M098 Carson Elementary and Intermediate Algebra 3e Section 11.1
|
|
|
- Eugene Higgins
- 5 years ago
- Views:
Transcription
1 M098 Carson Eleentary and Interediate Algebra e Section 11.1 Objectives 1. Use the square root principle to solve quadratic equations.. Solve quadratic equations by copleting the square. Vocabulary Pri Knowledge Solve quadratic equations by facting. New Concepts 1. Use the square root principle to solve quadratic equations. Until now, all quadratic equations have been carefully chosen so that they could be solved by facting. Obviously, not every quadratic equation can be facted, so in Chapter 11, we learn soe other ethods that will allow us to solve all quadratic equations. Reeber, we epect solutions to quadratic equations because of the ter. Again, let s look at soe eaples. If =, then = = -. If y = 6, then y = 6 y = -6. How can we get the answers if the proble isn t one that we can figure out in our head? Notice the f: a perfect square isolated on the left side of the equation and no variable to the first power. Then take the square root of both sides, reebering that there will be two answers. So we always use ± in front of the radical on the right side. Eaple 1: both sides to reove the square. Reeber the ± in front = ± 9 There are two solutions: 9 and -9. Eaple : a 0 a 0 both sides to reove the square. Reeber the ± in front a There are two solutions: 9 and -9. These are eaples of the Square Root Property. Reeber that solving an equation is a process of undoing the operations on the variable. Addition and subtraction undo each other, ultiplication and division undo each other and now powers and roots undo each other. V. Zabrocki 011 page 1
2 M098 Carson Eleentary and Interediate Algebra e Section 11.1 Eaple : The perfect square is not isolate. Clear the fractions and then isolate. 1 both sides to reove the square. Reeber the ± in front 1 Rationalize the radical and siplify The eaples we ve just looked at are the siplest kind: the perfect square was a onoial. The process is the sae if the base is a binoial. Eaple : a 6 9 a 6 9 The perfect square binoial is isolated, so take the square root of both sides to reove the square. Reeber the ± in front a + 6 = ± Solve f a. a = -6 ± Write the -6 in front of the ±. a = -6 + a = -6 a = - a = -9 Notice the shthand ethod (the ±) of writing the cases. When you use this shtcut, always put the nuber you bring to the right side in front of the ± sign. V. Zabrocki 011 page
3 M098 Carson Eleentary and Interediate Algebra e Section 11.1 Eaple : Isolate the perfect square binoial. 1 both sides to reove the square. Reeber the ± in front 1 1 Solve f. Place the -1 in front of the ± Eaple 6: t 8 t 8 both sides to reove the square. Reeber the ± in front t i Siplify the radical. Rewrite the iaginary nuber using i. t i Solve f t. t i t i. Solve quadratic equations by copleting the square. As was entioned earlier, very few quadratic equations can be facted. Even fewer of the coe in the f we saw in the last section: a perfect square equal to soe nuber. However, ALL quadratic equations can be changed into that f by a process called Copleting the Square. Let's start by looking at soe perfect square trinoials: ( + 6) = (y - ) = y - 8y + 16 (a + ) = a + 6a + 9 V. Zabrocki 011 page
4 M098 Carson Eleentary and Interediate Algebra e Section 11.1 Here are soe patterns you should notice: 1. the coefficient of the squared ter is always 1.. The constant ter is always equal to 1/ the coefficient of the iddle ter squared. 1/ (1) = 6 6 = 6 1/ (8) = = 16 1/ (6) = = 9. Notice the constant ter in the binoial on the left side. It is always equal to 1/ the iddle ter on the right side.. Notice the signs: The sign in the binoial is the sae as the first sign of the trinoial. The last sign of the trinoial is always positive. You can use this knowledge to rewrite ANY quadratic equation. Copleting the Square Procedure 1. The coefficient of squared ter ust be 1, so divide each ter by the coefficient of the ter.. Move the constant to the other side of the equation. (Leave space to add a ter on each side.). Take half of the coefficient of the ter and square it.. Add this aount to both sides of the equation.. Rewrite the left side as a perfect square using one half of the coefficient of the -ter as the nuber in the binoial and siplify the right side. 6. Use the square root property. (Reeber!! ). Finish the solution. Eaple : The coefficient of squared ter is already Move the constant to the other side of the equation. (Leave space to add a ter on each side.) 1 10 Take half of the coefficient of the ter and square it Add this aount to both sides of the equation. 1 Rewrite the left side as a perfect square using one half of the coefficient of the -ter as the nuber in the binoial and siplify the right side. 1 Use the square root property. (Reeber!! ) 1 Finish the solution. V. Zabrocki 011 page
5 M098 Carson Eleentary and Interediate Algebra e Section 11.1 Eaple 8: The coefficient of squared ter ust be 1, so divide each ter by the coefficient of the ter. 0 Move the constant to the other side of the equation. = (Leave space to add a ter on each side.) 1 9 Take half of the coefficient of the ter and square it. 9 Add this aount to both sides of the equation. Rewrite the left side as a perfect square using one half of the coefficient of the -ter as the nuber in the binoial and siplify the right side. Use the square root property. (Reeber!! ) Finish the solution. V. Zabrocki 011 page
a a a a a a a m a b a b
Algebra / Trig Final Exa Study Guide (Fall Seester) Moncada/Dunphy Inforation About the Final Exa The final exa is cuulative, covering Appendix A (A.1-A.5) and Chapter 1. All probles will be ultiple choice
M098 Carson Elementary and Intermediate Algebra 3e Section 11.3
Objectives. Solve equations by writing them in quadratic form.. Solve equations that are quadratic in form by using substitution. Vocabulary Prior Knowledge Solve rational equations: Clear the fraction.
Exponent Laws Unit 4 Exponential Functions - Lesson 18
(A) Lesson Contet BIG PICTURE of this UNIT: CONTEXT of this LESSON: What are & how & why do we use eponential and logarithic functions? proficiency with algeraic anipulations/calculations pertinent to
Quadratic Equations. Save My Exams! The Home of Revision For more awesome GCSE and A level resources, visit us at
Quadratic Equations Mark Schee 1 Level A LEVEL Ea Board Edecel GCE Subject Matheatics Module Core 1 Topic Algebra and Functions Sub-Topic Quadratic Equations Booklet Mark Schee 1 Tie Allowed: Score: Percentage:
Final Exam Practice: Part II Math MULTIPLE CHOICE Choose the one alternative that best completes the statement or answers the question.
MULTIPLE CHOICE Choose the one alternative that best copletes the stateent or answers the question. 1) Solve for y: y y 0 D) 4 9 ) Solve for : 0, 0 D) ) To Quig traveled 80 iles east of St. Louis. For
Finite fields. and we ve used it in various examples and homework problems. In these notes I will introduce more finite fields
Finite fields I talked in class about the field with two eleents F 2 = {, } and we ve used it in various eaples and hoework probles. In these notes I will introduce ore finite fields F p = {,,...,p } for
Analysis of Polynomial & Rational Functions ( summary )
Analysis of Polynoial & Rational Functions ( suary ) The standard for of a polynoial function is ( ) where each of the nubers are called the coefficients. The polynoial of is said to have degree n, where
QUADRATIC EQUATIONS. + 6 = 0 This is a quadratic equation written in standard form. x x = 0 (standard form with c=0). 2 = 9
QUADRATIC EQUATIONS A quadratic equation is always written in the form of: a + b + c = where a The form a + b + c = is called the standard form of a quadratic equation. Eamples: 5 + 6 = This is a quadratic
Divisibility of Polynomials over Finite Fields and Combinatorial Applications
Designs, Codes and Cryptography anuscript No. (will be inserted by the editor) Divisibility of Polynoials over Finite Fields and Cobinatorial Applications Daniel Panario Olga Sosnovski Brett Stevens Qiang
Feature Extraction Techniques
Feature Extraction Techniques Unsupervised Learning II Feature Extraction Unsupervised ethods can also be used to find features which can be useful for categorization. There are unsupervised ethods that
Congruences and Modular Arithmetic
Congruenes and Modular Aritheti 6-17-2016 a is ongruent to b od n eans that n a b. Notation: a = b (od n). Congruene od n is an equivalene relation. Hene, ongruenes have any of the sae properties as ordinary
Laurie s Notes. Overview of Section 7.5
Laurie s Notes Overview of Section 7. Introduction Students have developed a repertoire of equation-solving techniques throughout their study of algebra. In this lesson, additional techniques are presented.
Force and dynamics with a spring, analytic approach
Force and dynaics with a spring, analytic approach It ay strie you as strange that the first force we will discuss will be that of a spring. It is not one of the four Universal forces and we don t use
Lecture 21 Principle of Inclusion and Exclusion
Lecture 21 Principle of Inclusion and Exclusion Holden Lee and Yoni Miller 5/6/11 1 Introduction and first exaples We start off with an exaple Exaple 11: At Sunnydale High School there are 28 students
SOLVING LITERAL EQUATIONS. Bundle 1: Safety & Process Skills
SOLVING LITERAL EQUATIONS Bundle 1: Safety & Process Skills Solving Literal Equations An equation is a atheatical sentence with an equal sign. The solution of an equation is a value for a variable that
Let me just tell you how, then I ll use an example to make it more clear.
CHAPTER 15 Equations with Radicals Sec. 1 Simplifying Radicals From grade school, you can probably remember how to take square roots of numbers like 25, 64, and the 100. The number on the inside of the
ALGEBRA REVIEW. MULTINOMIAL An algebraic expression consisting of more than one term.
Page 1 of 6 ALGEBRAIC EXPRESSION A cobination of ordinary nubers, letter sybols, variables, grouping sybols and operation sybols. Nubers reain fixed in value and are referred to as constants. Letter sybols
Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?
Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using
PhysicsAndMathsTutor.com
. A raindrop falls vertically under gravity through a cloud. In a odel of the otion the raindrop is assued to be spherical at all ties and the cloud is assued to consist of stationary water particles.
LESSON 9.1 ROOTS AND RADICALS
LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical
Unit Essential Questions. What are the different representations of exponents? Where do exponents fit into the real number system?
Unit Essential Questions What are the different representations of exponents? Where do exponents fit into the real number system? How can exponents be used to depict real-world situations? REAL NUMBERS
Solving Quadratic Equations
Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic
The Weierstrass Approximation Theorem
36 The Weierstrass Approxiation Theore Recall that the fundaental idea underlying the construction of the real nubers is approxiation by the sipler rational nubers. Firstly, nubers are often deterined
Solving Linear Equations
Solving Linear Equations Golden Rule of Algebra: Do unto one side of the equal sign as you will do to the other Whatever you do on one side of the equal sign, you MUST do the same exact thing on the other
Derivative at a point
Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Derivative at a point Wat you need to know already: Te concept of liit and basic etods for coputing liits. Wat you can
Curious Bounds for Floor Function Sums
1 47 6 11 Journal of Integer Sequences, Vol. 1 (018), Article 18.1.8 Curious Bounds for Floor Function Sus Thotsaporn Thanatipanonda and Elaine Wong 1 Science Division Mahidol University International
GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME. PHYSICAL SCIENCE Grade 11 SESSION 11 (LEARNER NOTES)
PYSICAL SCIENCE Grade 11 SESSION 11 (LEARNER NOTES) MOLE CONCEPT, STOICIOMETRIC CALCULATIONS Learner Note: The ole concept is carried forward to calculations in the acid and base section, as well as in
Equations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero
Solving Other Types of Equations Rational Equations Examine each denominator to find values that would cause the denominator to equal zero Multiply each term by the LCD or If two terms cross-multiply Solve,
Dimensions and Units
Civil Engineering Hydraulics Mechanics of Fluids and Modeling Diensions and Units You already know how iportant using the correct diensions can be in the analysis of a proble in fluid echanics If you don
Problem Set 2. Chapter 1 Numerical:
Chapter 1 Nuerical: roble Set 16. The atoic radius of xenon is 18 p. Is that consistent with its b paraeter of 5.15 1 - L/ol? Hint: what is the volue of a ole of xenon atos and how does that copare to
The simplest method for constructing APN polynomials EA-inequivalent to power functions
The siplest ethod for constructing APN polynoials EA-inequivalent to power functions Lilya Budaghyan Abstract The first APN polynoials EA-inequivalent to power functions have been constructed in [7, 8]
y = (x + 2)(x + 3) y = (2x + 3)(x 4) 15x x 2
Go to butenhoffmath.com and check out the algebra notes video: A 8-1 AStd5a: I can rewrite quadratic equations in standard form, factored form, and vertex form. Factored form: y = (x + )(x + 3) y = (x
CH 54 PREPARING FOR THE QUADRATIC FORMULA
1 CH 54 PREPARING FOR THE QUADRATIC FORMULA Introduction W e re pretty good by now at solving equations like (3x 4) + 8 10(x + 1), and we ve had a whole boatload of word problems which can be solved by
Poly-Bernoulli Numbers and Eulerian Numbers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 21 (2018, Article 18.6.1 Poly-Bernoulli Nubers and Eulerian Nubers Beáta Bényi Faculty of Water Sciences National University of Public Service H-1441
Egyptian Mathematics Problem Set
(Send corrections to cbruni@uwaterloo.ca) Egyptian Matheatics Proble Set (i) Use the Egyptian area of a circle A = (8d/9) 2 to copute the areas of the following circles with given diaeter. d = 2. d = 3
Tactics Box 2.1 Interpreting Position-versus-Time Graphs
1D kineatic Retake Assignent Due: 4:32p on Friday, October 31, 2014 You will receive no credit for ites you coplete after the assignent is due. Grading Policy Tactics Box 2.1 Interpreting Position-versus-Tie
9.4 Radical Expressions
Section 9.4 Radical Expressions 95 9.4 Radical Expressions In the previous two sections, we learned how to multiply and divide square roots. Specifically, we are now armed with the following two properties.
Lesson 21 Not So Dramatic Quadratics
STUDENT MANUAL ALGEBRA II / LESSON 21 Lesson 21 Not So Dramatic Quadratics Quadratic equations are probably one of the most popular types of equations that you ll see in algebra. A quadratic equation has
m potential kinetic forms of energy.
Spring, Chapter : A. near the surface of the earth. The forces of gravity and an ideal spring are conservative forces. With only the forces of an ideal spring and gravity acting on a ass, energy F F will
NAME DATE PERIOD. Power and Radical Functions. New Vocabulary Fill in the blank with the correct term. positive integer.
2-1 Power and Radical Functions What You ll Learn Scan Lesson 2-1. Predict two things that you expect to learn based on the headings and Key Concept box. 1. 2. Lesson 2-1 Active Vocabulary extraneous solution
List Scheduling and LPT Oliver Braun (09/05/2017)
List Scheduling and LPT Oliver Braun (09/05/207) We investigate the classical scheduling proble P ax where a set of n independent jobs has to be processed on 2 parallel and identical processors (achines)
Lesson 9-5 Completing the Square Name
Lesson 9-5 Completing the Square Name In the next two lessons, you will be solving quadratic equations by a couple of new methods that often produce solutions involving square roots. In Lesson 9-5A, you
Lectures 8 & 9: The Z-transform.
Lectures 8 & 9: The Z-transfor. 1. Definitions. The Z-transfor is defined as a function series (a series in which each ter is a function of one or ore variables: Z[] where is a C valued function f : N
Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1
Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1 Example 1: Evaluate Numerical Expressions
Quantum Chemistry Exam 2 Take-home Solutions
Cheistry 60 Fall 07 Dr Jean M Standard Nae KEY Quantu Cheistry Exa Take-hoe Solutions 5) (0 points) In this proble, the nonlinear variation ethod will be used to deterine an approxiate solution for the
Controlling the Population
Lesson.1 Skills Practice Name Date Controlling the Population Adding and Subtracting Polynomials Vocabulary Match each definition with its corresponding term. 1. polynomial a. a polynomial with only 1
HONORS GEOMETRY Summer Skills Set
HONORS GEOMETRY Summer Skills Set Algebra Concepts Adding and Subtracting Rational Numbers To add or subtract fractions with the same denominator, add or subtract the numerators and write the sum or difference
We will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.).
College algebra We will review simplifying radicals, exponents and their rules, multiplying polynomials, factoring polynomials, greatest common denominators, and solving rational equations. Pre-requisite
Lecture 26. Quadratic Equations
Lecture 26 Quadratic Equations Quadratic polynomials....................................................... 2 Quadratic polynomials....................................................... 3 Quadratic equations
NAME DATE PERIOD. A negative exponent is the result of repeated division. Extending the pattern below shows that 4 1 = 1 4 or 1. Example: 6 4 = 1 6 4
Lesson 4.1 Reteach Powers and Exponents A number that is expressed using an exponent is called a power. The base is the number that is multiplied. The exponent tells how many times the base is used as
Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.
Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10
Solving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
Order of Operations Practice: 1) =
Order of Operations Practice: 1) 24-12 3 + 6 = a) 6 b) 42 c) -6 d) 192 2) 36 + 3 3 (1/9) - 8 (12) = a) 130 b) 171 c) 183 d) 4,764 1 3) Evaluate: 12 2-4 2 ( - ½ ) + 2 (-3) 2 = 4) Evaluate 3y 2 + 8x =, when
Functions: Review of Algebra and Trigonometry
Sec. and. Functions: Review o Algebra and Trigonoetry A. Functions and Relations DEFN Relation: A set o ordered pairs. (,y) (doain, range) DEFN Function: A correspondence ro one set (the doain) to anther
27 Oscillations: Introduction, Mass on a Spring
Chapter 7 Oscillations: Introduction, Mass on a Spring 7 Oscillations: Introduction, Mass on a Spring If a siple haronic oscillation proble does not involve the tie, you should probably be using conservation
A field trips costs $800 for the charter bus plus $10 per student for x students. The cost per student is represented by: 10x x
LEARNING STRATEGIES: Activate Prior Knowledge, Shared Reading, Think/Pair/Share, Note Taking, Group Presentation, Interactive Word Wall A field trips costs $800 for the charter bus plus $10 per student
EXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n
Algebra B: Chapter 6 Notes 1 EXPONENT REVIEW!!! Concept Byte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Property of Eponents: Product of Powers m n = m
Symbolic Analysis as Universal Tool for Deriving Properties of Non-linear Algorithms Case study of EM Algorithm
Acta Polytechnica Hungarica Vol., No., 04 Sybolic Analysis as Universal Tool for Deriving Properties of Non-linear Algoriths Case study of EM Algorith Vladiir Mladenović, Miroslav Lutovac, Dana Porrat
Basic ALGEBRA 2 SUMMER PACKET
Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout
1 Generalization bounds based on Rademacher complexity
COS 5: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #0 Scribe: Suqi Liu March 07, 08 Last tie we started proving this very general result about how quickly the epirical average converges
Chapter 10: Sinusoidal Steady-State Analysis
Chapter 0: Sinusoidal Steady-State Analysis Sinusoidal Sources If a circuit is driven by a sinusoidal source, after 5 tie constants, the circuit reaches a steady-state (reeber the RC lab with t = τ). Consequently,
2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY
2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you
P.5 Solving Equations
PRC Ch P_5.notebook P.5 Solving Equations What you should learn How to solve linear equations How to solve quadratic equations equations How to solve polynomial equations of degree three or higher How
Chapter 1.6. Perform Operations with Complex Numbers
Chapter 1.6 Perform Operations with Complex Numbers EXAMPLE Warm-Up 1 Exercises Solve a quadratic equation Solve 2x 2 + 11 = 37. 2x 2 + 11 = 37 2x 2 = 48 Write original equation. Subtract 11 from each
Math 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS:
Math 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS: 1 EXPONENT REVIEW PROBLEMS: 2 1. 2x + x x + x + 5 =? 2. (x 2 + x) (x + 2) =?. The expression 8x (7x 6 x 5 ) is equivalent to?.
Section 1.2 A Catalog of Essential Functions
Chapter 1 Section Page 1 of 6 Section 1 A Catalog of Essential Functions Linear Models: All linear equations have the form rise change in horizontal The letter m is the of the line, or It can be positive,
1 Quadratic Functions
Unit 1 Quadratic Functions Lecture Notes Introductory Algebra Page 1 of 8 1 Quadratic Functions In this unit we will learn many of the algebraic techniques used to work with the quadratic function fx)
Oscillations: Review (Chapter 12)
Oscillations: Review (Chapter 1) Oscillations: otions that are periodic in tie (i.e. repetitive) o Swinging object (pendulu) o Vibrating object (spring, guitar string, etc.) o Part of ediu (i.e. string,
CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic
CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic equations. They can be solved using a graph, a perfect square,
Graphing Radicals Business 7
Graphing Radicals Business 7 Radical functions have the form: The most frequently used radical is the square root; since it is the most frequently used we assume the number 2 is used and the square root
Unit 4 - Equations and Inequalities - Vocabulary
12/5/17 Unit 4 Unit 4 - Equations and Inequalities - Vocabulary Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line Review Vocabulary: Absolute
A Simplified Analytical Approach for Efficiency Evaluation of the Weaving Machines with Automatic Filling Repair
Proceedings of the 6th SEAS International Conference on Siulation, Modelling and Optiization, Lisbon, Portugal, Septeber -4, 006 0 A Siplified Analytical Approach for Efficiency Evaluation of the eaving
Topic 5a Introduction to Curve Fitting & Linear Regression
/7/08 Course Instructor Dr. Rayond C. Rup Oice: A 337 Phone: (95) 747 6958 E ail: rcrup@utep.edu opic 5a Introduction to Curve Fitting & Linear Regression EE 4386/530 Coputational ethods in EE Outline
Donald Fussell. October 28, Computer Science Department The University of Texas at Austin. Point Masses and Force Fields.
s Vector Moving s and Coputer Science Departent The University of Texas at Austin October 28, 2014 s Vector Moving s Siple classical dynaics - point asses oved by forces Point asses can odel particles
. The univariate situation. It is well-known for a long tie that denoinators of Pade approxiants can be considered as orthogonal polynoials with respe
PROPERTIES OF MULTIVARIATE HOMOGENEOUS ORTHOGONAL POLYNOMIALS Brahi Benouahane y Annie Cuyt? Keywords Abstract It is well-known that the denoinators of Pade approxiants can be considered as orthogonal
Eigenvalues & Eigenvectors
Eigenvalues & Eigenvectors Page 1 Eigenvalues are a very important concept in linear algebra, and one that comes up in other mathematics courses as well. The word eigen is German for inherent or characteristic,
Module #1: Units and Vectors Revisited. Introduction. Units Revisited EXAMPLE 1.1. A sample of iron has a mass of mg. How many kg is that?
Module #1: Units and Vectors Revisited Introduction There are probably no concepts ore iportant in physics than the two listed in the title of this odule. In your first-year physics course, I a sure that
AP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted
lecture 36: Linear Multistep Mehods: Zero Stability
95 lecture 36: Linear Multistep Mehods: Zero Stability 5.6 Linear ultistep ethods: zero stability Does consistency iply convergence for linear ultistep ethods? This is always the case for one-step ethods,
CH 55 THE QUADRATIC FORMULA, PART I
1 CH 55 THE QUADRATIC FORMULA, PART I Introduction I n the Introduction to the previous chapter we considered the quadratic equation 10 + 16 0. We verified in detail that this equation had two solutions:
Student Activity: Finding Factors and Prime Factors
When you have completed this activity, go to Status Check. Pre-Algebra A Unit 2 Student Activity: Finding Factors and Prime Factors Name Date Objective In this activity, you will find the factors and the
UNIT 4 NOTES: PROPERTIES & EXPRESSIONS
UNIT 4 NOTES: PROPERTIES & EXPRESSIONS Vocabulary Mathematics: (from Greek mathema, knowledge, study, learning ) Is the study of quantity, structure, space, and change. Algebra: Is the branch of mathematics
POLYNOMIAL EXPRESSIONS PART 1
POLYNOMIAL EXPRESSIONS PART 1 A polynomial is an expression that is a sum of one or more terms. Each term consists of one or more variables multiplied by a coefficient. Coefficients can be negative, so
Algebra Final Exam Review Packet
Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:
SOLVING EQUATIONS AND DEVELOPING THE FOUNDATION FOR PROOFS
12 SOLVING EQUATIONS AND DEVELOPING THE FOUNDATION FOR PROOFS INSTRUCTIONAL ACTIVITY Lesson 2 LEARNING GOAL Students will solve linear equations using concrete and semi-concrete models, algebraic procedures,
Does the Order Matter?
LESSON 6 Does the Order Mtter? LEARNING OBJECTIVES Tody I : writing out exponent ultipliction. So tht I cn: develop rules for exponents. I ll know I hve it when I cn: solve proble like ( b) = b 5 0. Opening
Revised: 2/19/09 Unit 1 Pre-Algebra Concepts and Operations Review
2/19/09 Unit 1 Pre-Algebra Concepts and Operations Review 1. How do algebraic concepts represent real-life situations? 2. Why are algebraic expressions and equations useful? 2. Operations on rational numbers
Solving Quadratic Equations by Formula
Algebra Unit: 05 Lesson: 0 Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots. In some cases, solutions involved a double root, but there were always
3.3 Dividing Polynomials. Copyright Cengage Learning. All rights reserved.
3.3 Dividing Polynomials Copyright Cengage Learning. All rights reserved. Objectives Long Division of Polynomials Synthetic Division The Remainder and Factor Theorems 2 Dividing Polynomials In this section
Review: Relativistic mechanics. Announcements. Relativistic kinetic energy. Kinetic energy. E tot = γmc 2 = K + mc 2. K = γmc 2 - mc 2 = (γ-1)mc 2
Announceents Reading for Monday: Chapters 3.7-3.12 Review session for the idter: in class on Wed. HW 4 due Wed. Exa 1 in 6 days. It covers Chapters 1 & 2. Roo: G1B30 (next to this classroo). Review: Relativistic
Online Courses for High School Students
Online Courses for High School Students 1-888-972-6237 Algebra I Course Description: Students explore the tools of algebra and learn to identify the structure and properties of the real number system;
Chapter 2A - Solving Equations
- Chapter A Chapter A - Solving Equations Introduction and Review of Linear Equations An equation is a statement which relates two or more numbers or algebraic expressions. For example, the equation 6
MULTIPLAYER ROCK-PAPER-SCISSORS
MULTIPLAYER ROCK-PAPER-SCISSORS CHARLOTTE ATEN Contents 1. Introduction 1 2. RPS Magas 3 3. Ites as a Function of Players and Vice Versa 5 4. Algebraic Properties of RPS Magas 6 References 6 1. Introduction
A REMARK ON PRIME DIVISORS OF PARTITION FUNCTIONS
International Journal of Nuber Theory c World Scientific Publishing Copany REMRK ON PRIME DIVISORS OF PRTITION FUNCTIONS PUL POLLCK Matheatics Departent, University of Georgia, Boyd Graduate Studies Research
Algebra I. Course Outline
Algebra I Course Outline I. The Language of Algebra A. Variables and Expressions B. Order of Operations C. Open Sentences D. Identity and Equality Properties E. The Distributive Property F. Commutative
TECHNIQUES IN FACTORISATION
TECHNIQUES IN FACTORISATION The process where brackets are inserted into an equation is referred to as factorisation. Factorisation is the opposite process to epansion. METHOD: Epansion ( + )( 5) 15 Factorisation
Distillation. The Continuous Column. Learning Outcomes. Recap - VLE for Meth H 2 O. Gavin Duffy School of Electrical Engineering DIT Kevin Street
Distillation The Continuous Colun Gavin Duffy School of Electrical Engineering DIT Kevin Street Learning Outcoes After this lecture you should be able to.. Describe how continuous distillation works List
A := A i : {A i } S. is an algebra. The same object is obtained when the union in required to be disjoint.
59 6. ABSTRACT MEASURE THEORY Having developed the Lebesgue integral with respect to the general easures, we now have a general concept with few specific exaples to actually test it on. Indeed, so far
Chapter 10: Sinusoidal Steady-State Analysis
Chapter 0: Sinusoidal Steady-State Analysis Sinusoidal Sources If a circuit is driven by a sinusoidal source, after 5 tie constants, the circuit reaches a steady-state (reeber the RC lab with t τ). Consequently,
Sect Complex Numbers
161 Sect 10.8 - Complex Numbers Concept #1 Imaginary Numbers In the beginning of this chapter, we saw that the was undefined in the real numbers since there is no real number whose square is equal to a