# Partial Fraction Decomposition for Inverse Laplace Transform

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Partial Fraction Decomposition for Inverse Laplace Transform Usually partial fractions method starts with polynomial long division in order to represent a fraction as a sum of a polynomial and an another fraction where the degree of the polynomial in the numerator of the new fraction is less than the degree of the polynomial in its denominator: s 3 + s 2 + = s + s + s 2 +. We however never have to do this polynomial long division when Partial Fraction Decomposition is applied to problems from Chapter 6. Another important fact in Chapter 6 is that we use only the following three types of fractions: s a. (s a) 2 + b 2 2. b (s a) 2 + b 2 3. (s a) n because we know the corresponding Inverse Laplace transforms [ ] [ ].L s a = e at cos(bt) 2.L b = e at sin(bt) () (s a) 2 + b 2 (s a) 2 + b 2 [ ] [ ] [ ] 3.L = e at L = te at L = t2 s a (s a) 2 (s a) 3 2 eat (2) [ ] L = t36 [ ] [ ] (s a) 4 eat L = t4 (s a) 5 24 eat... L = tn (s a) n+ n! eat We will call fractions 23 as standard fractions. The Partial Fraction Decomposition for Inverse Laplace Transform is as follows. Step Suitable decomposition. The objective of this step is to give the correct format of the partial fraction decomposition for a given fraction. Rules of suitable decomposition:. Numerator does not matter. 2. Number of standard fractions equals the degree of the denominator. 3. Number of undetermined constants equals the degree of the denominator. 4. All standard fractions involved should be different. Simplest Scenario. When you solve a homogeneous equation ay +by +cy = 0 you always have to solve Y = as 2 + bs + c where I put in the numerator (because by Rule the numerator does not matter in Step ) and in the denominator you will have the characteristic polynomial. Since the characteristic polynomial is quadratic you will need two different standard fractions (Rule 2 and 4) and two undetermined constants (Rule 3). There are three cases here. a) as 2 + bs + c = 0 has two distinct real roots. Example s 2 3s 4 = A s 4 + B s +.

2 where s 2 3s 4 = (s 4)(s + ). As in the example above here the rule is that the two (different) standard fractions should be s S s S 2 where S and S 2 are the roots of as 2 + bs + c = 0. s 2 + s = A s + B s + s 2 = A s + + B s. b) as 2 + bs + c = 0 has two repeated real roots. Example s 2 2s + = A s + B (s ) 2. where s 2 2s + = (s ) 2. As in the example above here the rule is that the two (different) standard fractions should be s S (s S) 2 where S is the repeated root of as 2 + bs + c = 0. s = A 2 s + B s 2 s 2 + 6s + 9 = A s B (s + 3) 2. Note that for the cases a) and b) you will need to use the first formula in (2) for the Inverse Laplace Transform. b) as 2 + bs + c = 0 has two complex roots. Example s 2 2s + 5 = A s (s ) B 2 (s ) where s 2 2s + 5 = (s ) As in the example above here the rule is that the two (different) standard fractions should be s k b (s k) 2 + m 2 (s k) 2 + m 2 where the denominator (s k) 2 + m 2 is obtained my completing the squares: as 2 + bs + c = a ( (s k) 2 + m 2). 4s 2 + 4s + 5 = A s + /2 (s + /2) B (s + /2) 2 + s = A s 5 s B s More Complicated Scenario. When you solve a nonhomogeneous equation ay + by + cy = g(t) you will have to deal with a fraction which denominator is of degree 3 or higher. The same four rules of suitable decomposition still apply here:. Numerator does not matter.

3 2. Number of standard fractions equals the degree of the denominator. 3. Number of undetermined constants equals the degree of the denominator. 4. All standard fractions involved should be different. The first preliminary step we have to take here is to decompose the denominator into a product standard polynomials. These standard polynomial are exactly of two types a) (s a) n b) and ( (s k) 2 + m 2). Examples (s )(s 2 +4s+3) = (s )(s+)(s+3) (s )(s 2 +4s+5) = (s )((s+2) 2 +) (s )(s 2 + 4s 5) = (s )(s )(s + 5) = (s ) 2 (s + 5). Note that in the last example above we have to combine two terms (s ) into one (s ) 2 so that all the involved standard polynomials have distinct roots. The standard fractions that arise in this more complicated scenario are identical to the ones that we discussed in the simplest case with one exception. We may now need to use (s S) 3 (s S) 4 (s S) For example (s 3) = A 4 s 3 + B (s 3) + C 2 (s 3) + D 3 (s 3) 4. For everything else we just use the rule that we need to get the appropriate number of fractions for each standard polynomial in our denominator and just add them up together. For example suppose you need to find a suitable form of a partial fractions decomposition for s 4 ((s + ) 2 + )(s 2 + )(s ) 2 (s 2 ) You first observe that there is one further decomposable fraction: s 2 = (s )(s+). Therefore s 4 ((s + ) 2 + )(s 2 + )(s ) 2 (s 2 ) = s 4 ((s + ) 2 + )(s 2 + )(s ) 3 (s + ) and the denominator is the product of five standard fractions: s 4 (s+) 2 + s 2 + (s ) 3 s +. For each one of them individually we know what we should do: s 4 = A s + B s 2 + C s 3 + D s 4 (s + ) 2 + = A s + (s + ) B (s + ) 2 + s 2 + = A s s B s 2 + (s ) = A 3 s + B (s ) + C 2 (s ) 3 s + = A s +.

4 Then for the product s 4 ((s + ) 2 + )(s 2 + )(s ) 3 (s + ) we should just add the individual suitable partial decompositions s 4 ((s + ) 2 + )(s 2 + )(s ) 3 (s + ) = A s + B s 2 + C s 3 + D s 4 + E s + (s + ) 2 + +F (s + ) G s s H s I s + J (s ) 2 + K (s ) 3 + L s +. Step 2 Evaluation of the undetermined constants. Here numerator matters. A first preliminary step here is to reduce the problem to an equality of polynomials. The bullet-proof way to do it is to multiply by the denominator of your given fraction in its standard form. s 2 + 2s + 3 (s + )(s 2 + 2s + 2) = A s + + B s + (s + ) + + C (s + ) + where (s + )(s 2 + 2s + 2) = (s + )((s + ) 2 + ). ((s + )((s + ) 2 + ))(s 2 + 2s + 3) (s + )(s 2 + 2s + 2) = A (s + )((s + )2 + ) s + +B (s + )(s + )((s + )2 + ) + C (s + )((s + )2 + ) (s + ) + (s + ) + ( (s + )((s + ) 2 + ) ) (s 2 + 2s + 3) = A((s + ) 2 + ) + B(s + )(s + ) + C(s + ). Then you need to find the undetermined constants. You do that by finding linear equations for these constants. The number of equations should be equal to the number of unknowns. Apart from small tricks there are two major methods: evaluation of your expression for some values of s and comparison of coefficients for of the polynomials. a) Evaluation at some points This method is very effective when the denominator has only distinct real roots. Hence s 2 + s(s + )(s ) = A s + B s + + C s s 2 + = A(s + )(s ) + Bs(s ) + Cs(s + ) s = 0 = A; s = 2 = 2C; s = 2 = 2B. A = B = C =. s 2 + s(s + )(s ) = s + s + s. b) Comparison of coefficients This method is bullet-proof but it is quite long. s 2 + s(s + )(s ) = A s + B s + + C s s 2 + = A(s + )(s ) + Bs(s ) + Cs(s + ) s 2 + = As 2 A + Bs 2 Bs + Cs 2 + Cs = (A + B + C)s 2 + (C B)s A

5 On the right-hand side the coefficients in front of s 2 s and are 0 and respectively. On the left-hand side the coefficients in front of s 2 s and are A + B + C C B and A respectively. Therefore = A + B + C 0 = C B = A Solving the above system we obtain Hence A = B = C =. s 2 + s(s + )(s ) = s + s + + s. As you see in the above example the evaluation method is somewhat better than the comparison method. Try to do the evaluation method for the next problem and compare with the following solution. s 3 (s 2 + )(s 2 + 4) = A s s B s C s s D 2 s s 3 = As(s 2 + 4) + B(s 2 + 4) + Cs(s 2 + ) + D(s 2 + ) = (A + C)s 3 + (B + D)s 2 + (4A + C)s + (4B + D). = A + C 0 = B + D 0 = 4A + C 0 = 4B + D From the second and the fourth equation above we obtain B = D = 0. The first and the third equations give A = /3 C = 4/3. s 3 (s 2 + )(s 2 + 4) = s 3 s s 3 s

### Lecture 5 Rational functions and partial fraction expansion

S. Boyd EE102 Lecture 5 Rational functions and partial fraction expansion (review of) polynomials rational functions pole-zero plots partial fraction expansion repeated poles nonproper rational functions

### Zero: 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

### So far, we have looked at homogeneous equations

Chapter 3.6: equations Non-homogeneous So far, we have looked at homogeneous equations L[y] = y + p(t)y + q(t)y = 0. Homogeneous means that the right side is zero. Linear homogeneous equations satisfy

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### Tool 1. Greatest Common Factor (GCF)

Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When

### Algebra Practice Problems for Precalculus and Calculus

Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials

### Partial Fractions Decomposition

Partial Fractions Decomposition Dr. Philippe B. Laval Kennesaw State University August 6, 008 Abstract This handout describes partial fractions decomposition and how it can be used when integrating rational

### 1.3 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,

### Discrete Mathematics: Homework 7 solution. Due: 2011.6.03

EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that

### Alum 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

### 5.1 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

### ORDINARY DIFFERENTIAL EQUATIONS

ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.

### Factoring 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

9.5 Quadratics - Build Quadratics From Roots Objective: Find a quadratic equation that has given roots using reverse factoring and reverse completing the square. Up to this point we have found the solutions

### System of First Order Differential Equations

CHAPTER System of First Order Differential Equations In this chapter, we will discuss system of first order differential equations. There are many applications that involving find several unknown functions

### Lesson 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.

### 3.1. 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

### CAHSEE on Target UC Davis, School and University Partnerships

UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

### Section 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) (

### ( ) FACTORING. x In this polynomial the only variable in common to all is x.

FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated

### Maths Workshop for Parents 2. Fractions and Algebra

Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)

### 9. Particular Solutions of Non-homogeneous second order equations Undetermined Coefficients

September 29, 201 9-1 9. Particular Solutions of Non-homogeneous second order equations Undetermined Coefficients We have seen that in order to find the general solution to the second order differential

### G(s) = Y (s)/u(s) In this representation, the output is always the Transfer function times the input. Y (s) = G(s)U(s).

Transfer Functions The transfer function of a linear system is the ratio of the Laplace Transform of the output to the Laplace Transform of the input, i.e., Y (s)/u(s). Denoting this ratio by G(s), i.e.,

### COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS

COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS BORIS HASSELBLATT CONTENTS. Introduction. Why complex numbers were introduced 3. Complex numbers, Euler s formula 3 4. Homogeneous differential equations 8 5.

FACTORING QUADRATIC EQUATIONS Summary 1. Difference of squares... 1 2. Mise en évidence simple... 2 3. compounded factorization... 3 4. Exercises... 7 The goal of this section is to summarize the methods

### QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS

QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS Content 1. Parabolas... 1 1.1. Top of a parabola... 2 1.2. Orientation of a parabola... 2 1.3. Intercept of a parabola... 3 1.4. Roots (or zeros) of a parabola...

### MATH 108 REVIEW TOPIC 10 Quadratic Equations. B. Solving Quadratics by Completing the Square

Math 108 T10-Review Topic 10 Page 1 MATH 108 REVIEW TOPIC 10 Quadratic Equations I. Finding Roots of a Quadratic Equation A. Factoring B. Quadratic Formula C. Taking Roots II. III. Guidelines for Finding

Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of

### Linear and quadratic Taylor polynomials for functions of several variables.

ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is

### Decomposing Rational Functions into Partial Fractions:

Prof. Keely's Math Online Lessons University of Phoenix Online & Clark College, Vancouver WA Copyright 2003 Sally J. Keely. All Rights Reserved. COLLEGE ALGEBRA Hi! Today's topic is highly structured and

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### Algebra 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

### Introduction to Matrix Algebra

Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

### 3.2 Sources, Sinks, Saddles, and Spirals

3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients

### MATH 4552 Cubic equations and Cardano s formulae

MATH 455 Cubic equations and Cardano s formulae Consider a cubic equation with the unknown z and fixed complex coefficients a, b, c, d (where a 0): (1) az 3 + bz + cz + d = 0. To solve (1), it is convenient

### north seattle community college

INTRODUCTION TO FRACTIONS If we divide a whole number into equal parts we get a fraction: For example, this circle is divided into quarters. Three quarters, or, of the circle is shaded. DEFINITIONS: The

### First, we show how to use known design specifications to determine filter order and 3dB cut-off

Butterworth Low-Pass Filters In this article, we describe the commonly-used, n th -order Butterworth low-pass filter. First, we show how to use known design specifications to determine filter order and

### Recursive Algorithms. Recursion. Motivating Example Factorial Recall the factorial function. { 1 if n = 1 n! = n (n 1)! if n > 1

Recursion Slides by Christopher M Bourke Instructor: Berthe Y Choueiry Fall 007 Computer Science & Engineering 35 Introduction to Discrete Mathematics Sections 71-7 of Rosen cse35@cseunledu Recursive Algorithms

### Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.

Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method

### SECTION 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

### ROUTH S STABILITY CRITERION

ECE 680 Modern Automatic Control Routh s Stability Criterion June 13, 2007 1 ROUTH S STABILITY CRITERION Consider a closed-loop transfer function H(s) = b 0s m + b 1 s m 1 + + b m 1 s + b m a 0 s n + s

### MESSAGE TO TEACHERS: NOTE TO EDUCATORS:

MESSAGE TO TEACHERS: NOTE TO EDUCATORS: Attached herewith, please find suggested lesson plans for term 1 of MATHEMATICS Grade 11 Please note that these lesson plans are to be used only as a guide and teachers

### A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

### Multiplying and Dividing Algebraic Fractions

. Multiplying and Dividing Algebraic Fractions. OBJECTIVES. Write the product of two algebraic fractions in simplest form. Write the quotient of two algebraic fractions in simplest form. Simplify a comple

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

### THE COMPLEX EXPONENTIAL FUNCTION

Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula

### Nonhomogeneous Linear Equations

Nonhomogeneous Linear Equations In this section we learn how to solve second-order nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form ay by cy G x where

### AIP Factoring Practice/Help

The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There

### 5. Factoring by the QF method

5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the

### EAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors.

EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an

### EL-9650/9600c/9450/9400 Handbook Vol. 1

Graphing Calculator EL-9650/9600c/9450/9400 Handbook Vol. Algebra EL-9650 EL-9450 Contents. Linear Equations - Slope and Intercept of Linear Equations -2 Parallel and Perpendicular Lines 2. Quadratic Equations

### Lesson 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.

### Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

### 2 When is a 2-Digit Number the Sum of the Squares of its Digits?

When Does a Number Equal the Sum of the Squares or Cubes of its Digits? An Exposition and a Call for a More elegant Proof 1 Introduction We will look at theorems of the following form: by William Gasarch

### NETWORK STRUCTURES FOR IIR SYSTEMS. Solution 12.1

NETWORK STRUCTURES FOR IIR SYSTEMS Solution 12.1 (a) Direct Form I (text figure 6.10) corresponds to first implementing the right-hand side of the difference equation (i.e. the eros) followed by the left-hand

### IOWA End-of-Course Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.

IOWA End-of-Course Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of \$2,000. She also earns \$500 for

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### Determinants can be used to solve a linear system of equations using Cramer s Rule.

2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

### 23. RATIONAL EXPONENTS

23. RATIONAL EXPONENTS renaming radicals rational numbers writing radicals with rational exponents When serious work needs to be done with radicals, they are usually changed to a name that uses exponents,

### 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

### Section 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

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

### Solving certain quintics

Annales Mathematicae et Informaticae 37 010) pp. 193 197 http://ami.ektf.hu Solving certain quintics Raghavendra G. Kulkarni Bharat Electronics Ltd., India Submitted 1 July 010; Accepted 6 July 010 Abstract

### Answer Key Building Polynomial Functions

Answer Key Building Polynomial Functions 1. What is the equation of the linear function shown to the right? 2. How did you find it? y = ( 2/3)x + 2 or an equivalent form. Answers will vary. For example,

### When factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF.

Factoring: reversing the distributive property. The distributive property allows us to do the following: When factoring, we look for greatest common factor of each term and reverse the distributive property

### Algebra II New Summit School High School Diploma Program

Syllabus Course Description: Algebra II is a two semester course. Students completing this course will earn 1.0 unit upon completion. Required Materials: 1. Student Text Glencoe Algebra 2: Integration,

### FACTORISATION YEARS. A guide for teachers - Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project

9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers - Years 9 10 June 2011 Factorisation (Number and Algebra : Module

### Math 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

### Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

### 10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

### INTRODUCTION TO MULTIPLE CORRELATION

CHAPTER 13 INTRODUCTION TO MULTIPLE CORRELATION Chapter 12 introduced you to the concept of partialling and how partialling could assist you in better interpreting the relationship between two primary

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### Domain of a Composition

Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Sect 6.7 - Solving Equations Using the Zero Product Rule

Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred

### Lecture 5 Principal Minors and the Hessian

Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and

### A Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles

A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...

### Prerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303.

Course Syllabus Math 1314 College Algebra Revision Date: 8-21-15 Catalog Description: In-depth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems

### Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

### Mathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework

Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal

### This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.

COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### a 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

### Multiplying Fractions

. Multiplying Fractions. OBJECTIVES 1. Multiply two fractions. Multiply two mixed numbers. Simplify before multiplying fractions 4. Estimate products by rounding Multiplication is the easiest of the four

### Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students

### Week 13 Trigonometric Form of Complex Numbers

Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working

Factorising quadratics An essential skill in many applications is the ability to factorise quadratic expressions. In this unit you will see that this can be thought of as reversing the process used to

### MEP Pupil Text 12. A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued.

MEP Pupil Text Number Patterns. Simple Number Patterns A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued. Worked Example Write down the

### Math 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction

Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### POLYNOMIALS and FACTORING

POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use

### FRACTIONS MODULE Part I

FRACTIONS MODULE Part I I. Basics of Fractions II. Rewriting Fractions in the Lowest Terms III. Change an Improper Fraction into a Mixed Number IV. Change a Mixed Number into an Improper Fraction BMR.Fractions

### FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.