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SCHAUM’S OUTLINE OF Mathematical Methods for Business and Economics Schaum’s Outline of Mathematical Methods for Business and Economics Edward T. Dowling, Ph.D. Chair and Professor Department of Economics Fordham University Schaum’s Outline Series Copyright © 1993 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN: 978-0-07-170246-1 MHID: 0-07-170246-6 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-163532-5, MHID: 0-07-163532-7. All trademarks are trademarks of their respective owners. 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Preface Students of undergraduate business and economics and candidates for the M.B.A. and M.A. degrees in economics today need a variety of mathematical skills to successfully complete their degree requirements and compete effectively in their chosen careers. Unfortunately, the requisite mathematical competence is not the subject of a single course in mathematics such as Calculus I or Linear Algebra I, and many students, pressed with the demands from business and economics courses, do not have space in their schedules for a series of math courses. Mathematical Methods for Business and Economics is designed to cull the mathematical tools, topics, and techniques essential for success in business and economics today. It is suitable for a one- or two-semester course in business mathematics, depending on the previous background of the students. It can also be used profitably in an introductory calculus or linear algebra course by professors and students interested in the business connections and applications. The theory-and-solved-problem format of each chapter provides concise explanations illustrated by examples, plus numerous problems with fully worked-out solutions. No mathematical proficiency beyond the high-school level is assumed. The learning-by-doing pedagogy will enable students to progress at their own rate and adapt the book to their own needs. Mathematical Methods for Business and Economics can be used by itself or as a supplement to other texts for undergraduate and graduate students in business and economics. It is largely selfcontained. Starting with a basic review of high-school algebra in Chapter 1, the book consistently explains all the concepts and techniques needed for the material in subsequent chapters. This book contains 1,066 problems, all of them solved in considerable detail. To derive the most from the book, students should strive as soon as possible to work independently of the solutions. This can be done by solving problems on individual sheets of paper with the book closed. If difficulties arise, the solution can then be checked in the book. For best results, students should never be satisfied with passive knowledge—the capacity merely to follow or comprehend the various steps presented in the book. Mastery of the subject and doing well on exams require active knowledge—the ability to solve any problem, in any order, without the aid of the book. Experience has proved that students of very different backgrounds and abilities can be successful in handling the subject matter of this text when the material is presented in the current format. I wish to express my gratitude for help with this book to Dr. Dominick Salvatore and Dr. Timothy Weithers of Fordham University; Maria Cristina Cacdac-Ampil, a doctoral candidate at Fordham; Professor Henry Mark Smith, the reviewer; and John Carleo, John Aliano, Maureen Walker, Pat Koch, and Patty Andrews of McGraw-Hill. EDWARD T. DOWLTNG Contents Chapter 1 REVIEW 1.1 Exponents 1.2 Polynomials 1.3 Factoring 1.4 Fractions 1.5 Radicals 1.6 Order of Mathematical Operations 1.7 Use of a Pocket Calculator Chapter 2 EQUATIONS AND GRAPHS 2.1 Equations 2.2 Cartesian Coordinate System 2.3 Linear Equations and Graphs 2.4 Slopes 2.5 Intercepts 2.6 The Slope-Intercept Form 2.7 Determining the Equation of a Straight-Line 2.8 Applications of Linear Equations in Business and Economics Chapter 3 FUNCTIONS 3.1 Concepts and Definitions 3.2 Graphing Functions 3.3 The Algebra of Functions 3.4 Applications of Linear Functions for Business and Economics 3.5 Solving Quadratic Equations 3.6 Facilitating Nonlinear Graphing 3.7 Applications of Nonlinear Functions in Business and Economics Chapter 4 SYSTEMS OF EQUATIONS 4.1 Introduction 4.2 Graphical Solutions 4.3 Supply-and-Demand Analysis 4.4 Break-Even Analysis 4.5 Elimination and Substitution Methods 4.6 Income Determination Models 4.7 IS-LM Analysis 4.8 Economic and Mathematical Modeling (Optional) 4.9 Implicit Functions and Inverse Functions (Optional) Chapter 5 LINEAR (OR MATRIX) ALGEBRA 5.1 Introduction 5.2 Definitions and Terms 5.3 Addition and Subtraction of Matrices 5.4 Scalar Multiplication 5.5 Vector Multiplication 5.6 Multiplication of Matrices 5.7 Matrix Expression of a System of Linear Equations 5.8 Augmented Matrix 5.9 Row Operations 5.10 Gaussian Method of Solving Linear Equations Chapter 6 SOLVING LINEAR EQUATIONS WITH MATRIX ALGEBRA 6.1 Determinants and Linear Independence 6.2 Third-Order Determinants 6.3 Cramer’s Rule for Solving Linear Equations 6.4 Inverse Matrices 6.5 Gaussian Method of Finding an Inverse Matrix 6.6 Solving Linear Equations with an Inverse Matrix 6.7 Business and Economic Applications 6.8 Special Determinants Chapter 7 LINEAR PROGRAMMING: USING GRAPHS 7.1 Use of Graphs 7.2 Maximization Using Graphs 7.3 The Extreme-Point Theorem 7.4 Minimization Using Graphs 7.5 Slack and Surplus Variables 7.6 The Basis Theorem Chapter 8 LINEAR PROGRAMMING: THE SIMPLEX ALGORITHM AND THE DUAL 8.1 The Simplex Algorithm 8.2 Maximization 8.3 Marginal Value or Shadow Pricing 8.4 Minimization 8.5 The Dual 8.6 Rules of Transformation to Obtain the Dual 8.7 The Dual Theorems 8.8 Shadow Prices in the Dual 8.9 Integer Programming 8.10 Zero-One Programming Chapter 9 DIFFERENTIAL CALCULUS: THE DERIVATIVE AND THE RULES OF DIFFERENTIATION 9.1 Limits 9.2 Continuity 9.3 The Slope of a Curvilinear Function 9.4 The Derivative 9.5 Differentiability and Continuity 9.6 Derivative Notation 9.7 Rules of Differentiation 9.8 Higher-Order Derivatives 9.9 Implicit Functions Chapter 10 DIFFERENTIAL CALCULUS: USES OF THE DERIVATIVE 10.1 Increasing and Decreasing Functions 10.2 Concavity and Convexity 10.3 Relative Extrema 10.4 Inflection Points 10.5 Curve Sketching 10.6 Optimization of Functions 10.7 The Successive-Derivative Test 10.8 Marginal Concepts in Economics 10.9 Optimizing Economic Functions for Business 10.10 Relationship Among Total, Marginal, and Average Functions Chapter 11 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 11.1 Exponential Functions 11.2 Logarithmic Functions 11.3 Properties of Exponents and Logarithms 11.4 Natural Exponential and Logarithmic Functions 11.5 Solving Natural Exponential and Logarithmic Functions 11.6 Logarithmic Transformation of Nonlinear Functions 11.7 Derivatives of Natural Exponential and Logarithmic Functions 11.8 Interest Compounding 11.9 Estimating Growth Rates from Data Points Chapter 12 INTEGRAL CALCULUS 12.1 Integration 12.2 Rules for Indefinite Integrals 12.3 Area under a Curve 12.4 The Definite Integral 12.5 The Fundamental Theorem of Calculus 12.6 Properties of Definite Integrals 12.7 Area between Curves 12.8 Integration by Substitution 12.9 Integration by Parts 12.10 Present Value of a Cash Flow 12.11 Consumers’ and Producers’ Surplus Chapter 13 CALCULUS OF MULTIVARIABLE FUNCTIONS 13.1 Functions of Several Independent Variables 13.2 Partial Derivatives 13.3 Rules of Partial Differentiation 13.4 Second-Order Partial Derivatives 13.5 Optimization of Multivariable Functions 13.6 Constrained Optimization with Lagrange Multipliers 13.7 Income Determination Multipliers 13.8 Optimizing Multivariable Functions in Business and Economics 13.9 Constrained Optimization of Multivariable Economic Functions 13.10 Constrained Optimization of Cobb-Douglas Production Functions 13.11 Implicit and Inverse Function Rules (Optional) INDEX Chapter 1 Review 1.1 EXPONENTS Given a positive integer n, xn signifies that x is multiplied by itself n number of times. Here x is referred to as the base; n is called an exponent. By convention an exponent of one is not expressed: x(1) = x, 8(1) = 8. By definition any nonzero number or variable raised to the zero power is equal to 1: x0 = 1, 30 = 1, and 00 is undefined. Assuming a and b are positive integers and x and y are real numbers for which the following exist, the rules of exponents are presented below, illustrated in Examples 1 and 2, and treated in Problems 1.1, 1.24, 1.26, and 1.27. EXAMPLE 1. In multiplication, exponents of the same variable are added; in division, exponents of the same variable are subtracted; when raised to a power, the exponents of a variable are multiplied, as indicated by the rules above and shown in the examples below followed by illustrations. (a) x2 · x5 = x2+5 = x7 ≠ x10 (Rule 1) x2 · x5 = (x · x)(x · x · x · x · x) = x7 (Rule 2) (c) (x3)2 = x3·2 = x6 ≠ x9 or x5 (Rule 3) (x3)2 = (x · x · x)(x · x · x) = x6 (d) (xy)3 = x3y3 ≠ xy3 or x3y (Rule 4) (xy)3 = (xy)(xy)(xy) = (x · x · x)(y · y · y) = x3y3 (Rule 5) (Rules 2 and 6) (Rule 7) Since and from Rule 1 exponents of a common base are added in multiplication, the exponent of , when added to itself, must equal 1. With the exponent of must equal . Thus, , . Just as so x1/4 · x1/4 · x1/4 · x1/4 = x1/4 + 1/4 + 1/4 + 1/4 = x1 = x. See Problems 1.1, 1.24, 1.26, and 1.27. EXAMPLE 2. From Rule 2, it can easily be seen why any variable or nonzero number raised to the zero power equals one. For example, x3/x3 = x3−3 = x0 = 1; 85/85 = 85−5 = 80 = 1. 1.2 POLYNOMIALS Given an expression such as 9x5, x is called a variable because it can assume any number of different values, and 9 is referred to as the coefficient of x. Expressions consisting simply of a real number or of a coefficient times one or more variables raised to the power of a positive integer are called monomials. Monomials can be added or subtracted to form polynomials. Each monomial constituting a polynomial is called a term. Terms that have the same variables and respective exponents are called like terms. The degree of a monomial is the sum of the exponents of its variables. The degree of a polynomial is the degree of its highest term. Rules for adding, subtracting, multiplying, and dividing polynomials are explained below, illustrated in Examples 3 to 5, and treated in Problems 1.3 and 1.4. 1.2.1 Addition and Subtraction of Polynomials Like terms in polynomials can be added or subtracted by adding or subtracting their coefficients. Unlike terms cannot be so added or subtracted. EXAMPLE 3. (a) 6x3 + 15x3 = 21x3 (b) 18xy − 7xy = 11xy (c) (4x3 + 13x2 − 7x) + (11x3 − 8x2 − 9x) = 15x3 + 5x2 − 16x (d) (22x − 19y) + (7x + 6z) = 29x − 19y) + 6z See also Problem 1.3. 1.2.2 Multiplication and Division of Terms Like and unlike terms can be multiplied or divided by multiplying or dividing both the coefficients and variables. EXAMPLE 4. (a) 20x4 · 7y6 = 140x4y6 (b) 6x2y3 · 8x4y6 = 48x6y9 (c) 12x3y2 · 5y4z5 = 60x3y6z5 (d) 3x3y2z5 · 15x4y3z4 = 45x7y5z9 1.2.3 Multiplication of Polynomials To multiply two polynomials, multiply each term in the first polynomial by each term in the second polynomial and then add their products together. EXAMPLE 5. See also Problem 1.4. 1.3 FACTORING Factoring reverses the process of polynomial multiplication in order to express a given polynomial as a product of simpler polynomials called factors. A monomial such as the number 14 is easily factored by expressing it as a product of its integer factors 1 · 14, 2 · 7, (−1) · (−14), or (−2) · (−7). A binomial such as 5x4 − 45x3 is easily factored by dividing or factoring out the greatest common factor, here 5x3, to obtain 5x3(x − 9). Factoring a trinomial such as mx2 + nx + p, however, generally requires the following rules: 1. Given (mx2 + nx + p), the factors are (ax + c)(bx + d), where (1) ab = m; (2) cd = p; and (3) ad + bc = n. 2. Given (mx2 + nxy + py2), the factors are (ax + cy)(bx + dy), where (1) ab = m; (2) cd = p; and (3) ad + bc = n, exactly as above. For proof of these rules, see Problems 1.28 and 1.29. EXAMPLE 6. To factor (x2 + 11x + 24), where in terms of Rule 1 (above) m = 1, n = 11, and p = 24, we seek integer factors such that: 1) a · b = 1. Integer factors: 1 · 1, (−1) · (−1). For simplicity we shall consider only positive sets of integer factors here and in step 2. 2) c · d = 24. Integer factors: 1 · 24, 2 · 12, 3 · 8, 4 · 6, 6 · 4, 8 · 3, 12 · 2, 24 · 1. 3) ad + bc = 11. With a = b = 1, c + d must equal 11. Adding the different combinations of factors from step 2, we have 1 + 24 = 25, 2 + 12 = 14, 3 + 8 = 11, 4 + 6 = 10, 6 + 4 = 10, 8 + 3 = 11, 12 + 2 = 14, and 21 + 1 = 25. Since only 3 + 8 and 8 + 3 = 11 in step 3, 3 and 8 are the only candidates for c and d from step 2 which, when used with a = b = 1 from step 1, fulfill all the above requirements, and the order does not matter. Hence (x2 + 11x + 24) = (x + 3)(x + 8) or (x + 8)(x + 3) See Problems 1.5 to 1.13. For derivation of the rules, see Problems 1.28 and 1.29. 1.4 FRACTIONS Fractions, or rational numbers, consist of polynomials in both numerator and denominator, assuming always that the denominator does not equal zero. Reducing a fraction to lowest terms involves the cancellation of all common factors from both the numerator and the denominator. Raising a fraction to higher terms means multiplying the numerator and denominator by the same nonzero polynomial. Assuming that A, B, C, and D are polynomials and C and D ≠ 0, fractions are governed by the following rules: The properties of fractions are illustrated in Example 7 and treated in Problems 1.14 to 1.21. EXAMPLE 7. (a) Multiplying or dividing both the numerator and the denominator of a fraction by the same nonzero number or polynomial leaves the value of the fraction unchanged. (Rule 1) Rule 1 provides the basis for reducing a fraction to its lowest terms as well as for raising a fraction to higher terms. (b) To multiply fractions, simply multiply the numerators and the denominators separately. The product of the numerators then forms the numerator of the product and the product of the denominators forms the denominator of the product. (Rule 2) (c) To divide fractions, simply invert the divisor and multiply. (Rule 3) (d) Fractions can be added or subtracted only if they have exactly the same denominator, called a common denominator. If a common denominator is present, simply add or subtract the numerators and set the result over the common denominator. Remember always to subtract all the terms within a given set of parentheses. (Rule 4) (e) To add or subtract fractions with different denominators, a common denominator must first be found. Multiplication of one denominator by the other will always produce a common denominator. Each fraction can then be restated in terms of the common denominator using Rule 1 and the numerators added as in (d). (Rule 5) (f) Similarly, (Rule 5) The least common denominator (LCD) of two or more fractions is the polynomial of lowest degree and smallest coefficient that is exactly divisible by the denominators of the original fractions. Use of the LCD helps simplify the final sum or difference. See Problems 1.19 to 1.21. Fractions are reviewed in Problems 1.14 to 1.21. 1.5 RADICALS If bn = a, where b > 0, then by taking the nth root of both sides of the equation, , where is a radical (sign), a is the radicami, and n is the index. For square roots, the index 2 is not expressed. Thus, . From Rules 7 and 8 in Section 1.1, we should also be aware that and . Assuming x and y are real nonnegative numbers and m and n are positive integers such that exist, the rules of radicals are given below. For proof of Rule 1, see Problem 1.30. and EXAMPLE 8. The laws of radicals are used to simplify the following expressions. Note that for evennumbered roots, positive and negative answers are equally valid. See also Problems 1.22, 1.23, and Problems 1.25 to 1.27. 1.6 ORDER OF MATHEMATICAL OPERATIONS Given an expression involving multiple mathematical operations, computations within parentheses are performed first. If there are parentheses within parentheses, computations on the innermost set take precedence. Within parentheses, all constants and variables are first raised to the powers of their respective exponents. Multiplication and division are then performed before addition and subtraction. In carrying out operations of the same priority, the procedure is from left to right. In sum, 1. Start within parentheses, beginning with the innermost. 2. Raise all terms to their respective exponents. 3. Multiply and divide before adding and subtracting. 4. For similar priorities, move from left to right. EXAMPLE 9. The following steps are performed to solve 1. 52 = 25 2. 25 · 6 = 150 4. 15 − 8 = 7 Thus 1.7 USE OF A POCKET CALCULATOR Pocket calculators are helpful for checking one’s ordinary calculations and performing arduous or otherwise time-consuming computations. Rules for the different mathematical operations are set forth and illustrated below, including some rales which will not be used or needed until later in the text. 1.7.1 Addition of Two Numbers To add two numbers, enter the first number, press the press the key, and enter the second number. Then key to find the total. EXAMPLE 10. (a) To find 139 + 216, enter 139, press ihe 139 + 216 = 355. key, enter 216, and press the key to find (b) To find 1025 + 38.75, enter 1025, press thethe key, then enter 38.75, and hit the key to find 1025 + 38.75 = 1063.75. Practice this and subsequent examples using simple numbers to which you already know the answers to see if you are doing the procedure correctly. 1.7.2 Addition of More Than Two Numbers To add more than two numbers, simply follow each entry of a number by pressing the all the numbers have been entered. Then press the key to find the total. Pressing the time after a number will give the subtotal at that point. EXAMPLE 11. To find 139 + 216 + 187, enter 139, press the key until key at any key, enter 216, press the again, enter 187, and hit the key to find 139 + 216 + 187 = 542. Hitting the reveal the subtotal of 139 + 216 is 355, as in the example above. key key after 216 would 1.7.3 Subtraction To find the difference A – B, enter A, press the key, and enter B. Then press the key to find the remainder. Multiple subtractions can be done as multiple additions in 1.7.2 above, with the key substituted for the key. EXAMPLE 12. (a) To find 315 − 708, enter 315, press the key, then enter 708 followed by the key to find 315 − 708 = −393. (b) To find 528 − 79.62, enter 528, hit the to find 528 − 79.62 = 448.38. key, then enter 79.62 followed by the key 1.7.4 Multiplication To multiply two numbers, enter the first number, press the press the key, enter the second number, and key to find the product. Serial multiplications can be done in the same way as multiple additions in 1.7.2. with the key substituted for the key. EXAMPLE 13. (a) To find 486 · 27, enter 486, press the 486 · 27 = 13,122. (b) To find 149 · −35, enter 149, press the make it negative, and hit the key, then enter 27, and hit the key to learn key, then enter 35 followed by the key to key to learn that 149 · −35 = −5215. Note: Be aware of the distinction between the key and the key. The key initiates the process of subtraction; the key simply changes the value of the previous entry from positive to negative or negative to positive. 1.7.5 Division Dividing A by B is accomplished by entering A, pressing the the key, then entering B and pressing key. EXAMPLE 14. (a) To find 6715 ÷ 79, enter 6715, hit the key, then enter 79 followed by the display will show 85, indicating that 6715 ÷ 79 = 85. (b) To find −297.36 ÷ 72.128, enter 297.36 followed by the press the key, enter 72.128, and hit the −4.1226708. key. The key to make it negative, then key to find −297.36 ÷ 72.128 = 1.7.6 Raising to a Power To raise a number to a power, enter the number, hit the the key, then enter the exponent and press key. EXAMPLE 15. (a) To find 85, enter 8, press the key, then enter 5 followed by the key to learn that 85 = 32,768. Continue to practice these and subsequent exercises by using simple numbers for which you already know the answers. (b) To find 360.25, enter 36, hit the 360.25 = 2.4494897. key, then enter 0.25, and press the key to see (c) To find 2−3, enter 2, hit the negative, and hit the key, then press 3 followed by the key to make it key to discover 2−3 = 0.125. See also Problem 1.24. 1.7.7 Finding a Square Root To find the square root of a number, enter the number, then press the key to find the square root immediately without having to press the key. Note that on many calculators the the inverse (shift, or second function) of the key, and to activate the the or key followed by the EXAMPLE 16. To find square root of 529. If the enter 529, then press the immediately that key) followed by the key, one must first press key. key to see immediately that ±23 is the key is the inverse, shift, or second function of the or key is key to activate the = 23 without having to press the key, enter 529, then press the key, and you will see key. 1.7.8 Finding the nth Root To find the nth root of a number, enter the number, press the root n and hit the key to find the root. If the is the inverse, shift, or second function of the key, enter the number, press the the value of the root n and hit the key, then enter the value of the , or key followed by the key, then enter key to find the answer. EXAMPLE 17. (a) To find hit the (b) To find hit the , enter 17,576, press the key to learn (c) From Rule 8 in Section 1.1, simply enter 32,768, press the key, enter 3, then . , enter 32,768, hit the key to learn key followed by the key followed by the key, then enter 5 and . = 32,7681/5 = 32,7680.2. To use this latter form, key, enter 0.2, and hit the key to find 32,7680.2 = 8. To make use of similar conversions, recall that , and so forth. See Problems 1.25 to 1.27. 1.7.9 Logarithms To find the value of the common logarithm log10x, enter the value of x and simply press the key. The answer will appear without the need to press the key. EXAMPLE 18. (a) To find the value of log 24, enter 24 and hit the key. The screen will immediately display 1.3802112, indicating that log 24 = 1.3802112. (b) To find log 175, enter 175 and hit the of log 175. key. You will see 2.243038, which is the value 1.7.10 Natural Logarithms To find the value of the natural logarithm lnx, enter the value of x and press the answer will appear immediately without the need to press the key. The key. EXAMPLE 19. (a) To find In 20, enter 20 and hit the key to see 2.9957323 = In 20. (b) For In 0.75, enter 0.75 and hit the key. You will find In 0.75 = −0.2876821. 1.7.11 Exponential Functions To find the value of an exponential function y = ax, enter the value of a and press the enter the value of x and hit the key, then key, similar to what was done in Section 1.7.6. EXAMPLE 20. (a) Given y = 1.53.2, enter 1.5, press the 1.53.2 = 3.6600922. (b) For y = 256−1.25, enter 256, hit the key, then enter 3.2, and hit the key to get key, then enter 1.25 followed immediately by the key to make it negative, and then press the key to leam 256−1.25 = 0.0009766. 1.7.12 Natural Exponential Functions To find the value of a natural exponential function y = ex, enter the value of x press the the answer will appear immediately without the need of hitting the key, enter the value of x, hit the inverse, shift, or second function of the the key. If the key, and key is the key followed by key, and the answer will also appear immediately. EXAMPLE 21. (a) Given y = e1.4, enter 1.4, press the that e1.4 = 4.0552. (b) For e−0.65, enter 0.65, hit the followed by the key followed by the key, and you will see key to make it negative, then press the key to find e−0.65 = 0.5220458. See also Problems 1.24 to 1.27. Solved Problems key EXPONENTS 1.1. Simplify the following expressions using the rules of exponents from Section 1.1. (a) x3 · x4 x3 · x4 = x3+4 = x7 (Rule 1) (b) x5 · x−3 (Rule 1) (c) x−2 · x−4 (Rule 1) (d) x1/2 · x3 (Rules 7 and 1) (Rule 2) (Rules 2 and 6) (Rule 2) (Rules 2 and 7) (i) (x4)−3 (Rules 3 and 6) (Rules 8 and 3) (Rules 6 and 4) (Rules 3 and 5) See also Problems 1.24, 1.26, and 1.27. POLYNOMIALS 1.2. Perform the indicated arithmetic operations on the following polynomials: (a) 35xy + 52xy 35xy + 52xy = 87xy (b) 22yz2 − 46yz2 22yz2 − 46yz2 = −24yz2 (c) 79x2y3 − 46x2y3 79x2y3 − 46x2y3 = 33x2y3 (d) 16x1x2 + 62x1x2 16x1x2 + 62x1x2 = 78x1x2 (e) 57y1y2 − 70y1y2 57y1y2 − 70y1y2 = −13y1y2 (f) 0.5x2y3z5 + 0.9x2y3z5 0.5x2y3z5 + 0.9x2y3z5 = 1.4x2y3z5 1.3. Add or subtract the following polynomials as indicated. Note that in subtraction the sign of every term within the parentheses must be changed before corresponding elements are added. (a) (25x − 9y) + (32x + 16y) (25x − 9y) + (32x + 16y) = 57x + 7y (b) (84x − 31 y) − (76x + 43y) Multiplying each term in the second set of parentheses by −1, which in effect changes the sign of the said terms, and then simply adding, we have (84x − 31y) − (76x + 43y) = 84x − 31y − 76x − 43y = 8x − 74y (c) (9x2 + 7x) − (3x2 − 4x) Changing the signs of each of the terms in the second set of parentheses and then adding, we have (9x2 + 7x) − (3x2 − 4x) = 9x2 + 7x − 3x2 + 4x = 6x2 + 11x (d) (42x2 + 23x) − (5x2 + 11x − 82) (42x2 + 23x) − (5x2 + 11x −82) = 42x2 + 23x − 5x2 − 11x + 82 = 37x2 + 12x + 82 1.4. Perform the indicated operations, recalling that each term in the first polynomial must be multiplied by each term in the second and their products summed. (a) (2x + 7)(4x − 5) (2x + 7)(4x − 5) = 8x2 − 10x + 28x − 35 = 8x2 + 18x − 35 (b) (5x − 6y)(4x − 3y) (5x − 6y)(4x − 3y) = 20x2 − 15xy − 18y2 = 20x2 − 39xy + 18y2 (c) (2x − 9)2 (2x − 9)2 = (2x − 9)(2x − 9) = 4x2 − 18x − 18x + 81 = 4x2 − 36x + 81 (d) (2x + 3y)(2x − 3y) (2x + 3y)(2x − 3y) = 4x2 − 6xy + 6xy − 9y2 = 4x2 − 9y2 (e) (4x + 3y)(5x2 − 2xy + 6y2) (4x + 3y)(5x2 − 2xy + 6y2) = 20x3 − 8x2y + 24xy2 + 15x2y − 6xy2 + 18y3 = 20x3 + 7x2y + 18xy2 + 18y3 (f) (3x2 − 5x2y2 − 2y3)(7x − 4y) (3x3 − 5x2y2 − 2y3)(7x − 4y) = 21x4 − 12x3y − 35x3y2 + 20x2y3 − 14xy3 + 8y4 FACTORING 1.5. Simplify each of the following polynomials by factoring out the greatest common factor: (a) 32x − 8 32x − 8 = 8(4x − 1) (b) 18x2 + 27x 18x2 + 27x = 9x(2x + 3) (c) 14x5 − 35x4 14x5 − 35x4 = 7x4(2x − 5) (d) 45x2y5 − 75x4y3 45x2y5 − 75x4y3 = 15x2y3(3y2 − 5x2) (e) 55x8y9 − 22x6y4 − 99x5y7 55x8y9 − 22x6y4 − 99x5y7 = 11x5y4(5x3y5 − 2x − 9y3) 1.6. Factor each of the following using integer coefficients: (a) x2 + 10x + 21 Here, using the notation from Rule 1 in Section 1.3, m = 1, n = 10, and p = 21. For simplicity we limit our search to positive integers such that: (1) a · b = 1 [1, 1] Henceforth, when a = b = 1, this step will be omitted and only one order will be considered. (2) c · d = 21 [1,21; 3,7; 7,3; 21,1] (3) ad + bc = 10. With a = 1 = b, (c + d) must equal 10. Since only 3 + 7 and 7 + 3 = 10, x2 + 10x + 21 = (x + 3)(x + 7) or (x + 7)(x + 3) Whenever a · b = 1, the order of factors is unimportant and only one ordering of a given pair will be mentioned. (b) x2 + 8x + 16 (1) c · d = 16 [1, 16; 2, 8; 4, 4] (2) c + d = 8 [1 + 16 ≠ 8; 2 + 8 ≠ 8; 4 + 4 = 8] x2 + 8x + 16 = (x + 4)(x + 4)