, SOLUTION MANUAL FOR Algebra & Trig 12th Edition Larson
Important Notes
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provided.
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, CONTENTS
Part I Solutions to All Exercises .................................................................... 1
Chapter P Prerequisites............................................................................................ 1
Chapter 1 Equations, Inequalities, and Mathematical Modeling ........................ 94
Chapter 2 Functions and Their Graphs............................................................... 308
Chapter 3 Polynomial Functions ........................................................................ 498
Chapter 4 Rational Functions and Conics .......................................................... 691
Chapter 5 Exponential and Logarithmic Functions ........................................... 858
Chapter 6 Trigonometry .................................................................................... 1010
Chapter 7 Analytic Trigonometry..................................................................... 1223
Chapter 8 Additional Topics in Trigonometry................................................. 1365
Chapter 9 Systems of Equations and Inequalities ............................................ 1520
Chapter 10 Matrices and Determinants .............................................................. 1720
Chapter 11 Sequences, Series, and Probability .................................................. 1880
Appendix A Errors and the Algebra of Calculus ................................................. 2018
Appendix B Concepts in Statistics ....................................................................... 2030
Getting Ready Solutions .......................................................................................... 2061
Solutions to Checkpoints ............................................................... 2085
Part II Solutions to Chapter Tests ............................................................ 2342
,
, Solution and Answer Guide: Larson, Algebra & Trig, 12e, 2027, 9798214407227;
Chapter 0P: Prerequisites
Solution and Answer Guide
LARSON, ALGEBRA & TRIG, 12E, 2027, 9798214407227;
CHAPTER 0P: PREREQUISITES
TABLE OF CONTENTS
Section P.1 Real Numbers and Their Properties ................................................................ 2
Section P.2 Exponents and Radicals .................................................................................. 11
Section P.3 Polynomials and Factoring ............................................................................. 21
Section P.4 Factoring Polynomials ................................................................................... 34
Section P.5 Rational Expressions ...................................................................................... 47
Section P.6 The Rectangular Coordinate System and Graphs ........................................61
Review Exercises for Chapter P ........................................................................................ 72
Problem Solving for Chapter P .......................................................................................... 89
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 1
accessible website, in whole or in part.
, Solution and Answer Guide: Larson, Algebra & Trig, 12e, 2027, 9798214407227;
Chapter 0P: Prerequisites
SECTION P.1 REAL NUMBERS AND THEIR PROPERTIES
P.1.1.
terms
P.1.2.
Zero-Factor Property
P.1.3.
Yes. 3 − 10 = −7 = 7 and 10 − 3 = 7 = 7.
P.1.4.
(a) iii
(b) iv
(c) ii
(d) v
(e) i
P.1.5.
−9, − 72 , 5, 32 , 3, 0, 3, − 4, 2, −11
(a) Natural numbers: 5, 8, 2
(b) Whole numbers: 5, 0, 8, 2
(c) Integers: −9, 5, 0, 8, − 4, 2, − 11
(d) Rational numbers: −9, − 7 , 5, 2, 0, 8, − 4, 2, − 11
2 3
(e) Irrational numbers: 3
P.1.6.
5, − 7, − 73 , 0, 3.14, 45 , − 3, 12, 5
(a) Natural numbers: 12, 5
(b) Whole numbers: 0, 12, 5
(c) Integers: −7, 0, − 3, 12, 5
(d) Rational numbers: −7, − 7 , 0, 3.14, 5, − 3, 12, 5
3 4
(e) Irrational numbers: 5
P.1.7.
2.01, 0.6, − 13, 0.010110111 , 1, − 6
(a) Natural numbers: 1
(b) Whole numbers: 1
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 2
accessible website, in whole or in part.
, Solution and Answer Guide: Larson, Algebra & Trig, 12e, 2027, 9798214407227;
Chapter 0P: Prerequisites
(c) Integers: −13, 1, − 6
(d) Rational numbers: 2.01, 0.6, − 13, 1, − 6
(e) Irrational numbers: 0.010110111
P.1.8.
25, − 17, − 12 , 9, 3.12, 1 , 18, − 11.1, 13
5 2
(a) Natural numbers: 25, 9 = 3, 18, 13
(b) Whole numbers: 25, 9 = 3, 18, 13
(c) Integers: 25, − 17, 9 = 3, 18, 13
(d) Rational numbers: 25, − 17, − 12 , 9 = 3, 3.12, 18, − 11.1, 13
5
(e) Irrational numbers: 1
2
P.1.9.
5 2
6 3
P.1.10.
−4 −8
P.1.11.
−5.2 −8.5
P.1.12.
− 43 −4.75
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 3
accessible website, in whole or in part.
, Solution and Answer Guide: Larson, Algebra & Trig, 12e, 2027, 9798214407227;
Chapter 0P: Prerequisites
P.1.13.
The inequality x 5 denotes the set of all real numbers less than or equal to 5.
P.1.14.
The inequality x 0 denotes the set of all real numbers less than zero.
P.1.15.
The inequality −2 x 2 denotes the set of all real numbers greater than −2
and less than 2.
P.1.16.
The inequality 0 x 6 denotes the set of all positive real numbers less than
or equal to 6.
P.1.17.
)
The interval 4, denotes the set of all real numbers greater than or equal to
4; x 4.
P.1.18.
( )
The interval −, 2 denotes the set of all real numbers less than 2; x 2.
P.1.19.
)
The interval −5, 2 denotes the set of all real numbers greater than or equal
to − 5 and less than 2; −5 x 2.
P.1.20.
(
The interval −1, 2 denotes the set of all real numbers greater than −1 and
less than or equal to 2; −1 x 2.
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 4
accessible website, in whole or in part.
, Solution and Answer Guide: Larson, Algebra & Trig, 12e, 2027, 9798214407227;
Chapter 0P: Prerequisites
P.1.21.
( −, 0 ; y 0
P.1.22.
( −, 25 ; y 25
P.1.23.
10, 22 ; 10 t 22
P.1.24.
)
−3, 5 ; − 3 k 5
P.1.25.
( )
−10 = − −10 = 10
P.1.26.
0 =0
P.1.27.
−1 − −2 = 1 − 2 = −1
P.1.28.
−3 − −3 = −3 − 3 = −6 ()
P.1.29.
()
5 − 5 = 5 5 = 25
P.1.30.
( )
− 4 − 4 = − 4 4 = − 16
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 5
accessible website, in whole or in part.
, Solution and Answer Guide: Larson, Algebra & Trig, 12e, 2027, 9798214407227;
Chapter 0P: Prerequisites
P.1.31.
If x −2, then x + 2 is negative.
So,
x +2
=
(
− x +2 ) = −1.
x +2 x +2
P.1.32.
If x 1, then x − 1 is positive.
x−1 x−1
So, = = 1.
x−1 x−1
P.1.33.
−4 = 4 because −4 = 4 and 4 = 4.
P.1.34.
−5 = − 5 because −5 = −5.
P.1.35.
− −6 −6 because −6 = 6 and
()
− −6 = − 6 = −6.
P.1.36.
− −2 = − 2 because −2 = −2.
P.1.37.
( )
d 126, 75 = 75 − 126 = 51
P.1.38.
( )
d −20, 30 = 30 − −20 = 50 = 50 ( )
P.1.39.
( )
d − 52 , 0 = 0 − − ( )= 5
2
5
2
P.1.40.
(
d − 41 , − 11
4 ) = − − (− ) = −
11
4
1
4
5
2
= 5
2
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 6
accessible website, in whole or in part.
Important Notes
The file includes the complete test bank, organized chapter by chapter.
A sample of selected pages has been provided for preview.
All available appendices and Excel files (if included in the original resources) are
provided.
We continuously update our files to ensure you receive the latest and most accurate
editions.
New editions are added regularly – stay connected for updates!
⚠️Note on Answer Keys: If the answer key is not included within the chapter
questions, you will find the complete answers and solutions at the end of each
chapter.
✅ Why Buy From Us?
📚 Complete & organized chapter-by-chapter – no missing content, no guessing.
⚡ Instant digital delivery – get your file the moment you pay, no waiting.
📅 Always up to date – we track new editions so you always get the latest version.
💬 Friendly support – real humans ready to help, anytime you need us.
🔒 Safe & secure – thousands of satisfied students trust us every semester.
🛡️Our Guarantees
💰 Money-Back Guarantee: Not satisfied? We offer a full refund – no questions asked.
🔄 Wrong File? No Problem: Contact us and we will replace it immediately with the
correct version, free of charge.
⏰ 24/7 Support: We are always here – reach out anytime and expect a fast response.
Contact Email:
, CONTENTS
Part I Solutions to All Exercises .................................................................... 1
Chapter P Prerequisites............................................................................................ 1
Chapter 1 Equations, Inequalities, and Mathematical Modeling ........................ 94
Chapter 2 Functions and Their Graphs............................................................... 308
Chapter 3 Polynomial Functions ........................................................................ 498
Chapter 4 Rational Functions and Conics .......................................................... 691
Chapter 5 Exponential and Logarithmic Functions ........................................... 858
Chapter 6 Trigonometry .................................................................................... 1010
Chapter 7 Analytic Trigonometry..................................................................... 1223
Chapter 8 Additional Topics in Trigonometry................................................. 1365
Chapter 9 Systems of Equations and Inequalities ............................................ 1520
Chapter 10 Matrices and Determinants .............................................................. 1720
Chapter 11 Sequences, Series, and Probability .................................................. 1880
Appendix A Errors and the Algebra of Calculus ................................................. 2018
Appendix B Concepts in Statistics ....................................................................... 2030
Getting Ready Solutions .......................................................................................... 2061
Solutions to Checkpoints ............................................................... 2085
Part II Solutions to Chapter Tests ............................................................ 2342
,
, Solution and Answer Guide: Larson, Algebra & Trig, 12e, 2027, 9798214407227;
Chapter 0P: Prerequisites
Solution and Answer Guide
LARSON, ALGEBRA & TRIG, 12E, 2027, 9798214407227;
CHAPTER 0P: PREREQUISITES
TABLE OF CONTENTS
Section P.1 Real Numbers and Their Properties ................................................................ 2
Section P.2 Exponents and Radicals .................................................................................. 11
Section P.3 Polynomials and Factoring ............................................................................. 21
Section P.4 Factoring Polynomials ................................................................................... 34
Section P.5 Rational Expressions ...................................................................................... 47
Section P.6 The Rectangular Coordinate System and Graphs ........................................61
Review Exercises for Chapter P ........................................................................................ 72
Problem Solving for Chapter P .......................................................................................... 89
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 1
accessible website, in whole or in part.
, Solution and Answer Guide: Larson, Algebra & Trig, 12e, 2027, 9798214407227;
Chapter 0P: Prerequisites
SECTION P.1 REAL NUMBERS AND THEIR PROPERTIES
P.1.1.
terms
P.1.2.
Zero-Factor Property
P.1.3.
Yes. 3 − 10 = −7 = 7 and 10 − 3 = 7 = 7.
P.1.4.
(a) iii
(b) iv
(c) ii
(d) v
(e) i
P.1.5.
−9, − 72 , 5, 32 , 3, 0, 3, − 4, 2, −11
(a) Natural numbers: 5, 8, 2
(b) Whole numbers: 5, 0, 8, 2
(c) Integers: −9, 5, 0, 8, − 4, 2, − 11
(d) Rational numbers: −9, − 7 , 5, 2, 0, 8, − 4, 2, − 11
2 3
(e) Irrational numbers: 3
P.1.6.
5, − 7, − 73 , 0, 3.14, 45 , − 3, 12, 5
(a) Natural numbers: 12, 5
(b) Whole numbers: 0, 12, 5
(c) Integers: −7, 0, − 3, 12, 5
(d) Rational numbers: −7, − 7 , 0, 3.14, 5, − 3, 12, 5
3 4
(e) Irrational numbers: 5
P.1.7.
2.01, 0.6, − 13, 0.010110111 , 1, − 6
(a) Natural numbers: 1
(b) Whole numbers: 1
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 2
accessible website, in whole or in part.
, Solution and Answer Guide: Larson, Algebra & Trig, 12e, 2027, 9798214407227;
Chapter 0P: Prerequisites
(c) Integers: −13, 1, − 6
(d) Rational numbers: 2.01, 0.6, − 13, 1, − 6
(e) Irrational numbers: 0.010110111
P.1.8.
25, − 17, − 12 , 9, 3.12, 1 , 18, − 11.1, 13
5 2
(a) Natural numbers: 25, 9 = 3, 18, 13
(b) Whole numbers: 25, 9 = 3, 18, 13
(c) Integers: 25, − 17, 9 = 3, 18, 13
(d) Rational numbers: 25, − 17, − 12 , 9 = 3, 3.12, 18, − 11.1, 13
5
(e) Irrational numbers: 1
2
P.1.9.
5 2
6 3
P.1.10.
−4 −8
P.1.11.
−5.2 −8.5
P.1.12.
− 43 −4.75
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 3
accessible website, in whole or in part.
, Solution and Answer Guide: Larson, Algebra & Trig, 12e, 2027, 9798214407227;
Chapter 0P: Prerequisites
P.1.13.
The inequality x 5 denotes the set of all real numbers less than or equal to 5.
P.1.14.
The inequality x 0 denotes the set of all real numbers less than zero.
P.1.15.
The inequality −2 x 2 denotes the set of all real numbers greater than −2
and less than 2.
P.1.16.
The inequality 0 x 6 denotes the set of all positive real numbers less than
or equal to 6.
P.1.17.
)
The interval 4, denotes the set of all real numbers greater than or equal to
4; x 4.
P.1.18.
( )
The interval −, 2 denotes the set of all real numbers less than 2; x 2.
P.1.19.
)
The interval −5, 2 denotes the set of all real numbers greater than or equal
to − 5 and less than 2; −5 x 2.
P.1.20.
(
The interval −1, 2 denotes the set of all real numbers greater than −1 and
less than or equal to 2; −1 x 2.
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 4
accessible website, in whole or in part.
, Solution and Answer Guide: Larson, Algebra & Trig, 12e, 2027, 9798214407227;
Chapter 0P: Prerequisites
P.1.21.
( −, 0 ; y 0
P.1.22.
( −, 25 ; y 25
P.1.23.
10, 22 ; 10 t 22
P.1.24.
)
−3, 5 ; − 3 k 5
P.1.25.
( )
−10 = − −10 = 10
P.1.26.
0 =0
P.1.27.
−1 − −2 = 1 − 2 = −1
P.1.28.
−3 − −3 = −3 − 3 = −6 ()
P.1.29.
()
5 − 5 = 5 5 = 25
P.1.30.
( )
− 4 − 4 = − 4 4 = − 16
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 5
accessible website, in whole or in part.
, Solution and Answer Guide: Larson, Algebra & Trig, 12e, 2027, 9798214407227;
Chapter 0P: Prerequisites
P.1.31.
If x −2, then x + 2 is negative.
So,
x +2
=
(
− x +2 ) = −1.
x +2 x +2
P.1.32.
If x 1, then x − 1 is positive.
x−1 x−1
So, = = 1.
x−1 x−1
P.1.33.
−4 = 4 because −4 = 4 and 4 = 4.
P.1.34.
−5 = − 5 because −5 = −5.
P.1.35.
− −6 −6 because −6 = 6 and
()
− −6 = − 6 = −6.
P.1.36.
− −2 = − 2 because −2 = −2.
P.1.37.
( )
d 126, 75 = 75 − 126 = 51
P.1.38.
( )
d −20, 30 = 30 − −20 = 50 = 50 ( )
P.1.39.
( )
d − 52 , 0 = 0 − − ( )= 5
2
5
2
P.1.40.
(
d − 41 , − 11
4 ) = − − (− ) = −
11
4
1
4
5
2
= 5
2
© 2027 Cengage Learning, Inc. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly 6
accessible website, in whole or in part.
