra
4. 21 43 06
x − =
5. 297 136 289 0x x+ + = 6. 210 7 0x x− =
7. 249 14 1 09 15 25
x x+ + =
Application of Quadratic Equations 1. The length of a rectangle is 4 feet more than the width. If the area of the rectangle is 136 ft2 what are the dimensions of the rectangle? 2. The area of some rectangle is 35 in2. Four times the width of this rectangle is the same as 3 inches more than twice the length. What are the dimensions of the rectangle? 3. The area of a triangle is 28 m2 and the height of the triangle is 2 meters less than 5 times the base. What are the height and base of this triangle? 4. Two cars start out at the same spot. One car starts to drive north at 18 mph 5 hours before the second car starts driving to the east at 32 mph. How long after the first car starts driving does it take for the two cars to be 350 miles apart? 5. Two cars start out at the same point and at the same time one starts driving north while the other starts driving east at a speed that is 4 mph faster than the car driving north. Twelve hours after the cars start driving they are 600 miles apart. What was the speed of each car? 6. Two people can paint a house in 21 hours. Working individually one of the people can paint the house in 6 hours more than it takes the other person to paint the house. How long would it take each person working individually to paint the house?
Equations Reducible to Quadratic Form Solve each of the following equations. 1. 6 38 215 27 0x x+ − =
2. 4 23 313 36 0x x− + =
3. 10 532 31 1 0x x− −− − =
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College Algebra
4. 8 15 0x x− + =
5. 1 12 413 30 0x x
− −− + =
6. 6 33 28 0x x− −− − = 7. 10 1024 0x − = 8. 4 28 5 0x x− + =
9. 4 2
1 10 22 0x x
+ + =
Equations with Radicals Solve each of the following equations. 1. 4 3x x= − 2. 2 3x x= − − 3. 4 6x x− + = − 4. ( )3 11 3x x+ = + 5. 8 22 3x x= + − 6. 2 8 7x x− = − 7. 1 3 4 5x x+ = + − 8. 3 1 2x x− + + =
Linear Inequalities For problems 1 – 6 solve each of the following inequalities. Give the solution in both inequality and interval notations. 1. ( ) ( )7 2 4 12 3 5 6x x x+ − < − +
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College Algebra
2. ( ) ( )10 3 9 2 4w w+ ≥ − 3. ( ) ( )2 4 5 12 6 1 3y y y+ ≤ − −
4. 1 1 1 72 4 23 6 9 18
z z z − > + −
5. ( )2 2 4 3 6x≤ + − ≤ 6. 4 7 8 1x− < + ≤
7. 1 1 1 322 4 8 4
t < + <
8. 12 4 11 3m− ≤ − ≤
9. 3 5 107 14 2
x≤ − <
10. ( ) ( )8 2 3 4 4 1 3 3x x− < + − + ≤ 11. If 7 6x− < ≤ determine a and b for the inequality : 3 8a x b≤ + < 12. If 3 1x− ≤ ≤ − determine a and b for the inequality : 6 2a x b≤ − <
Polynomial Inequalities Solve each of the following inequalities. 1. 2 11 24 0z z− + < 2. 22 3 5x x− ≥ 3. 2 30 7t t> − 4. 2 7 8m m− ≤ 5. 2 6 9x x+ ≥ − 6. 2 1u u+ ≤ 7. 2 4 12 0w w+ − >
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College Algebra
8. 2 49 14x x+ > 9. 2t t≤ 10. 2 8 14x x− > − 11. 29 6 1 0u u− + < 12. 6 5 48 12 0z z z+ + ≥ 13. 3 22 3 14w w w− >
Rational Inequalities Solve each of the following inequalities.
1. 6 01
tt+
<−
2. 4 2 03x
x+
≥−
3. 2 3 0
6u
u+
>+
4. 3 2
1z
z−
< −+
5. 9 32
ww+
≤+
6. 2 9 14 0
1x x
x+ +
>−
7. 24 3 10 0z z
z+ −
≥
8. 2
2
10 16 04 3
t tt t+ +
≤− +
9. 2 6 4 4
5z z
z− +
<−
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College Algebra
10. 235 ww
w−
− ≥
11. 2 8 16 0x x
x+ +
>
12. 4 5
8 03
uu u−
≤−
13. 2
2 02 1x x
≥− +
Absolute Value Equations For problems 1 – 10 solve each of the equation. 1. 2 9 7x + = 2. 5 2 3w− = 3. 6 7 10t− =
4. 1 124 3
m= −
5. 8 9 9u + = 6. 3 4 1x x+ = + 7. 2 7 3 10z z− = − 8. 3 9 10y y+ = − 9. 6 12 1w w+ = + 10. 8 3 0x + = For problems 11 – 13 find all the real valued solutions to the equation. 11. 2 1 4x + = −
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College Algebra
12. 2 7 12u u− = 13. 2 6z z− =
Absolute Value Inequalities Solve each of the following inequalities. 1. 3 1 9x + ≤ 2. 10 4 2w− < 3. 8 5 0t − ≤ 4. 9 14z− < 5. 2 7 20u− ≤ 6. 4 2 1x + < − 7. 1 4 1z− > 8. 3 15 4w+ ≥ 9. 6 10 12t − > 10. 8 2 5x− ≥ 11. 4 1 1u − > −
Graphing and Functions
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College Algebra
Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. . Graphing Lines Circles The Definition of a Function Graphing Functions Combining functions Inverse Functions .
Graphing For problems 1 – 7 construct a table of at least 4 ordered pairs of points on the graph of the equation and use the ordered pairs from the table to sketch the graph of the equation.
1. 1 32 2
y x= +
2. 4y x= − 3. 23y x= 4. ( )23y x= + 5. 2y x= + 6. y x=
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College Algebra
7. 3y x= For problems 8 – 18 determine the x-intercepts and y-intercepts for the equation. Do not sketch the graph.
8. 7 23
y x= +
9. 6 11 2y x+ = − 10. 210y x= 11. 2 10 25y x x= − + 12. 216 8 17y x x= − + 13. 2 25 24y x x= − − − 14. 22 6 7y x x= − + 15. 24 3y x= − − 16. 36 48y x= + 17. 4 7y x= + − 18. 4 2y x= − −
Lines For problems 1 – 5 determine the slope of the line containing the two points and sketch the graph of the line. 1. ( ) ( )2,10 , 2,14 2. ( ) ( )6,0 , 1,3− − 3. ( ) ( )2,12 , 6,10 4. ( ) ( )5,7 , 1, 11− −
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College Algebra
5. ( ) ( )1, 6 , 4, 6− − − For problems 6 – 12 write down the equation of the line that passes through the two points. Give your answer in point-slope form and slope-intercept form. 6. ( ) ( )2,10 , 2,14 7. ( ) ( )6,0 , 1,3− − 8. ( ) ( )2,12 , 6,10 9. ( ) ( )5,7 , 1, 11− − 10. ( ) ( )1, 6 , 4, 6− − − 11. ( ) ( )0,10 , 4,2 12. ( ) ( )9,2 , 3,24− For problems 13 – 17 determine the slope of the line and sketch the graph of the line. 13. 6 8x y− = 14. 2 3y x+ = − 15. 3 1x y− = 16. 5 4 7y x+ = 17. 6 13 4y x− = − For problems 18 - 20 determine if the two given lines are parallel, perpendicular or neither. 18. The line containing the two points ( )0,0 , ( )3,18 and the line containing the two points
( )1, 5− − , ( )1,7 . 19. 4 9y x− = and 4 3y x− = −
20. 2 43
y x= − and the line containing the two points ( )4,7− , ( )2, 2−
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College Algebra
21. Find the equation of the line through ( )6, 1− and is parallel to the line 9 2 1x y+ = . 22. Find the equation of the line through ( )6, 1− and is perpendicular to the line 9 2 1x y+ = . 23. Find the equation of the line through ( )4, 9− − and is parallel to the line 8 43y x− − = . 24. Find the equation of the line through ( )4, 9− − and is perpendicular to the line 8 43y x− − = .
Circles 1. Write the equation of the circle with radius 1 and center ( )11,4 . 2. Write the equation of the circle with radius 10 and center ( )6,0− . 3. Write the equation of the circle with radius 19 and center ( )7, 2− .
4. Write the equation of the circle with radius 73
and center 1 3,2 4
−
.
For problems 5 – 10 determine the radius and center of the circle and sketch the graph of the circle. 5. ( )2 28 36x y+ + = 6. ( ) ( )2 21 7 16x y− + − = 7. ( ) ( )2 210 6 25x y+ + − =
8. ( )22 494144
x y+ + =
9. ( ) ( )2 22 1 3x y+ + − = 10. ( ) ( )2 25 3 11x y− + − = For problems 11 – 17 determine the radius and center of the circle. If the equation is not the equation of a circle clearly explain why not. 11. 2 2 8 0x y y+ − =
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College Algebra
12. 2 2 6 4 12 0x y x y+ − − − = 13. 2 2 12 2 28 0x y x y+ + + + = 14. 2 216 16 16 8 11 0x y x y+ − + − = 15. 2 22 2 3 1 0x y x+ − + = 16. 2 2 2 2 11 0x y x y+ + − + = 17. 2 2 10 4 29 0x y x y+ − + + =
The Definition of a Function For problems 1 – 6 determine if the given relation is a function. 1. ( ) ( ) ( ) ( ) ( ){ }0,1 , 2,6 , 9,4 , 7,2 , 12,3 2. ( ) ( ) ( ) ( ){ }4,1 , 2,1 , 0,1 , 3,1− − 3. ( ) ( ) ( ){ }0,4 , 0,6 , 0,8 4. ( ) ( ) ( ) ( ){ }1,6 , 3,4 , 7,6 , 2, 10− − 5. ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }0,1 , 2,3 , 4,5 , 6,7 , 8,9 , 10,11 , 12,13 6. ( ) ( ) ( ) ( ) ( ){ }7,0 , 4, 2 , 4,1 , 2,3 , 6,0− − For problems 7 – 13 determine if the given equation is a function.
7. 2 75 5
y x= +
8. 23 4 1y x x= + + 9. 42y x= − 10. 2 10 3y x= −
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College Algebra
11. 2 2 1y x= + 12. 4 3 1y x+ = 13. 3 4 1y x+ = 14. Given ( ) 7 2A t t= + determine each of the following.
(a) ( )9A − (b) ( )0A (c) ( )2A (d) ( )6A x (e) ( )2 1A t +
15. Given ( ) 3f xx
= determine each of the following.
(a) ( )4f − (b) 13
f
(c) 67
f
(d) ( )4 2f t + (e) 6fx
16. Given ( ) 2 10h w w= + determine each of the following.
(a) ( )1h − (b) ( )0h (c) ( )3h (d) ( )2h t− (e) ( )4h w+ 17. Given ( ) 23 2P x x x= − − determine each of the following.
(a) ( )6P − (b) ( )0P (c) ( )3P (d) ( )2P z (e) ( )4P x− 18. Given ( ) 3 22f z z z= − determine each of the following.
(a) ( )1f − (b) ( )0f (c) ( )4f (d) 12
f t
(e) ( )1f z −
19. Given ( )2 if 10
7 if 10t t
g tt t+ ≥
= − < determine each of the following.
(a) ( )14g (b) ( )10g (c) ( )1g −
20. Given ( )24 if 4
6 if 4x x
f xx x
< −=
≥ − determine each of the following.
(a) ( )6f − (b) ( )4f − (c) ( )3f
21. Given ( )12
2
if 71 if 7 11
3 if 11
x xg x x x
x x
≤= + < < − ≥
determine each of the following.
(a) ( )2g (b) ( )7g (c) ( )8g (d) ( )11g (e) ( )14g
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College Algebra
22. Given ( )12 if 82 3 if 10 8
1 if 10
wA w w w
w
> −= + − ≤ ≤ −− < −
determine each of the following.
(a) ( )12A − (b) ( )10A − (c) ( )9A − (d) ( )8A − (e) ( )0A
23. Given ( )2
2 if 64 if 6
if 6
x xf x x x
x x
<= + = >
determine each of the following.
(a) ( )0f (b) ( )2f (c) ( )6f (d) ( )8f (e) ( )10f For problems 24 – 28 compute the difference quotient for the given function. The difference quotient for the function ( )f x is defined to be,
( ) ( )f x h f xh
+ −
24. ( ) 8 1f x x= − 25. ( ) 23f x x= 26. ( ) 27f x x= − 27. ( ) 23 7 4f x x x= + −
28. ( ) 2f xx
=
For problems 29 – 39 determine the domain of the function. 29. ( ) 9f x x= − 30. ( ) 2 4P z z= −
31. ( ) 28 1
xh xx+
=−
32. ( )2
2
46 7
tA tt t
−=
+ −
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College Algebra
33. ( )2
2
3 212 36
w wh ww w
+ +=
+ +
34. ( ) 10 15g x x= −
35. ( ) 106 4
tf tt
=−
36. ( ) 72wf w
w+
=−
37. ( ) 2 9A z z z= −
38. ( ) 2 20h z z z= − −
39. ( ) 65 10
tg tt+
=−
Graphing Functions For problems 1 – 13 construct a table of at least 4 ordered pairs of points on the graph of the function and use the ordered pairs from the table to sketch the graph of the function. 1. ( ) 6 1f x x= − 2. ( ) 3 5f x x= − 3. ( ) 22f x x= 4. ( ) 2 7f x x= + 5. ( ) 3f x x= + 6. ( ) 6f x x= −
7. ( ) 1f xx
= , use only positive x’s
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College Algebra
8. ( ) 1f xx
= , use only negative x’s
9. ( )3 if 04 if 0
xf x
x x≥
= − <
10. ( )4 if 23 2 if 2
x xf x
x x≤ −
= − > −
11. ( )( )
2
2
2 if 1
2 if 1
x xf x
x x
− <= − ≥
12. ( )
2 if 34 if 2 31 if 2
x xf x x
x x
>= − ≤ ≤ − < −
13. ( ) 2
1 if 11 if 1 1
1 if 1
x xf x x x
x x
− ≥= − − < <− − ≤ −
Combining Functions 1. Given ( ) 12f x x= + and ( ) 9 4g x x= + compute each of the following.
(a) f g+ (b) ( )( )1f g− (c) ( )( )f g x (d) fg
2. Given ( ) 2 4h w w w= − and ( ) 22f w w= + compute each of the following.
(a) ( )( )h f w− (b) ( )( )4f h+ − (c) f h (d) ( )h wf
3. Given ( ) 6 1A x x= − and ( ) 14
P xx
=−
compute each of the following.
(a) ( )( )0A P+ (b) ( )( )2P A− − (c) AP (d) ( )A xP
4. Given ( ) 2 9f t t= + and ( ) 2 1g t t= − compute each of the following.
(a) ( )( )f g t (b) ( )( )f g t (c) ( )( )g f t (d) ( )( )g g t
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College Algebra
5. Given ( ) 2 1h x x= + and ( ) 6 4g x x= − compute each of the following.
(a) ( )( )g h x (b) ( )( )g h x (c) ( )( )h g x (d) ( )( )h h x 6. Given ( ) 22 9A w w= + and ( ) 21 2R w w w= − − compute each of the following.
(a) ( )( )A R w (b) ( )( )A R w (c) ( )( )R A w (d) ( )( )A A w 7. Given ( ) 29 10 12f x x x= + + and ( ) 2g x = compute each of the following.
(a) ( )( )f g x (b) ( )( )g f x (c) ( )( )f g x (d) ( )( )g g x
8. Given ( ) 1g t t= + and ( ) 23
h tt
=−
compute each of the following.
(a) ( )( )g h t (b) ( )( )g h t (c) ( )( )h g t (d) ( )( )h h x
9. Given ( ) 1 32
f x x= − and ( ) 2 6g x x= + , 0t ≥ compute each of the following.
(a) ( )( )f g x (b) ( )( )g f x
10. Given ( ) 13
h ww
=−
and ( ) 1 3wf ww+
= compute each of the following.
(a) ( )( )h f w (b) ( )( )f h w
Inverse Functions 1. Given ( ) 12 7P x x= − find ( )1P x− . 2. Given ( ) 7g x x= find ( )1g x− .
3. Given ( ) 3 94 7
h x x= − find ( )1h x− .
4. Given ( ) ( )54 3A x x= − + find ( )1A x− . 5. Given ( ) ( )32 1 4 1f x x= − + find ( )1f x− . 6. Given ( ) 7 5 8P x x= − find ( )1P x− .
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College Algebra
7. Given ( ) 31 3 4g x x= + + find ( )1g x− .
8. Given ( ) 10 38
xf xx−
= find ( )1f x− .
9. Given ( ) 6 74xg x
x−
=+
find ( )1g x− .
10. Given ( ) 39 7
xf xx
−=
− find ( )1f x− .
Common Graphs
Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. Lines, Circles and Piecewise Functions Parabolas Ellipses Hyperbolas Miscellaneous Functions Transformations Symmetry Rational Functions
Lines, Circles and Piecewise Functions We looked at these topics in the previous chapter. Problems for these topics can be found in the following sections. Lines : Graphing and Functions – Lines Circles : Graphing and Functions – Circles
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College Algebra
Piecewise Functions : Graphing and Functions – Graphing Functions
Parabolas For problems 1 – 18 sketch the graph of the following parabolas. The graph should contain the vertex, the y-intercept, x-intercepts (if any) and at least one point on either side of the vertex. 1. ( ) 24f x x= − 2. ( ) ( )26 1f x x= − + 3. ( ) ( )22 4f x x= + − 4. ( ) ( )23 1 12f x x= − + 5. ( ) ( )26 5 54f x x= − + + 6. ( ) ( )27 3f x x= − − − 7. ( ) ( )22 3 6f x x= + − 8. ( ) 2 8f x x= − 9. ( ) 24 1f x x= − − 10. ( ) 2 16 55f x x x= − + 11. ( ) 2 2 5f x x x= − + 12. ( ) 24 16f x x x= + 13. ( ) 2 10 25f x x x= + + 14. ( ) 22 24 64f x x x= − + − 15. ( ) 23 6 12f x x x= + −
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College Algebra
16. ( ) 24 12 9f x x x= − + − 17. ( ) 2 6 16f x x x= − + − 18. ( ) 2 8 5f x x x= + + For problems 19 – 25 convert the following equations into the form ( )2y a x h k= − + . 19. ( ) 2 4f x x x= + 20. ( ) 2 6 19f x x x= − + 21. ( ) 2 2 6f x x x= − + + 22. ( ) 27 56 111f x x x= + + 23. ( ) 23 60 306f x x x= − + 24. ( ) 225 10 1f x x x= + + 25. ( ) 22 16 18f x x x= − − −
Ellipses For problems 1 – 7 sketch the ellipse.
1. ( ) ( )2 25 2
14 9
x y+ −+ =
2. ( )2
24 116yx − + =
3. ( ) ( )2 21 6
125 4
x y+ ++ =
4. ( ) ( )2 23 1
15 12
x y− ++ =
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College Algebra
5. ( ) ( )2 29 2 4 3 1x y− + − =
6. ( ) ( )
223
2 4 19
xy
−+ + =
7. ( ) ( )2 24 1
19 9
x y− −+ =
For problems 8 – 10 complete the square on the x and y portions of the equation and write the equation into the standard form of the equation of the ellipse. 8. 2 24 16 2 13 0x x y y− + + + = 9. 2 26 4 16 9 0x x y y+ + + + = 10. 2 25 10 3 6 7 0x x y y+ + − − =
Hyperbolas For problems 1 – 5 sketch the hyperbola.
1. 2 2
19 4x y
− =
2. ( ) ( )2 23 2
136 16
y x+ +− =
3. ( )2 25
149 64
y x−− =
4. ( ) ( )22 1
9 4 14
yx
−− − =
5. ( ) ( )2 21 1 15 3 125
y x+ − − =
For problems 6 – 8 complete the square on the x and y portions of the equation and write the equation into the standard form of the equation of the hyperbola. 6. 2 29 4 48 180 0x y y− + − =
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College Algebra
7. 2 26 4 8 11 0y y x x− − − − = 8. 2 27 28 4 40 100 0x x y y− − + − =
Miscellaneous Functions The sole purpose of this section was to get you familiar with the basic shape of some miscellaneous functions for the next section. As such there are no problems for this section. You will see quite a few problems utilizing these functions in the Transformations section.
Transformations Use transformations to sketch the graph of the following functions. 1. ( ) 4f x x= − 2. ( ) 3f x x= − 3. ( ) 2 7f x x= + 4. ( ) 2f x x= + 5. ( ) ( )23f x x= + 6. ( ) 1f x x= − 7. ( )f x x= − 8. ( )f x x= − 9. ( ) ( )3f x x= − 10. ( )f x x= − 11. ( ) 2 3f x x= − − 12. ( ) ( )21 4f x x= + −
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College Algebra
13. ( ) 2 4f x x= + + 14. ( ) ( )35 2f x x= − +
Symmetry Determine the symmetry of each of the following equations. 1. 5 35 2x y y+ = 2. 2 34 5 1y y x+ = +
3. 2
2 428 1xy x
y= + −
4. 24 7 1y x x= − + 5. 5 8y x= + 6. 29 4x y= − 7. 4 28 5 1y y x+ = − 8. 2 24 1x xy y− + =
9. 2
2 1xy
x=
+
Rational Functions Sketch the graph of each of the following functions. Clearly identify all intercepts and asymptotes.
1. ( ) 75 10
f xx
=+
2. ( ) 63xf x
x−
=−
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College Algebra
3. ( ) 8 64 2
xf xx
+=
−
4. ( ) 2
25
f xx x−
=−
5. ( ) 2
34 5
xf xx x
+=
+ −
6. ( ) 2
212
f xx x
=− −
7. ( )2
2
5 12 32
xf xx
+=
−
8. ( )2
2
5 42 15
x xf xx x
− +=
+ −
Polynomial Functions
Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. Dividing Polynomials Zeroes/Roots of Polynomials Graphing Polynomials Finding Zeroes of Polynomials Partial Fractions
Dividing Polynomials For problems 1 – 6 use long division to perform the indicated division.
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College Algebra
1. Divide 27 4 9x x+ − by 1x − 2. Divide 38 4 1x x− + by 6x + 3. Divide 4 22 7x x x− + by 4x − 4. Divide 4 32 9 2 8x x x− + + by 3x + 5. Divide 4 3 28 3 1x x x+ − + by 2 2x − 6. Divide 5 3 24 7 4 2x x x x− + − + by 22 3 6x x− − For problems 7 – 11 use synthetic division to perform the indicated division. 7. Divide 3 28 10x x x− − + + by 2x + 8. Divide 310 9x x− by 10x − 9. Divide 4 33 5 2x x x+ + − by 7x + 10. Divide 4 32 9 11x x x+ − + by 3x + 11. Divide 4 3 25 4 3 2 1x x x x− + − + by 1x −
Zeroes/Roots of Polynomials For problems 1 – 6 list all of the zeros of the polynomial and give their multiplicities. 1. ( ) 2 2 120f x x x= + − 2. ( ) 2 12 32R x x x= + + 3. ( ) 3 24 3h x x x x= + − 4. ( ) ( ) ( )( )2 25 4 3 22 35 92 92 32 1 8 2A x x x x x x x x x= + − + − + = − + − 5. ( ) ( ) ( )3 210 9 8 7 6 5 517 115 387 648 432 3 4Q x x x x x x x x x x= + + + + + = + + 6. ( ) ( )( ) ( )4 38 7 6 5 4 3 22 14 16 49 62 44 88 32 4 1 2g x x x x x x x x x x x x= + − − + + − − − = + + − For problems 7 – 11 x r= is a root of the given polynomial. Find the other two roots and write the polynomial in fully factored form.
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College Algebra
7. ( ) 4 3 23 18P x x x x= − − ; 6r = 8. ( ) 3 2 46 80P x x x x= + − + ; 8r = − 9. ( ) 3 29 26 24P x x x x= − + − ; 3r = 10. ( ) 3 212 13 1P x x x= − − ; 1r = − 11. ( ) 3 24 11 134 105P x x x x= + − − ; 5r = For problems 12 – 14 determine the smallest possible degree for a polynomial with the given zeros and their multiplicities. 12. 1 2r = − (multiplicity 1), 2 1r = (multiplicity 1), 3 4r = (multiplicity 1) 13. 1 3r = (multiplicity 4), 2 5r = − (multiplicity 1) 14. 1 7r = (multiplicity 2), 2 4r = (multiplicity 7), 3 10r = − (multiplicity 5) 15. A 7th degree polynomial has roots 1 9r = − (multiplicity 2) and 2 3r = (multiplicity 1). What is the maximum number of remaining roots for the polynomial?
Graphing Polynomials Sketch the graph of each of the following polynomials. 1. ( ) ( )( )( )3 2 17 15 1 3 5f x x x x x x x= − − + − = − − − + 2. ( ) 3 22 3A x x x x= + − 3. ( ) 4 3 22 3h x x x x= + − 4. ( ) ( ) ( )( )24 3 214 68 136 96 2 4 6g x x x x x x x x= + + + + = + + + 5. ( ) ( ) ( ) ( )2 25 4 3 28 13 22 32 32 4 1 2Q x x x x x x x x x= − + − − + + = − − + − 6. ( ) ( ) ( )34 3 25 6 4 8 2 1P x x x x x x x= − + − − + = − − + 7. ( ) ( ) ( ) ( )2 25 4 3 25 18 58 145 75 1 5 3h x x x x x x x x x= + − − + − = − + −
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College Algebra
8. ( ) ( ) ( )2 26 5 4 3 2 22 11 12 36 2 3R x x x x x x x x x= − − + + = + −
Finding Zeroes of Polynomials Find all the zeroes of the following polynomials. 1. ( ) 3 22 11 12h x x x x= − − + 2. ( ) 3 210 29 20f x x x x= + + + 3. ( ) 3 22 15 34 24h x x x x= − + − 4. ( ) 4 36 22 15g x x x x= − + + 5. ( ) 4 3 23 7 15 18f x x x x x= − − + + 6. ( ) 4 3 24 35 24 36Q x x x x x= + − − + 7. ( ) 4 3 29 15 11 11 2h x x x x x= + − − − 8. ( ) 5 4 3 22 19 68 114 90 27A x x x x x x= + + + + + 9. ( ) 5 4 3 216 48 24 40 39 9P x x x x x x= − + + − +
Partial Fractions Determine the partial fraction decomposition of each of the following expressions.
1. 2
22 75 4
xx x
++ +
2. 2
7 444 25 21
xx x
−+ −
3. 2
4711 24x
x x− −− +
4. 2
5 388 2 1
xx x−+ −
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College Algebra
5. ( )( )( )
26 50 161 2 7x x
x x x+ +
− + +
6. ( )( )( )
232 39 81 2 2 3
x xx x x
+ −+ + −
7. ( )( )( )
236 115 193 5 4 3
x xx x x
+ −+ − −
8. ( )23 5
3x
x−
−
9. ( )224 413 5
xx+
+
10. ( )210 93
10x
x+
+
11. ( )( )
2
27 31 107
4 3x xx x+ +
− +
12. ( )( )
2
29 58 37
7 2x xx x
− −
+ −
13. ( )( )
2
221 43 203 2 1
x xx x
− +
− −
14. ( )
2
2
7 108 119 1
x xx x x
− + −− +
15. ( )
2
2
24 2 1172 13x x
x x x+ ++ +
16. ( )
2 3
22
2 11 7
2
x x x
x
− + −
+
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College Algebra
17. ( )
3 2
22
4 3 5 5
1
x x x
x
− − −
+
Exponential and Logarithm Functions
Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. Exponential Functions Logarithm Functions Solving Exponential Equations Solving Logarithm Equations Applications
Exponential Functions 1. Given the function ( ) 9xf x = evaluate each of the following.
(a) ( )3f − (b) ( )1f − (c) ( )0f (d) ( )12f (e) ( )3
2f 2. Given the function ( ) 8xf x = evaluate each of the following.
(a) ( )23f − (b) ( )1f − (c) ( )0f (d) ( )2f (e) ( )5
3f 3. Given the function ( ) ( )1
7xf x = evaluate each of the following.
(a) ( )2f − (b) ( )1f − (c) ( )0f (d) ( )2f (e) ( )4f 4. Given the function ( ) ( )1
16xf x = evaluate each of the following.
(a) ( )2f − (b) ( )14f − (c) ( )0f (d) ( )2f (e) ( )1
4f
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College Algebra
5. Sketch each of the following. (a) ( ) ( )1
3xf x = (b) ( ) ( )1
3 2xg x = + (c) ( ) ( ) 413
xg x +=
6. Sketch each of the following. (a) ( ) 5xf x = (b) ( ) 5 4xg x = − (c) ( ) 35xg x −= 7. Sketch the graph of ( ) 210 6xf x −= + . 8. Sketch the graph of ( ) ( ) 41
7 1xf x += − .
9. Sketch the graph of ( ) 1 2xf x += −e . 10. Sketch the graph of ( ) 4 1xf x −= −e .
Logarithm Functions For problems 1 – 5 write the expression in logarithmic form.
1. 3 1111331
− =
2. 74 16384=
3. 32 343
7 8
− =
4. 3225 125=
5. 53 127
243−
=
For problems 6 – 10 write the expression in exponential form. 6. 1
6
log 36 2= −
7. 12log 20736 4=
8. 95log 2432
=
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College Algebra
9. 41 7log
128 2= −
10. 8log 32768 5= For problems 11 – 18 determine the exact value of each of the following without using a calculator. 11. 7log 343 12. 4log 1024
13. 38
27log512
14. 111log
121
15. 0.1log 0.0001 16. 16log 4 17. log10000
18. 5
1lne
For problems 19 – 20 write each of the following in terms of simpler logarithms 19. ( )7 3 8
7log 10a b c−
20. ( )32 2log 4z x +
21. 2 34
ln w tt w
+
For problems 22 – 24 combine each of the following into a single logarithm with a coefficient of one. 22. 7 ln 6ln 5lnt s w− +
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College Algebra
23. ( )1 log 1 2log 4log 3log2
z x y z+ − − −
24. ( )3 312 log 6log3
x y x+ + −
For problems 25 & 26 use the change of base formula and a calculator to find the value of each of the following. 25. 7log 100
26. 57
1log8
For problems 27 – 31 sketch each of the given functions. 27. ( ) ( )lng x x= − 28. ( ) ( )ln 3g x x= − 29. ( ) ( )ln 7g x x= + 30. ( ) ( )ln 2 4g x x= + − 31. ( ) ( )ln 6 2g x x= − +
Solving Exponential Equations Solve each of the following equations. 1. 4 7 1011 11x x+ −= 2. 4 73 3x x= 3. 2 312 2x x−− = 4.
2 12 49 9x x−= 5.
2 3 20 56 6x x x− +=
6. 14 2
1636
xx
++=
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College Algebra
7. 29 27x x+= 8. 4 18 1x+ = 9. 9 23 14 x−= 10. 2 8 26 8x x+ += 11. 5 7 313 2x x+ −= 12. 710 3x = 13. 2 316 10 x+= 14. 4 96 x+= e 15. 69 0x− =e 16.
2 2 4x − =e
Solving Logarithm Equations Solve each of the following equations. 1. ( ) ( )2
11 11log 3 log 3 16x x x+ = + 2. ( ) ( ) ( )ln 4 3 ln 7 ln 11x x− − = 3. ( ) ( ) ( )log log 12 log 10x x x+ + = − 4. ( ) ( ) ( )ln ln 15 ln 1x x x= − − + 5. ( )8log 4 1 1x + = − 6. ( ) ( )6 6log 3 log 5 1x x− + = 7. ( ) ( )3 3log log 6 3x x+ + = 8. ( ) ( )2
2 2log 2 log 8x x= + −
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College Algebra
9. ( ) ( )4 4log 2 log 6x x= − + 10. ( ) ( )log log 15 2x x− + − = 11. ( ) ( )ln ln 2 3x x+ − = 12. ( ) ( )22 log log 4 1 0x x x− + + =
Applications 1. We have $2,500 to invest and 80 months. How much money will we have if we put the money into an account that has an annual interest rate of 9% and interest is compounded (a) quarterly (b) monthly (c) continuously 2. We are starting with $60,000 and we’re going to put it into an account that earns an annual interest rate of 7.5%. How long will it take for the money in the account to reach $100,000 if the interest is compounded (a) quarterly (b) monthly (c) continuously 3. Suppose that we put some money in an account that has an annual interest rate of 10.25%. How long will it take to triple our money if the interest is compounded (a) twice a year (b) 8 times a year (c) continuously 4. A population of bacteria initially has 90,000 present and in 2 weeks there will be 200,000 bacteria present. (a) Determine the exponential growth equation for this population. (b) How long will it take for the population to grow from its initial population of 90,000 to a population of 150,000? 5. We initially have 2 kg grams of some radioactive element and in 7250 years there will be 1.5 kg left. (a) Determine the exponential decay equation for this element. (b) How long will it take for half of the element to decay? (c) How long will it take until there is 250 grams of the element left? 6. For a particular radioactive element the value of k in the exponential decay equation is given by
0.000825k = . (a) How long will it take for a quarter of the element to decay? (b) How long will it take for half of the element to decay? (c) How long will it take 90% of the element to decay?
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College Algebra
Systems of Equations
Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. Linear Systems with Two Variables Linear Systems with Three Variables Augmented Matrices More on the Augmented Matrix Nonlinear Systems
Linear Systems with Two Variables For problems 1 – 5 use the Method of Substitution to find the solution to the given system or to determine if the system is inconsistent or dependent. 1. 8 13
3 4 6x y
x y+ =+ = −
2. 3 7
2 6 4x yx y− =
− + =
3. 12 6 12
4 2 2x yx y
− + = −+ = −
4. 3 6 12
4 7 12x yx y+ =
− − = −
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College Algebra
5. 12 6 18
4 2 6x yx y− =− =
For problems 6 – 10 use the Method of Elimination to find the solution to the given system or to determine if the system is inconsistent or dependent. 6. 5 10 1
2 8x yx y
− + =− = −
7. 7 6 0
2 3 0x yx y+ =+ =
8. 8 24 12
10 30 15x yx y
− + =− = −
9. 2 3 24
3 8 57x yx y
− + =− = −
10. 6 4 20
7 3 35x yx y+ = −+ = −
Linear Systems with Three Variables Find the solution to each of the following systems of equations. 1. 3 7 2 8
2 5 108 2 3 38
x y zx y z
x y z
− + + = −− + − = −
− + =
2. 6 4 8 56
4 53 9 10
x y zx y zx y z
+ − = −− − + =
+ + =
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College Algebra
3. 2 6 12 9 19
4 3 7 25
x y zx y zx y z
+ − =− + + = −
+ − =
Augmented Matrices 1. For the following augmented matrix perform the indicated elementary row operations.
9 0 7 43 2 1 72 4 1 2
− − −
(a) 24R− (b) 3 1R R↔ (c) 1 3 110R R R− → 2. For the following augmented matrix perform the indicated elementary row operations.
9 3 11 62 7 4 31 1 1 1
− − − −
(a) 15R (b) 2 3R R↔ (c) 3 2 32R R R− → 3. For the following augmented matrix perform the indicated elementary row operations.
4 12 8 09 2 1 31 5 1 10
− − − − −
(a) 313
R (b) 1 2R R↔ (c) 2 1 252
R R R+ →
4. For the following augmented matrix perform the indicated elementary row operations.
1 5 6 23 15 18 34 2 7 1
− − − − − −
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College Algebra
(a) 37R− (b) 1 3R R↔ (c) 2 1 23R R R+ → Note : Problems using augmented matrices to solve systems of equations are in the next section.
More on the Augmented Matrix For each of the following systems of equations convert the system into an augmented matrix and use the augmented matrix techniques to determine the solution to the system or to determine if the system is inconsistent or dependent. 1. 8 13
3 4 6x y
x y+ =+ = −
2. 3 7
2 6 4x yx y− =
− + =
3. 12 6 12
4 2 2x yx y
− + = −+ = −
4. 3 6 12
4 7 12x yx y+ =
− − = −
5. 12 6 18
4 2 6x yx y− =− =
6. 5 10 1
2 8x yx y
− + =− = −
7. 7 6 0
2 3 0x yx y+ =+ =
8. 8 24 12
10 30 15x yx y
− + =− = −
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College Algebra
9. 2 3 243 8 57
x yx y
− + =− = −
10. 6 4 20
7 3 35x yx y+ = −+ = −
11. 3 7 2 8
2 5 108 2 3 38
x y zx y z
x y z
− + + = −− + − = −
− + =
12. 6 4 8 56
4 53 9 10
x y zx y zx y z
+ − = −− − + =
+ + =
13. 2 6 1
2 9 194 3 7 25
x y zx y zx y z
+ − =− + + = −
+ − =
Non-Linear Systems Find the solution to each of the following system of equations. 1. 2 5 16
7 8y x xy x= − + += −
2. 2
2
38 2
y xy x= −
= +
3. 2
2 14
4 4
yx
y x
+ =
= −
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College Algebra
4. 2 2
2
9
15
x yxy
+ =
= +
5. 2 2
22
16
115
x yxy
+ =
− =
6.
22
2
125
xyyx
= −
+ =
7. 2 2
22
1
14
x yx y
+ =
+ =
8. 2 2
22
3
19
x yx y
+ =
+ =
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