Schaum's Outline of College Algebra

416 Pages·1997·18.59 MB·English

by  Spiegel

SCHAUM’S OUTLINE OF
THEORY AND PROBLEMS
of
COLLEGE
ALGEBRA
Second Edition
MURRAY R. SPIEGEL, Ph.D.
Former Profdsor and Chairman
Mathematics Department
Rensselaer Polytechnic Institute
Hartford Graduate Center
ROBERT E. MOYER, Ph.D.
Professor of Mathematics
Fort Valley State University
SCHAUM’S OUTLINE SERIES
McGRAW-HILL
New York San Francisco Washington, D.C. Auckland Bogotci CaracuJ Lisbon
London Madrid Mexico City Milan Montreal New Dehli
San Juan Singapore Sydney Tokyo Toronto
MURRAY R. SPIEGEL received the M.S.d egree in Physics and the Ph.D.
in Mathematics from Cornell University. He had positions at Harvard
University, Columbia University, Oak Ridge, and Rensselaer Polytechnic
Institute, and had served as a mathematical consultant at several large
companies. His last position was Professor and Chairman of Mathematics
at the Rensselaer Polytechnic Institute, Hartford Graduate Center. He was
interested in most branches of mathematics, especially those which
involved applications to physics and engineering problems. He was the
author of numerous journal articles and 14 books on various topics in
mat hematics.
ROBERT E. MOYER has been teaching mathematics at Fort Valley State
University in Fort Valley, Georgia, since 1985. He served as head of the
Department of Mathematics and Physics from 1992 to 1994. Before joining
the FVSU faculty, he served as the mathematics consultant for a
five-county public school cooperative. His experiences include 12 years of
high school mathematics teaching in Illinois. He received his Doctor of
Philosophy in Mathematics Education from the University of Illinois in
1974. From Southern Illinois University he received his Master of Science
in 1967 and his Bachelor of Science in 1964, both in Mathematics
Education.
Schaum’s Outline of Theory and Problems of
COLLEGE ALGEBRA
Copyright @ 1998, 1956 by The McGraw-Hill Companies, Inc. All rights reserved.
Printed in the United States of America. Except as permitted under the Copyright
Act of 1976, no part of this publication may be reproduced or distributed in any
forms or by any means, or stored in a data base or retrieval system, without the
prior written permission of the publisher.
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 PRS PRS 9 0 2 1 0 9
ISBN 0-07-060266-2
Sponsoring Editor: Barbara Gilson
Production Supervisor: Tina Cameron
Editing Supervisor: Maureen B. Walker
Library of Congress Cataloging-in-Publication Data
Spiegel, MmyR .
Schaum’s outline of theory and problems of college algebral
--
Murray R. Spiegel, Robert E. Moyer. 2nd ed.
p. cm. -- (Schaum’s outline series)
Includes index.
ISBN 0-07-060266-2
1. Algebra -- Problems, excercises, etc. 2. Algebra -- Outlines,
syllabi, etc. I. Moyer. Robert E. 11. Title.
QA157S725 1997
5 12.9’076-dc21 97-31223
CIP
McGraw-Hi22 a
A Division of The McGmw-HiUC ompanies
In revising this book, the strengths of the first edition were retained while
reflecting the changes in the study of algebra since the first edition was written.
Many of the changes were based on changes in terminology and notation. This
edition focuses only on college algebra, includes the use of a graphing calculator,
provides tables for both common and natural logarithms, expands the numbers
of graphs and the material on graphing, adds material on matrices, and expands
the material on, analytic geometry.
The book is complete in itself and can be used equally well by those who are
studying college algebra for the first time as well as those who wish to review the
fundamental principles and procedures of college algebra. Students who are
studying advanced algebra in high school will be able to use the book as a source
of additional examples, explanations, and problems. The thorough treatment of
the topics of algebra allows an instructor to use the book as the textbook for a
course, as a resource for material on a specific topic, or as a source for additional
problems.
Each chapter contains a summary of the necessary definitions and theorems
followed by a set of solved problems. These solved problems include the proofs
of theorems and the derivations of formulas. The chapters end with a set of
supplementary problems and their answers.
The choice of whether to use a calculator or not is left to the student. A
calculator is not required, but its use is explained. Problem-solving procedures for
using logarithmic tables and for using a calculator are included. Procedures
for graphing expressions are discussed both using a graphing calculator and
manually.
ROBERTE . MOVER
Professor of Mathematics
Fort Valley State University
iii
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Contents
PROBLEMS ALSO FOUND IN THE COMPANION SCHAUM'S ELECTRONIC TUTOR x
Chapter 1 FUNDAMENTAL OPERATIONS WITH NUMBERS . . . . . . . . . . . . 1
1.1 Four Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 System of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Graphical Representation of Real Numbers . . . . . . . . . . . . . . . . . 2
1.4 Properties of Addition and Multiplication of Real Numbers . . . . . . . . 2
1.5 Rules of Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Exponents and Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.7 Operations with Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Chapter 2 FUNDAMENTAL OPERATIONS WITH ALGEBRAIC EXPRESSIONS 12
2.1 Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Computation with Algebraic Expressions . . . . . . . . . . . . . . . . . . 13
Chapter 3 PROPERTIES OF NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1 SetsofNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Additional Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Chapter 4 SPECIAL PRODUCTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1 Special Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Products Yielding Answers of the Form a" k b" . . . . . . . . . . . . . . 27
Chapter 5 FACTORING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.1 Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Factorization Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.3 Greatest Common Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.4 Least Common Multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Chapter 6 FRACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.1 Rational Algebraic Fractions . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Operations with Algebraic Fractions . . . . . . . . . . . . . . . . . . . . . 42
6.3 Complex Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Chapter 7 EXPONENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.1 Positive Integral Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.2 Negative Integral Exponent . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.3 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.4 Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.5 General Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . 49
7.6 Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
V
vi CONTENTS
Chapter 8 RADICALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8.1 Radical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8.2 Laws for Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8.3 Simplifying Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8.4 Operations with Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8.5 Rationalizing Binomial Denominators . . . . . . . . . . . . . . . . . . . . 60
Chapter 9 SIMPLE OPERATIONS WITH COMPLEX NUMBERS . . . . . . . . . . 67
9.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9.2 Graphical Representation of Complex Numbers . . . . . . . . . . . . . . 67
9.3 Algebraic Operations with Complex Numbers . . . . . . . . . . . . . . . 68
Chapter 10 EQUATIONS IN GENERAL . . . . . . . . . . . . . . . . . . . . . . . . . . 73
10.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
10.2 Operations Used in Transforming Equations . . . . . . . . . . . . . . . . 73
10.3 Equivalent Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
10.4 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
10.5 Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Chapter 11 RATIO. PROPORTION. AND VARIATION . . . . . . . . . . . . . . . . 81
11.1 Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
11.2 Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
11.3 Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
11.4 Unit Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
11.5 Best Buy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Chapter 12 FUNCTIONS AND GRAPHS . . . . . . . . . . . . . . . . . . . . . . . . . 89
12.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
12.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
12.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
12.4 Function Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
12.5 Rectangular Coordinate System . . . . . . . . . . . . . . . . . . . . . . . 90
12.6 Function of Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . 91
12.7 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
12.8 Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
12.9 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
12.10 Using a Graphing Calculator . . . . . . . . . . . . . . . . . . . . . . . . 93
Chapter 13 LINEAR EQUATIONS IN ONE VARIABLE . . . . . . . . . . . . . . . . 113
13.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
13.2 Literal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
13.3 Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Chapter 14 EQUATIONS OF LINES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
14.1 Slope of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
14.2 Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . 127
14.3 Slope-Intercept Form of Equation of a Line . . . . . . . . . . . . . . . 128
14.4 Slope-Point Form of Equation of a Line . . . . . . . . . . . . . . . . . 129
14.5 Two-Point Form of Equation of a Line . . . . . . . . . . . . . . . . . . 129
14.6 Intercept Form of Equation of a Line . . . . . . . . . . . . . . . . . . . 130
CONTENTS vii
Chapter 15 SIMULTANEOUS LINEAR EQUATIONS . . . . . . . . . . . . . . . . . . 136
15.1 Systems of Two Linear Equations . . . . . . . . . . . . . . . . . . . . . . 136
15.2 Systems of Three Linear Equations . . . . . . . . . . . . . . . . . . . . . 137
Chapter I6 QUADRATIC EQUATIONS IN ONE VARIABLE . . . . . . . . . . . . . 149
16.1 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
16.2 Methods of Solving Quadratic Equations . . . . . . . . . . . . . . . . . . 149
16.3 Sum and Product of the Roots . . . . . . . . . . . . . . . . . . . . . . . 150
16.4 Nature of the Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
16.5 Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
16.6 Quadratic-Type Equations . . . . . . . . . . . . . . . . . . . . . . . . . 151
Chapter 17 CONIC SECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
17.1 General Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . 167
17.2 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
17.3 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
17.4 Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
17.5 Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
17.6 Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
17.7 Graphing Conic Sections with a Calculator . . . . . . . . . . . . . . . . . 177
Chapter 18 SYSTEMS OF EQUATIONS INVOLVING QUADRATICS . . . . . . . . 188
18.1 Graphical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
18.2 Algebraic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Chapter 19 INEQUALITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
19.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
19.2 Principles of Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . 195
19.3 Absolute Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 196
19.4 Higher Degree Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 196
19.5 Linear Inequalities in Two Variables . . . . . . . . . . . . . . . . . . . . 197
19.6 Systems of Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 198
19.7 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Chapter 20 POLYNOMIAL FUNCTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . 210
20.1 Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
20.2 Zeros of Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . . 210
20.3 Solving Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . . 211
20.4 Approximating Real Zeros . . . . . . . . . . . . . . . . . . . . . . . . . 213
Chapter 21 RATIONAL FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
21.1 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
21.2 Vertical Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
21.3 Horizontal Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
21.4 Graphing Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . 232
21.5 Graphing Rational Functions Using a Graphing Calculator . . . . . . . . 233
Chapter 22 SEQUENCES AND SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . 241
22.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
22.2 Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
22.3 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
viii CONTENTS
22.4 Infinite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . 242
22.5 Harmonic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
22.6 Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Chapter 23 LOGARITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
23.1 Definition of a Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . 259
23.2 Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
23.3 Common Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
23.4 Using a Common Logarithm Table . . . . . . . . . . . . . . . . . . . . . 260
23.5 Natural Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
23.6 Using a Natural Logarithm Table . . . . . . . . . . . . . . . . . . . . . 261
23.7 Finding Logarithms Using a Calculator . . . . . . . . . . . . . . . . . . . 263
Chapter 24 APPLICATIONS OF LOGARITHMS AND EXPONENTS . . . . . . . . . 274
24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
24.2 Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
24.3 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
24.4 Applications of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . 276
24.5 Applications of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . 277
Chapter 25 PERMUTATIONS AND COMBINATIONS . . . . . . . . . . . . . . . . . . 287
25.1 Fundamental Counting Principle . . . . . . . . . . . . . . . . . . . . . . 287
25.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
25.3 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
25.4 Using a Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Chapter 26 THE BINOMIAL THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . 303
26.1 Combinatorial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
26.2 Expansion of (a + x ) ~. . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Chapter 27 PROBABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
27.1 Simple Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
27.2 Compound Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
27.3 Mathematical Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . 312
27.4 Binomial Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
27.5 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
Chapter 28 DETERMINANTS AND SYSTEMS OF LINEAR EQUATIONS . . . . . 325
28.1 Determinants of Second Order . . . . . . . . . . . . . . . . . . . . . . . 325
28.2 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
28.3 Determinants of Third Order . . . . . . . . . . . . . . . . . . . . . . . . 326
Chapter 29 DETERMINANTS OF ORDER n . . . . . . . . . . . . . . . . . . . . . . . 335
29.1 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
29.2 Determinants of Order n . . . . . . . . . . . . . . . . . . . . . . . . . . 335
29.3 Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . 336
29.4 Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
29.5 Value of a Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
29.6 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
29.7 Homogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . . . 338
CONTENTS ix
Chapter 30 MATRICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
30.1 Definition of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
30.2 Operations with Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 353
30.3 Elementary Row Operations . . . . . . . . . . . . . . . . . . . . . . . . 355
30.4 Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
30.5 Matrix Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
30.6 Matrix Solution of a System of Equations . . . . . . . . . . . . . . . . . 357
Chapter 31 MATHEMATICAL INDUCTION . . . . . . . . . . . . . . . . . . . . . . . . 366
31.1 Principle of Mathematical Induction . . . . . . . . . . . . . . . . . . . . 366
31.2 Proof by Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . 366
Chapter 32 PARTIAL FRACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
32.1 Rational Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
32.2 Proper Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
32.3 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
32.4 Identically Equal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 372
32.5 Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
32.6 Finding the Partial Fraction Decomposition . . . . . . . . . . . . . . . . 374
Appendix A TABLE OF COMMON LOGARITHMS . . . . . . . . . . . . . . . . . . . 379
Appendix B TABLE OF NATURAL LOGARITHMS . . . . . . . . . . . . . . . . . . . 382
c
Appendix SAMPLE SCREENS FROM THE COMPANION
SCHAUM’S ELECTRONIC TUTOR . . . . . . . . . . . . . . . . . . . . . . 385
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401