The Fibonacci Sequence and Recursion

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "The Fibonacci Sequence and Recursion"

Transcription

1 The Fibonacci Sequence and Recursion (Handout April 6, 0) The Fibonacci sequence,,,3,,8,3,,34,,89,44,... is defined recursively by {, for n {0,}; F n = F n + F n, for n. Note that the first two terms in the sequence are defined explicitly; thereafter, each term is the sum of the previous two terms. This is an example of a recursive definition. In order to evaluate F 00 using this definition, we would first have to evaluate the previous hundred terms in the sequence. We will soon find a closed form expression for F n which does not require evaluation of all the previous terms in the sequence in order to find a given term. Both recursion and induction may be described as chain reactions; the difference is that mathematical induction is used to prove an infinite sequence of statements, whereas recursion is used to define an infinite sequence of quantities. The connection is a close one; and in order to prove facts about recursively defined sequences, it is natural to try to use induction. The following is an example of a fact about Fibonacci numbers, which we prove by induction. Theorem. For every n, we have Fn = F n+ F n + ( ) n. Proof. The result holds for n = since F = = = = F F 0 + ( ). Now assume that the result holds for n = k, i.e. Fk = F k+f k + ( ) k where k is some positive integer. Then Fk+ = (F k + F k )F k+ (using the recursive formula for F n ) = F k+ F k + F k+ F k = F k+ F k + Fk ( ) k (by the inductive hypothesis) = (F k+ + F k )F k + ( ) k+ = F k+ F k + ( ) k+ (again using the recursive formula for F n )

2 so that the result holds for n = k + whenever it holds for n = k. By induction, the identity holds for all n as required. We now derive a closed formula for F n, i.e. an explicit formula not requiring recursion. The entire sequence is embodied in the series F n x n = + x + x + 3x 3 + x 4 + 8x + 3x 6 + x 7 +. The idea is to find a simple way of evaluating f(x), from which we can identify all coefficients in the series for f(x). In order to make use of the recursion formula for F n, we write which can be solved for f(x): + x + = + x + = + x + = + x + x F n+ x n+ (F n+ + F n )x n+ F n+ x n+ + F n x n+ F n+ x n+ + x F n x n = + x + x ( f(x) ) + x f(x) x x. Thus f(x) is a rational function in x, i.e. a quotient of two polynomials in x. If we can compute the coefficients in the Taylor series of this function, then we will have a formula for F n. However, it is not necessary to know anything about Taylor series to solve this problem; all we need is a little algebra skill, and one very basic series, the geometric series x = + x + x + x 3 + x 4 +. Even if you haven t seen (or don t remember) this series, you can quickly verify it by cross-multiplying: ( x)( + x + x + x 3 + ) = x + x x + x x 3 + x 3 + =.

3 This suggests possibly writing (x + x ) = + (x+x ) + (x+x ) + (x+x ) 3 + (x+x ) 4 + = + (x+x ) + (x +x 3 +x 4 ) + (x 3 +3x 4 +3x +x 6 ) + (x 4 + ) + = + x + x + 3x 3 + x 4 + which is encouraging since we see the Fibonacci numbers appearing as coefficients; however, this doesn t seem to lead to an explicit formula for the Fibonacci numbers. A better idea is to find a partial fraction decomposition of f(x). We first factor the denominator as where x x = ( αx)( βx) α = , β = We may describe α and β as the reciprocal roots of the quadratic, rather than the roots themselves; the reason why they are easier to use will soon become apparent. We try to find constants A and B such that x x = ( αx)( βx) = A αx + B βx. You may have seen this technique used in Calculus II; if not, keep in mind the basic idea, which is as follows. The right hand side of the latter expression can be combined using a common denominator to obtain a rational function whose denominator is ( αx)( βx). Instead what we want is the reverse process: to split up the rational function into two terms, each having a single linear factor in the denominator. Multiplying both sides by ( αx)( βx) in order to clear denominators, we obtain the identity = ( βx)a + ( αx)b. This is a polynomial identity! so equality must hold if x is replaced by any constant. In particular we evaluate at x = α since this causes the B term to vanish, thereby allowing us to solve for A: ( = β ) A = α β α α A = α A, so A = α. Similarly, evaluating at x = β causes the A term to vanish, allowing us to solve for B: = ( α ) B = β α β β B = 3 β B, so B = β.

4 This gives us the desired partial fraction decomposition of f(x): x x = A αx + B βx = ( α αx β ). βx This is just what we need to expand f(x) as a power series: we expand both terms as geometric series to obtain ] [α( + αx + α x + α 3 x 3 + ) β( + βx + β x + β 3 x 3 + ) = (α n+ β n+ )x n. Reading off the coefficient of x n gives F n = αn+ β n+ for all n 0. All of this is demonstrated in the attached Maple worksheet. Recall that the set of polynomials in x with real coefficients, forms the polynomial ring R[x]. The set of rational functions in x with real coefficients is the field of rational functions R(x) = { f(x) g(x) } : f(x),g(x) R[x], g(x) 0. This is a field, not just a ring: it is closed under division. Here we see yet another analogy between the ring Z of integers and the ring R[x] of polynomials: both rings can be extended to fields by formally introducing quotients. We call Q the fraction field of Z; and R(x) is the fraction field of R[x]. Note that R[x] R(x). We also consider the set of all power series in x with real coefficients: R[[x]] = { a 0 +a x+a x +a 3 x 3 + : a 0,a,a,a 3,... R }. This is not a field; it is a ring with R[x] R[[x]]. The latter inclusion follows from the fact that every polynomial may be regarded as a power series where most of the coefficients (all but finitely many) are zero. Elements of R[x], or of R(x), or R[[x]], are first and foremost regarded as formal objects. In some (but not all) cases, they may represent actual functions. You are quite accustomed to using elements of R[x] and R(x) to represent functions; but we have tried to also highlight applications where they are used in other ways without serving as functions. An example is x x R(x) used above to find a closed formula for the Fibonacci numbers. Never did we evaluate f(a) for any number a R; so the popular term rational function is misleading. 4

5 This point is even more important when studying power series. In order to study an arbitrary sequence a 0,a,a,a 3,..., it is often useful to study instead the associated power series g(x) = a n x n = a 0 + a x + a x + a 3 x 3 +, popularly known as the generating function of the original sequence. In many cases this does not represent a function at all; so in general, the name generating function is quite a misnomer. For example the factorial sequence gives rise to a power series p(x) = n!x n = + x + x + 6x 3 + 4x 4 + 0x +. The values p(a) are undefined for any nonzero real number a, although the series itself is a perfectly useful and interesting object. An important part of Calculus II is devoted to the study of when a given power series converges, and so represents an actual function; but none of this is relevant to our discussion! In the case of the series for f(x) defined above, it turns out that the series does converge on a suitable open interval centered at 0; but we don t care about this. Some sources may refer to power series in our setting as formal power series to emphasize the role of power series as purely formal objects without using them to represent functions. However, the term formal is redundant, and used for emphasis only: a power series is, without any added considerations, a purely formal object. Some additional examples of infinite sequences and their generating functions appear below: The sequence,,,,,... has generating function x n = + x + x + x 3 + x 4 + = x. The sequence,,4,8,6,3,... generated recursively by a 0 = ; a n+ = a n for all n 0 has generating function n x n = + x + 4x + 8x 3 + 6x 4 + = x. The sequence,, 3, 4,,... has generating function (n + )x n = + x + 3x + 4x 3 + x 4 + = The sequence, 0,, 0,, 0,... has generating function + x + x 4 + x 6 + x 8 + = x. The sequence,3,6,0,,,8,... defined by a n = ( n+ n 0 has generating function ( n + ) x n = + 3x + 6x + 0x 3 + x 4 + = ( x). ) = ( n+ )( n+ ) for all ( x) 3.

6 We first demonstrate how to list the Fibonacci numbers through recursively: > F:=array(0..00): > F[0]:=: F[]:=: > for n from to 00 do F[n]:=F[n-]+F[n-]: od: > seq(f[n],..00); () Next, we expand in a power series and observe the appearance of the Fibonacci numbers as its sequence of coefficients: > f:=/(-x-x^); () > series(f,x=0,0); (3)

7 Finally, we demonstrate how the formula for the n th Fibonacci number is used to find the 00 th Fibonacci number: > alpha:=(+sqrt())/; beta:=(-sqrt())/; (4) > Fib:=n->simplify((alpha^(n+)-beta^(n+))/sqrt()); () > Fib(00); (6)

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra: Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than

More information

Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 126 Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

More information

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

Differentiation and Integration

Differentiation and Integration This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

More information

1 Lecture: Integration of rational functions by decomposition

1 Lecture: Integration of rational functions by decomposition Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

More information

3.1. RATIONAL EXPRESSIONS

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

More information

2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

More information

3.4 Complex Zeros and the Fundamental Theorem of Algebra

3.4 Complex Zeros and the Fundamental Theorem of Algebra 86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and

More information

The Division Algorithm for Polynomials Handout Monday March 5, 2012

The Division Algorithm for Polynomials Handout Monday March 5, 2012 The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,

More information

Integrals of Rational Functions

Integrals of Rational Functions Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t

More information

Polynomials and Vieta s Formulas

Polynomials and Vieta s Formulas Polynomials and Vieta s Formulas Misha Lavrov ARML Practice 2/9/2014 Review problems 1 If a 0 = 0 and a n = 3a n 1 + 2, find a 100. 2 If b 0 = 0 and b n = n 2 b n 1, find b 100. Review problems 1 If a

More information

Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

More information

Partial Fractions. p(x) q(x)

Partial Fractions. p(x) q(x) Partial Fractions Introduction to Partial Fractions Given a rational function of the form p(x) q(x) where the degree of p(x) is less than the degree of q(x), the method of partial fractions seeks to break

More information

Worksheet on induction Calculus I Fall 2006 First, let us explain the use of for summation. The notation

Worksheet on induction Calculus I Fall 2006 First, let us explain the use of for summation. The notation Worksheet on induction MA113 Calculus I Fall 2006 First, let us explain the use of for summation. The notation f(k) means to evaluate the function f(k) at k = 1, 2,..., n and add up the results. In other

More information

The Method of Partial Fractions Math 121 Calculus II Spring 2015

The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

More information

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

Real Roots of Univariate Polynomials with Real Coefficients

Real Roots of Univariate Polynomials with Real Coefficients Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials

More information

Proofs are short works of prose and need to be written in complete sentences, with mathematical symbols used where appropriate.

Proofs are short works of prose and need to be written in complete sentences, with mathematical symbols used where appropriate. Advice for homework: Proofs are short works of prose and need to be written in complete sentences, with mathematical symbols used where appropriate. Even if a problem is a simple exercise that doesn t

More information

Zeros of Polynomial Functions

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

More information

7. Some irreducible polynomials

7. Some irreducible polynomials 7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of

More information

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year. This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

More information

it is easy to see that α = a

it is easy to see that α = a 21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

More information

Math Common Core Sampler Test

Math Common Core Sampler Test High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests

More information

Partial Fractions. (x 1)(x 2 + 1)

Partial Fractions. (x 1)(x 2 + 1) Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

More information

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)! Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

More information

x if x 0, x if x < 0.

x if x 0, x if x < 0. Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

More information

0.8 Rational Expressions and Equations

0.8 Rational Expressions and Equations 96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

More information

Zeros of a Polynomial Function

Zeros 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 information

2.3 Solving Equations Containing Fractions and Decimals

2.3 Solving Equations Containing Fractions and Decimals 2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions

More information

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions. Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + 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 information

Solutions to Self-Test for Chapter 4 c4sts - p1

Solutions to Self-Test for Chapter 4 c4sts - p1 Solutions to Self-Test for Chapter 4 c4sts - p1 1. Graph a polynomial function. Label all intercepts and describe the end behavior. a. P(x) = x 4 2x 3 15x 2. (1) Domain = R, of course (since this is a

More information

Mathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships

Mathematics. Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships Georgia Standards of Excellence Frameworks Mathematics Accelerated GSE Analytic Geometry B/Advanced Algebra Unit 7: Rational and Radical Relationships These materials are for nonprofit educational purposes

More information

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

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

More information

Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

More information

3. Mathematical Induction

3. Mathematical Induction 3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

More information

5.4 The Quadratic Formula

5.4 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 information

Lies My Calculator and Computer Told Me

Lies My Calculator and Computer Told Me Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing

More information

Equations and Inequalities

Equations 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 information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

Solving Rational Equations

Solving 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 information

Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

More information

Guide to SRW Section 1.7: Solving inequalities

Guide to SRW Section 1.7: Solving inequalities Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is -6-5 -4-3 -2-1 0

More information

Fractions and Decimals

Fractions and Decimals Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first

More information

LEAST SQUARES APPROXIMATION

LEAST SQUARES APPROXIMATION LEAST SQUARES APPROXIMATION Another approach to approximating a function f(x) on an interval a x b is to seek an approximation p(x) with a small average error over the interval of approximation. A convenient

More information

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

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

More information

Taylor and Maclaurin Series

Taylor and Maclaurin Series Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

More information

Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012 Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

More information

Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)

Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x)

More information

minimal polyonomial Example

minimal polyonomial Example Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

Arithmetic 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 information

Core Maths C1. Revision Notes

Core Maths C1. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

More information

z = i ± 9 2 2 so z = 2i or z = i are the solutions. (c) z 4 + 2z 2 + 4 = 0. By the quadratic formula,

z = i ± 9 2 2 so z = 2i or z = i are the solutions. (c) z 4 + 2z 2 + 4 = 0. By the quadratic formula, 91 Homework 8 solutions Exercises.: 18. Show that Z[i] is an integral domain, describe its field of fractions and find the units. There are two ways to show it is an integral domain. The first is to observe:

More information

MATH 132: CALCULUS II SYLLABUS

MATH 132: CALCULUS II SYLLABUS MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early

More information

Factoring Polynomials

Factoring Polynomials Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent

More information

Main page. Given f ( x, y) = c we differentiate with respect to x so that

Main page. Given f ( x, y) = c we differentiate with respect to x so that Further Calculus Implicit differentiation Parametric differentiation Related rates of change Small variations and linear approximations Stationary points Curve sketching - asymptotes Curve sketching the

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

More information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

More information

Discrete Mathematics: Homework 7 solution. Due: 2011.6.03

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

More information

(x + a) n = x n + a Z n [x]. Proof. If n is prime then the map

(x + a) n = x n + a Z n [x]. Proof. If n is prime then the map 22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find

More information

Quotient Rings and Field Extensions

Quotient Rings and Field Extensions Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

More information

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

More information

SOLVING POLYNOMIAL EQUATIONS

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

More information

2.3. Finding polynomial functions. An Introduction:

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

More information

SECTION 10-2 Mathematical Induction

SECTION 10-2 Mathematical Induction 73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

More information

H/wk 13, Solutions to selected problems

H/wk 13, Solutions to selected problems H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.

More information

In this lesson you will learn to find zeros of polynomial functions that are not factorable.

In this lesson you will learn to find zeros of polynomial functions that are not factorable. 2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: A polynomial of n th degree has

More information

MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity

MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x

More information

MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform

MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish

More information

Legendre fractional differential equation and Legender fractional polynomials

Legendre fractional differential equation and Legender fractional polynomials International Journal of Applied Mathematical Research, 3 (3) (2014) 214-219 c Science Publishing Corporation www.sciencepubco.com/index.php/ijamr doi: 10.14419/ijamr.v3i3.2747 Research Paper Legendre

More information

is identically equal to x 2 +3x +2

is identically equal to x 2 +3x +2 Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3

More information

CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

More information

3.2 The Factor Theorem and The Remainder Theorem

3.2 The Factor Theorem and The Remainder Theorem 3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial

More information

Continued Fractions. Darren C. Collins

Continued Fractions. Darren C. Collins Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history

More information

Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES

Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 0-1 Sets There are no state-mandated Precalculus 0-2 Operations

More information

Name: where Nx ( ) and Dx ( ) are the numerator and

Name: where Nx ( ) and Dx ( ) are the numerator and Oblique and Non-linear Asymptote Activity Name: Prior Learning Reminder: Rational Functions In the past we discussed vertical and horizontal asymptotes of the graph of a rational function of the form m

More information

Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Copyrighted Material. Chapter 1 DEGREE OF A CURVE Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

More information

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

March 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 information

Actually, 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

Actually, 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 information

Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.

Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of. Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:

More information

The Deadly Sins of Algebra

The Deadly Sins of Algebra The Deadly Sins of Algebra There are some algebraic misconceptions that are so damaging to your quantitative and formal reasoning ability, you might as well be said not to have any such reasoning ability.

More information

PRINCIPLES OF PROBLEM SOLVING

PRINCIPLES OF PROBLEM SOLVING PRINCIPLES OF PROBLEM SOLVING There are no hard and fast rules that will ensure success in solving problems. However, it is possible to outline some general steps in the problem-solving process and to

More information

Answer Key for California State Standards: Algebra I

Answer 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 information

4.3 Lagrange Approximation

4.3 Lagrange Approximation 206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

1 Review of complex numbers

1 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 information

To discuss this topic fully, let us define some terms used in this and the following sets of supplemental notes.

To discuss this topic fully, let us define some terms used in this and the following sets of supplemental notes. INFINITE SERIES SERIES AND PARTIAL SUMS What if we wanted to sum up the terms of this sequence, how many terms would I have to use? 1, 2, 3,... 10,...? Well, we could start creating sums of a finite number

More information

TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

More information

Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

More information

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

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

More information

The Topsy-Turvy World of Continued Fractions [online]

The Topsy-Turvy World of Continued Fractions [online] Chapter 47 The Topsy-Turvy World of Continued Fractions [online] The other night, from cares exempt, I slept and what d you think I dreamt? I dreamt that somehow I had come, To dwell in Topsy-Turveydom!

More information

1.6 The Order of Operations

1.6 The Order of Operations 1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

More information

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x) SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

More information

Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)

Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes) NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain

More information