1.3 Complex Numbers; Quadratic Equations in the Complex Number System*


 Rosamond O’Connor’
 2 years ago
 Views:
Transcription
1 04 CHAPTER Equations and Inequalities Explaining Conepts: Disussion and Writing 7. Whih of the following pairs of equations are equivalent? Explain. x 2 9; x 3 (b) x 29; x 3 () x  2x  22 x ; x  2 x  8. Desribe three ways that you might solve a quadrati equation. State your preferred method; explain why you hose it. 9. Explain the benefits of evaluating the disriminant of a quadrati equation before attempting to solve it. 20. Create three quadrati equations: one having two distint solutions, one having no real solution, and one having exatly one real solution. 2. The word quadrati seems to imply four (quad), yet a quadrati equation is an equation that involves a polynomial of degree 2. Investigate the origin of the term quadrati as it is used in the expression quadrati equation. Write a brief essay on your findings. Are You Prepared? Answers True 5. x 2 + 5x ax x  62x + 2 2x  32x + 2 e  5 3, 3 f 2 b.3 Complex Numbers; Quadrati Equations in the Complex Number System* PREPARING FOR THIS SECTION Before getting started, review the following: Classifiation of Numbers (Setion R., pp. 4 5) Rationalizing Denominators (Setion R.8, p. 45) Now Work the Are You Prepared? problems on page. OBJECTIVES Add, Subtrat, Multiply, and Divide Complex Numbers (p. 05) 2 Solve Quadrati Equations in the Complex Number System (p. 09) Complex Numbers One property of a real number is that its square is nonnegative. For example, there is no real number x for whih x 2  To remedy this situation, we introdue a new number alled the imaginary unit. DEFINITION The imaginary unit, whih we denote by i, is the number whose square is That is, . i 2  This should not surprise you. If our universe were to onsist only of integers, there would be no number x for whih 2x. This unfortunate irumstane was 2 remedied by introduing numbers suh as and the rational numbers. If our 2 3, universe were to onsist only of rational numbers, there would be no x whose square equals 2. That is, there would be no number x for whih x 2 2. To remedy this, we introdued numbers suh as 2 and 3 5, the irrational numbers. The real numbers, you will reall, onsist of the rational numbers and the irrational numbers. Now, if our universe were to onsist only of real numbers, then there would be no number x whose square is . To remedy this, we introdue a number i, whose square is . *This setion may be omitted without any loss of ontinuity.
2 SECTION.3 Complex Numbers; Quadrati Equations in the Complex Number System 05 In the progression outlined, eah time we enountered a situation that was unsuitable, we introdued a new number system to remedy this situation. The number system that results from introduing the number i is alled the omplex number system. DEFINITION Complex numbers are numbers of the form a bi, where a and b are real numbers. The real number a is alled the real part of the number a + bi; the real number b is alled the imaginary part of a + bi; and i is the imaginary unit, so i 2 . For example, the omplex number i has the real part 5 and the imaginary part 6. When a omplex number is written in the form a + bi, where a and b are real numbers, we say it is in standard form. However, if the imaginary part of a omplex number is negative, suh as in the omplex number i, we agree to write it instead in the form 32i. Also, the omplex number a + 0i is usually written merely as a. This serves to remind us that the real numbers are a subset of the omplex numbers. The omplex number 0 + bi is usually written as bi. Sometimes the omplex number bi is alled a pure imaginary number. Add, Subtrat, Multiply, and Divide Complex Numbers Equality, addition, subtration, and multipliation of omplex numbers are defined so as to preserve the familiar rules of algebra for real numbers. Two omplex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is, Equality of Complex Numbers a + bi + di if and only if a and b d () Two omplex numbers are added by forming the omplex number whose real part is the sum of the real parts and whose imaginary part is the sum of the imaginary parts. That is, Sum of Complex Numbers a + bi2 + + di2 a b + d2i (2) To subtrat two omplex numbers, use this rule: Differene of Complex Numbers a + bi2  + di2 a b  d2i (3) EXAMPLE Adding and Subtrating Complex Numbers (b) 3 + 5i i i + 8i 6 + 4i i i i 32i Now Work PROBLEM 3
3 06 CHAPTER Equations and Inequalities Produts of omplex numbers are alulated as illustrated in Example 2. EXAMPLE 2 Multiplying Complex Numbers 5 + 3i2 # 2 + 7i2 5 # 2 + 7i2 + 3i2 + 7i i + 6i + 2i 2 Distributive Property Distributive Property 0 + 4i i i Based on the proedure of Example 2, the produt of two omplex numbers is defined as follows: Produt of Complex Numbers a + bi2 # + di2 a  bd2 + ad + b2i (4) Do not bother to memorize formula (4). Instead, whenever it is neessary to multiply two omplex numbers, follow the usual rules for multiplying two binomials, as in Example 2, remembering that i 2 . For example, 2 + i2  i2 22i + i  i i Now Work PROBLEM 9 2i22i2 4i 24 Algebrai properties for addition and multipliation, suh as the ommutative, assoiative, and distributive properties, hold for omplex numbers. The property that every nonzero omplex number has a multipliative inverse, or reiproal, requires a loser look. DEFINITION If z a + bi is a omplex number, then its onjugate, denoted by z, is defined as z a + bi a  bi For example, 2 + 3i 23i and 62i i. EXAMPLE 3 Multiplying a Complex Number by Its Conjugate Find the produt of the omplex number z 3 + 4i and its onjugate z. Sine z 34i, we have zz 3 + 4i234i2 92i + 2i  6i The result obtained in Example 3 has an important generalization. THEOREM The produt of a omplex number and its onjugate is a nonnegative real number. That is, if z a + bi, then zz a 2 + b 2 (5)
4 SECTION.3 Complex Numbers; Quadrati Equations in the Complex Number System 07 Proof If z a + bi, then zz a + bi2a  bi2 a 2  bi2 2 a 2  b 2 i 2 a 2 + b 2 To express the reiproal of a nonzero omplex number z in standard form, multiply the numerator and denominator of by z. That is, if z a + bi is a z nonzero omplex number, then a + bi z # z z z z zz a  bi a 2 + b 2 Use (5). a a 2 + b 2  b a 2 + b 2 i EXAMPLE 4 Writing the Reiproal of a Complex Number in Standard Form Write in standard form a + bi; that is, find the reiproal of 3 + 4i i The idea is to multiply the numerator and denominator by the onjugate of 3 + 4i, that is, by the omplex number 34i. The result is 3 + 4i # 34i 3 + 4i 34i 34i i To express the quotient of two omplex numbers in standard form, multiply the numerator and denominator of the quotient by the onjugate of the denominator. EXAMPLE 5 Writing the Quotient of Two Complex Numbers in Standard Form Write eah of the following in standard form. + 4i 52i (b) 23i 43i (b) + 4i 52i + 4i # 5 + 2i 52i 5 + 2i 23i 43i 23i # 4 + 3i 43i 4 + 3i i i i 5 + 2i + 20i + 48i i 8 + 6i  2i  9i Now Work PROBLEM 27 EXAMPLE 6 Writing Other Expressions in Standard Form If z 23i and w 5 + 2i, write eah of the following expressions in standard form. z (b) z + w () z + z w
5 08 CHAPTER Equations and Inequalities z w z # w w # w 23i252i i252i2 04i  5i + 6i i i (b) z + w 23i i2 7  i 7 + i () z + z 23i i2 4 The onjugate of a omplex number has ertain general properties that we shall find useful later. For a real number a a + 0i, the onjugate is a a + 0i a  0i a. That is, THEOREM The onjugate of a real number is the real number itself. Other properties of the onjugate that are diret onsequenes of the definition are given next. In eah statement, z and w represent omplex numbers. THEOREM The onjugate of the onjugate of a omplex number is the omplex number itself. z2 z (6) The onjugate of the sum of two omplex numbers equals the sum of their onjugates. z + w z + w (7) The onjugate of the produt of two omplex numbers equals the produt of their onjugates. z # w z # w (8) We leave the proofs of equations (6), (7), and (8) as exerises. Powers of i The powers of i follow a pattern that is useful to know. i i i 2  i 6 i 4 # i 2  i 3 i 2 # i  # i i i 7 i 4 # i 3 i i 4 i 2 # i And so on. The powers of i repeat with every fourth power. i 5 i 4 # i # i i i 8 i 4 # i 4 EXAMPLE 7 Evaluating Powers of i i 27 i 24 # i 3 i # i 3 6 # i 3 i (b) i 0 i 00 # i i # i 25 # i i
6 SECTION.3 Complex Numbers; Quadrati Equations in the Complex Number System 09 EXAMPLE 8 Writing the Power of a Complex Number in Standard Form Write 2 + i2 3 in standard form. Use the speial produt formula for x + a2 3. NOTE Another way to find (2 + i) 3 is to multiply out (2 + i) 2 (2 + i). x + a2 3 x 3 + 3ax 2 + 3a 2 x + a 3 Using this speial produt formula, 2 + i # i # # i 2 # 2 + i i i2 2 + i. Now Work PROBLEM 4 2 Solve Quadrati Equations in the Complex Number System Quadrati equations with a negative disriminant have no real number solution. However, if we extend our number system to allow omplex numbers, quadrati equations will always have a solution. Sine the solution to a quadrati equation involves the square root of the disriminant, we begin with a disussion of square roots of negative numbers. DEFINITION WARNING In writing N N i be sure to plae i outside the symbol. If N is a positive real number, we define the prinipal square root of denoted by N, as 2N 2N i where i is the imaginary unit and i 2 . N, EXAMPLE 9 Evaluating the Square Root of a Negative Number  i i (b) 4 4 i 2i () 8 8 i 22 i EXAMPLE 0 Solving Equations Solve eah equation in the omplex number system. x 2 4 (b) x 29 x 2 4 x ;4 ;2 The equation has two solutions, 2 and 2. The solution set is {2, 2}. (b) x 29 x ;9 ;9 i ;3i The equation has two solutions, 3i and 3i. The solution set is {3i, 3i}. Now Work PROBLEMS 49 AND 53
7 0 CHAPTER Equations and Inequalities WARNING When working with square roots of negative numbers, do not set the square root of a produt equal to the produt of the square roots (whih an be done with positive numbers). To see why, look at this alulation: We know that However, it is also true that , so Z A 225 iba24 ib 5i22i2 0i 20 Here is the error. Beause we have defined the square root of a negative number, we an now restate the quadrati formula without restrition. THEOREM Quadrati Formula In the omplex number system, the solutions of the quadrati equation ax 2 + bx + 0, where a, b, and are real numbers and a Z 0, are given by the formula x b ; 4 b24a (9) EXAMPLE Solving a Quadrati Equation in the Complex Number System Solve the equation x 24x in the omplex number system. Here a, b 4, 8, and b 24a Using equation (9), we find that x 42 ; ; 26 i 2 4 ; 4i 2 2 ; 2i The equation has two solutions 22i and 2 + 2i. The solution set is 522i, 2 + 2i6. Chek: 2 + 2i: 2 + 2i i i + 4i i i: 22i i i + 4i i Now Work PROBLEM 59 The disriminant b 24a of a quadrati equation still serves as a way to determine the harater of the solutions. Charater of the s of a Quadrati Equation In the omplex number system, onsider a quadrati equation ax 2 + bx + 0 with real oeffiients.. If b 24a 7 0, the equation has two unequal real solutions. 2. If b 24a 0, the equation has a repeated real solution, a double root. 3. If b 24a 6 0, the equation has two omplex solutions that are not real. The solutions are onjugates of eah other.
8 SECTION.3 Complex Numbers; Quadrati Equations in the Complex Number System The third onlusion in the display is a onsequene of the fat that if b 24a N 6 0 then, by the quadrati formula, the solutions are x b + 4 b24a b + 2N b + 2N i b + 2N i and x b  4 b24a b  2N b  2N i b  2N i whih are onjugates of eah other. EXAMPLE 2 Determining the Charater of the s of a Quadrati Equation Without solving, determine the harater of the solutions of eah equation. 3x 2 + 4x (b) 2x 2 + 4x + 0 () 9x 26x + 0 Conepts and Voabulary Here a 3, b 4, and 5, so b 24a The solutions are two omplex numbers that are not real and are onjugates of eah other. (b) Here a 2, b 4, and, so b 24a The solutions are two unequal real numbers. () Here a 9, b 6, and, so b 24a The solution is a repeated real number, that is, a double root. Now Work PROBLEM 73.3 Assess Your Understanding Are You Prepared? Answers are given at the end of these exerises. If you get a wrong answer, read the pages listed in red.. Name the integers and the rational numbers in the set b 3, 0, 2, 6 (pp. 4 5) 5, p r. 2. True or False Rational numbers and irrational numbers are in the set of real numbers. (pp. 4 5) 3 3. Rationalize the denominator of (p. 45) In the omplex number 5 + 2i, the number 5 is alled the 6. True or False The onjugate of 2 + 5i is 25i. part; the number 2 is alled the 7. True or False All real numbers are omplex numbers. part; the number i is alled the. 8. True or False If 23i is a solution of a quadrati equation 5. The equation x 24 has the solution set. with real oeffiients, then i is also a solution. Skill Building In Problems 9 46, write eah expression in the standard form a + bi i i i i i i i i i i i i i i2 7. 2i23i2 8. 3i3 + 4i i22 + i i22  i i26  i i23 + i i i 29. a ib i 2i 30. a ib i 6  i + i i 2 + 3i  i 3. + i i2 2
9 2 CHAPTER Equations and Inequalities 33. i i i i i i i 34i i 32i i i i 7 + i i 4 + i i 6 + i 4 + i i 7 + i 5 + i 3 + i In Problems 47 52, perform the indiated operations and express your answer in the form a + bi i24i i23i  42 In Problems 53 72, solve eah equation in the omplex number system. 53. x x x x x 26x x 2 + 4x x 26x x 22x x 24x x 2 + 6x x 2 + 2x 64. 3x 2 + 6x 65. x 2 + x x 2  x x x x x 4 7. x 4 + 3x x 4 + 3x In Problems 73 78, without solving, determine the harater of the solutions of eah equation in the omplex number system x 23x x 24x x 2 + 3x x x 77. 9x 22x x 2 + 2x i is a solution of a quadrati equation with real i is a solution of a quadrati equation with real oeffiients. Find the other solution. oeffiients. Find the other solution. In Problems 8 84, z 34i and w 8 + 3i. Write eah expression in the standard form a + bi. 8. z + z 82. w  w 83. zz 84. z  w Appliations and Extensions 85. Eletrial Ciruits The impedane Z, in ohms, of a iruit element is defined as the ratio of the phasor voltage V, in volts, aross the element to the phasor urrent I, in amperes, through the elements. That is, Z V. If the voltage aross a I iruit element is 8 + i volts and the urrent through the element is 34i amperes, determine the impedane. 86. Parallel Ciruits In an a iruit with two parallel pathways, the total impedane Z, in ohms, satisfies the formula where Z is the impedane of the first pathway Z +, Z Z 2 Explaining Conepts: Disussion and Writing 9. Explain to a friend how you would add two omplex numbers and how you would multiply two omplex numbers. Explain any differenes in the two explanations. 92. Write a brief paragraph that ompares the method used to rationalize the denominator of a radial expression and the method used to write the quotient of two omplex numbers in standard form. 93. Use an Internet searh engine to investigate the origins of omplex numbers. Write a paragraph desribing what you find and present it to the lass. Are You Prepared? Answers and Z 2 is the impedane of the seond pathway. Determine the total impedane if the impedanes of the two pathways are Z 2 + i ohms and Z 2 43i ohms. 87. Use z a + bi to show that z + z and z  z 2bi. 88. Use z a + bi to show that z z. 89. Use z a + bi and w + di to show that z + w z + w. 90. Use z a + bi and w + di to show that z # w z # w. 94. Explain how the method of multiplying two omplex numbers is related to multiplying two binomials. 95. What Went Wrong? A student multiplied 29 and 29 as follows: 29 # 29 2(9)(9) 28 9 The instrutor marked the problem inorret. Why?. Integers: 53, 06; rational numbers: b 3, 0, 6 2. True 3. 3A223B 5 r
4.1. COMPLEX NUMBERS
4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers
More informationchapter > Make the Connection Factoring CHAPTER 4 OUTLINE Chapter 4 :: Pretest 374
CHAPTER hapter 4 > Make the Connetion 4 INTRODUCTION Developing seret odes is big business beause of the widespread use of omputers and the Internet. Corporations all over the world sell enryption systems
More informationSet Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos)
A.1 Set Theory and Logi: Fundamental Conepts (Notes by Dr. J. Santos) A.1. Primitive Conepts. In mathematis, the notion of a set is a primitive notion. That is, we admit, as a starting point, the existene
More informationSequences CheatSheet
Sequenes CheatSheet Intro This heatsheet ontains everything you should know about real sequenes. It s not meant to be exhaustive, but it ontains more material than the textbook. Definitions and properties
More informationSTUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS
STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an
More informationLecture 5. Gaussian and GaussJordan elimination
International College of Eonomis and Finane (State University Higher Shool of Eonomis) Letures on Linear Algebra by Vladimir Chernyak, Leture. Gaussian and GaussJordan elimination To be read to the musi
More informationCOMPLEX NUMBERS. 2+2i
COMPLEX NUMBERS Cartesian Form of Complex Numbers The fundamental complex number is i, a number whose square is 1; that is, i is defined as a number satisfying i 1. The complex number system is all numbers
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationcos t sin t sin t cos t
Exerise 7 Suppose that t 0 0andthat os t sin t At sin t os t Compute Bt t As ds,andshowthata and B ommute 0 Exerise 8 Suppose A is the oeffiient matrix of the ompanion equation Y AY assoiated with the
More information5.2 The Master Theorem
170 CHAPTER 5. RECURSION AND RECURRENCES 5.2 The Master Theorem Master Theorem In the last setion, we saw three different kinds of behavior for reurrenes of the form at (n/2) + n These behaviors depended
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationAlgebra 1 Chapter 3 Vocabulary. equivalent  Equations with the same solutions as the original equation are called.
Chapter 3 Vocabulary equivalent  Equations with the same solutions as the original equation are called. formula  An algebraic equation that relates two or more reallife quantities. unit rate  A rate
More informationAlgebra I Notes Review Real Numbers and Closure Unit 00a
Big Idea(s): Operations on sets of numbers are performed according to properties or rules. An operation works to change numbers. There are six operations in arithmetic that "work on" numbers: addition,
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationActually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationChemical Equilibrium. Chemical Equilibrium. Chemical Equilibrium. Chemical Equilibriu m. Chapter 14
Chapter 14 Chemial Equilibrium Chemial Equilibriu m Muh like water in a Ushaped tube, there is onstant mixing bak and forth through the lower portion of the tube. reatants produts It s as if the forward
More informationMath 1111 Journal Entries Unit I (Sections , )
Math 1111 Journal Entries Unit I (Sections 1.11.2, 1.41.6) Name Respond to each item, giving sufficient detail. You may handwrite your responses with neat penmanship. Your portfolio should be a collection
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
More informationLecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties
Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an
More informationRestricted Least Squares, Hypothesis Testing, and Prediction in the Classical Linear Regression Model
Restrited Least Squares, Hypothesis Testing, and Predition in the Classial Linear Regression Model A. Introdution and assumptions The lassial linear regression model an be written as (1) or (2) where xt
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More information1.1 Solving a Linear Equation ax + b = 0
1.1 Solving a Linear Equation ax + b = 0 To solve an equation ax + b = 0 : (i) move b to the other side (subtract b from both sides) (ii) divide both sides by a Example: Solve x = 0 (i) x = 0 x = (ii)
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More information3.7 Complex Zeros; Fundamental Theorem of Algebra
SECTION.7 Complex Zeros; Fundamental Theorem of Algebra 2.7 Complex Zeros; Fundamental Theorem of Algebra PREPARING FOR THIS SECTION Before getting started, review the following: Complex Numbers (Appendix,
More informationCOMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i
COMPLEX NUMBERS _4+i _i FIGURE Complex numbers as points in the Arg plane i _i +i i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with
More informationMarch 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
More informationSummer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2
Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationSECTION 16 Quadratic Equations and Applications
58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be
More informationThis is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0).
This is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationAlgebra 12. A. Identify and translate variables and expressions.
St. Mary's College High School Algebra 12 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used
More information5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0
a p p e n d i x e ABSOLUTE VALUE ABSOLUTE VALUE E.1 definition. The absolute value or magnitude of a real number a is denoted by a and is defined by { a if a 0 a = a if a
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationAlgebra Tiles Activity 1: Adding Integers
Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting
More informationSOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The OddRoot Property
498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1
More informationEE 201 ELECTRIC CIRCUITS LECTURE 25. Natural, Forced and Complete Responses of First Order Circuits
EE 201 EECTRIC CIUITS ECTURE 25 The material overed in this leture will be as follows: Natural, Fored and Complete Responses of First Order Ciruits Step Response of First Order Ciruits Step Response of
More informationReview Session #5 Quadratics
Review Session #5 Quadratics Discriminant How can you determine the number and nature of the roots without solving the quadratic equation? 1. Prepare the quadratic equation for solving in other words,
More informationElectrician'sMathand BasicElectricalFormulas
Eletriian'sMathand BasiEletrialFormulas MikeHoltEnterprises,In. 1.888.NEC.CODE www.mikeholt.om Introdution Introdution This PDF is a free resoure from Mike Holt Enterprises, In. It s Unit 1 from the Eletrial
More informationAlgebra. Indiana Standards 1 ST 6 WEEKS
Chapter 1 Lessons Indiana Standards  11 Variables and Expressions  12 Order of Operations and Evaluating Expressions  13 Real Numbers and the Number Line  14 Properties of Real Numbers  15 Adding
More informationMathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)
( : Focus on free Education) Miscellaneous Exercise on chapter 5 Question 1: Evaluate: Answer 1: 1 ( : Focus on free Education) Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) =
More informationThe numbers that make up the set of Real Numbers can be classified as counting numbers whole numbers integers rational numbers irrational numbers
Section 1.8 The numbers that make up the set of Real Numbers can be classified as counting numbers whole numbers integers rational numbers irrational numbers Each is said to be a subset of the real numbers.
More informationALGEBRA I A PLUS COURSE OUTLINE
ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationPre Cal 2 1 Lesson with notes 1st.notebook. January 22, Operations with Complex Numbers
0 2 Operations with Complex Numbers Objectives: To perform operations with pure imaginary numbers and complex numbers To use complex conjugates to write quotients of complex numbers in standard form Complex
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationSection 25 Quadratic Equations and Inequalities
5 Quadratic Equations and Inequalities 5 a bi 6. (a bi)(c di) 6. c di 63. Show that i k, k a natural number. 6. Show that i k i, k a natural number. 65. Show that i and i are square roots of 3 i. 66.
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationIdentity: a statement that is true for all values for which both sides are defined Example from algebra: 3 x x csc sec cot 1 1 1
Sllaus Ojetives:. The student will simplif trigonometri epressions and prove trigonometri identities (fundamental identities)..4 The student will solve trigonometri equations with and without tehnolog.
More informationREDUCTION FACTOR OF FEEDING LINES THAT HAVE A CABLE AND AN OVERHEAD SECTION
C I E 17 th International Conferene on Eletriity istriution Barelona, 115 May 003 EUCTION FACTO OF FEEING LINES THAT HAVE A CABLE AN AN OVEHEA SECTION Ljuivoje opovi J.. Elektrodistriuija  Belgrade 
More informationPhase Velocity and Group Velocity for Beginners
Phase Veloity and Group Veloity for Beginners Rodolfo A. Frino January 015 (v1)  February 015 (v3) Eletronis Engineer Degree from the National University of Mar del Plata  Argentina Copyright 014015
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationThe Product Property of Square Roots states: For any real numbers a and b, where a 0 and b 0, ab = a b.
Chapter 9. Simplify Radical Expressions Any term under a radical sign is called a radical or a square root expression. The number or expression under the the radical sign is called the radicand. The radicand
More informationClass XI Chapter 5 Complex Numbers and Quadratic Equations Maths. Exercise 5.1. Page 1 of 34
Question 1: Exercise 5.1 Express the given complex number in the form a + ib: Question 2: Express the given complex number in the form a + ib: i 9 + i 19 Question 3: Express the given complex number in
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationCHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS
CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided
More information1 Review of complex numbers
1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely
More informationMagnetic Materials and Magnetic Circuit Analysis
Chapter 7. Magneti Materials and Magneti Ciruit Analysis Topis to over: 1) Core Losses 2) Ciruit Model of Magneti Cores 3) A Simple Magneti Ciruit 4) Magneti Ciruital Laws 5) Ciruit Model of Permanent
More informationMATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.
MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationMth 95 Module 2 Spring 2014
Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More informationSebastián Bravo López
Transfinite Turing mahines Sebastián Bravo López 1 Introdution With the rise of omputers with high omputational power the idea of developing more powerful models of omputation has appeared. Suppose that
More informationChannel Assignment Strategies for Cellular Phone Systems
Channel Assignment Strategies for Cellular Phone Systems Wei Liu Yiping Han Hang Yu Zhejiang University Hangzhou, P. R. China Contat: wliu5@ie.uhk.edu.hk 000 Mathematial Contest in Modeling (MCM) Meritorious
More informationCHAPTER 1 NUMBER SYSTEMS POINTS TO REMEMBER
CHAPTER NUMBER SYSTEMS POINTS TO REMEMBER. Definition of a rational number. A number r is called a rational number, if it can be written in the form p, where p and q are integers and q 0. q Note. We visit
More informationAlgebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , )
Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.11.4, 1.61.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationMultiplying Polynomials 5
Name: Date: Start Time : End Time : Multiplying Polynomials 5 (WS#A10436) Polynomials are expressions that consist of two or more monomials. Polynomials can be multiplied together using the distributive
More informationSolving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
More informationCapacity at Unsignalized TwoStage Priority Intersections
Capaity at Unsignalized TwoStage Priority Intersetions by Werner Brilon and Ning Wu Abstrat The subjet of this paper is the apaity of minorstreet traffi movements aross major divided fourlane roadways
More informationPortable Assisted Study Sequence ALGEBRA IIA
SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of
More informationThis is Solving Equations and Inequalities, chapter 6 from the book Advanced Algebra (index.html) (v. 1.0).
This is Solving Equations and Inequalities, chapter 6 from the book Advanced Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More information6.3 Point of View: Integrating along the yaxis
math appliation: area between urves 7 EXAMPLE 66 Find the area of the region in the first quadrant enlosed by the graphs of y =, y = ln x, and the x and yaxes SOLUTION It is easy to sketh the region
More informationA.1 Radicals and Rational Exponents
APPENDIX A. Radicals and Rational Eponents 779 Appendies Overview This section contains a review of some basic algebraic skills. (You should read Section P. before reading this appendi.) Radical and rational
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationExample 1 Example 2 Example 3. The set of the ages of the children in my family { 27, 24, 21, 19 } The set of Counting Numbers
Section 0 1A: The Real Number System We often look at a set as a collection of objects with a common connection. We use brackets like { } to show the set and we put the objects in the set inside the brackets
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationAlgebra II Pacing Guide First Nine Weeks
First Nine Weeks SOL Topic Blocks.4 Place the following sets of numbers in a hierarchy of subsets: complex, pure imaginary, real, rational, irrational, integers, whole and natural. 7. Recognize that the
More informationChapter 3. Algebra. 3.1 Rational expressions BAa1: Reduce to lowest terms
Contents 3 Algebra 3 3.1 Rational expressions................................ 3 3.1.1 BAa1: Reduce to lowest terms...................... 3 3.1. BAa: Add, subtract, multiply, and divide............... 5
More informationINTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS
INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28
More informationpp. 4 8: Examples 1 6 Quick Check 1 6 Exercises 1, 2, 20, 42, 43, 64
Semester 1 Text: Chapter 1: Tools of Algebra Lesson 11: Properties of Real Numbers Day 1 Part 1: Graphing and Ordering Real Numbers Part 1: Graphing and Ordering Real Numbers Lesson 12: Algebraic Expressions
More informationPrecalculus Workshop  Functions
Introduction to Functions A function f : D C is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set C. D is called the domain of f. C is called the codomain of f.
More informationLecture Notes on Polynomials
Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationSection 6.1 Factoring Expressions
Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what
More informationMontana Common Core Standard
Algebra 2 Grade Level: 10(with Recommendation), 11, 12 Length: 1 Year Period(s) Per Day: 1 Credit: 1 Credit Requirement Fulfilled: Mathematics Course Description This course covers the main theories in
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More information6.4 Special Factoring Rules
6.4 Special Factoring Rules OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor a sum of cubes. By reversing the rules for multiplication
More information