Polynomials with Rational Roots that Differ by a Non-zero Constant. Generalities

Size: px
Start display at page:

Download "Polynomials with Rational Roots that Differ by a Non-zero Constant. Generalities"

Transcription

1 Polyomials with Ratioal Roots that Differ by a No-zero Costat Philip Gibbs The problem of fidig two polyomials P(x) ad Q(x) of a give degree i a sigle variable x that have all ratioal roots ad differ by a o-zero costat is ivestigated. It is show that the problem reduces to cosiderig oly polyomials with iteger roots. The cases < are solved geerically. For = the case of polyomials whose roots come i pairs (a,-a) is solved. For = 5 a ifiite umber of iequivalet solutios are foud with the asatz P(x) = -Q(-x). For = 6 a ifiite umber of solutios are also foud. Fially for = 8 we fid solitary examples. This also solves the problem of fidig two polyomials of degree that fully factorise ito liear factors with iteger coefficiets such that the differece is oe. Geeralities Let P(x) ad Q(x) be two polyomials of degree such that all roots of P(x) ad Q(x) are ratioal. For which degrees ca we fid two such polyomials which differ oly by a o-zero costat idepedet of x? Sice all roots are ratioal we ca factorise both polyomials P x = a (x r i ) Q x = b (x s i ) Where r i ad s i are the ratioal roots. For > 0, if the differece P(x) Q(x) = c is a costat idepedet of x the we must have a = b ad without loss of geerality we ca assume a = b = 1. Coditios o the roots The coditio for a solutio ca be writte i terms of the roots r i k k = s i, for 1 k < r i s i Equivalece to problems over itegers Take N to be the multiple of the deomiators of all roots for both equatios the P x = N P x/n = (x Nr i ) So P (x) is a polyomial with iteger roots. Similarly for Q (x) = N Q(x/N). But P (x) - Q (x) = N c is costat. It follows that if we ca fid a solutio with ratioal roots the we ca also fid oe with

2 iteger roots. The coverse is trivial. Therefore we eed oly search for polyomials with iteger roots r i ad s i. This problem is also equivalet to fidig two polyomials of degree that ca be fully factorised ito liear factors with iteger coefficiets ad that differ by oe. If we have two such polyomials the they have ratioal roots. Coversely, if we have two polyomials P(x) ad Q(x) with iteger roots that differ by a costat P(x) Q(x) = c, the substitute x = x s 1 where s 1 is a root of Q(x) so that Q (x ) = Q(x + s 1 ) has a factor of x ad therefore the product of the roots of P (x ) = P(x + s 1 ) is c. Now make a secod substitutio to defie P (x ) = (1/c)P (cx ) ad Q (x ) = (1/c)Q (cx ). It ca be verified that P (x ) ad Q (x ) factorise ito liear factors with iteger coefficiets ad that P (x ) - Q (x ) = 1 No coicidet roots If a root r of P(x) were also a root of Q(x) the x-r would be a factor of P(x)-Q(x) which could the ot be costat. It follows that oe of the roots of P(x) ca coicide with roots of Q(x). Trasformatios ad dualities Give oe solutio as roots r i ad s i, aother ca be formed by traslatig usig a iteger costat traslatio r i = r i + t, s i = s i + t, or by multiplyig by a costat r i = k.r i, s i = k.s i. Solutios which differ by combiatios of such trasformatios will be regarded as equivalet. Whe a solutio is equivalet to itself uder a o-trivial trasformatio we call it self-dual. This ca happe i essetially two ways; (1) r i = t r j ad s i = t s j () r i = t s j By usig the trasformatios we ca assume t = 0 i either case. Self-dual type (1) ca oly arise for eve sice otherwise it require a zero root for both polyomials. It is the equivalet to P(x) = P(-x) ad Q(x) = Q(-x) Self-dual type () ca arise for odd or eve. We the get P(x) = (-1) Q(-x) Whe we impose self duality o the solutios we automatically fulfil about half of the required coditios o the roots. This ca help us i the search at higher values of. = Geeral case The quadratic case is simple to solve. Usig the trasformatios we ca assume that the roots are (r, -r) ad (s, -s) from polyomials P(x) = x -r ad Q(x) = x -s. This provides solutios whe r s. All other solutios are equivalet to oe of these.

3 = Type () self -daul The cubic case is more challegig, but it is ot difficult to fid some type () self-dual solutios with This automatically gives us that The remaiig requiremet is that r 1 = -s 1, r = -s, r = -s r 1 + r + r = s 1 + s + s r 1 + r + r = 0 with oe of the roots equal to zero. This is easy to satisfy, i.e. take r 1 = a-b, r = b-c, r = c-a, where a, b ad c are ay distict itegers. This icludes a ifiite umber of iequivalet solutios e.g. by fixig a ad b, the varyig c. Geeral case For = it is also possible to costruct a more complete solutio. To see this first set r 1 = -s 1 This ca be doe without loss of geerality sice ay solutio is equivalet to oe with this coditio. The we require just, r + r = s + s s 1 = r + r - s - s The first equatio is well kow with geeral solutio i four parameters a, b, c, d, based o complex umber orms is give by r = ab + cd, r = ad - bc, s = ab - cd, s = ad + bc, The we ca complete the solutio by solvig the secod equatio with s 1 = cd - bc, r 1 = bc - cd To esure that the differece i the polyomials is o-zero, we eed r 1 r r - s 1 s s = bcd(b-d)(a-c)(a+c) 0 Although this solutio is complete up to traslatios it does ot reflect the permutatio symmetries of the origial problem. To fid a more symmetric solutio, first examie the matrix of differeces Each of these compoets factorises as follows δ ij = r i - s j

4 = c(b d) b(c a) d(c + a) b(a + c) cd a c (b d) d(a c) a + c (d b) bc Reame the factors as follows The the matrix takes a more symmetric form p = -c, q = c a, t = c + a, u = d b, v = b, w = -d = pu qv tw tv pw qu qw tu pv With the extra coditios p + q + t = u + v + w = 0 A further observatio is that a solutio ca also be derived from this matrix form by takig r i = j δ ij, s j = δ ij i It ca the be verified that the required equatios for the roots are satisfied without the extra coditios. To esure that the differece is o-zero we require r 1 r r - s 1 s s = (p-q)(p-t)(q-t)(u-v)(v-w)(u-w) 0 = Type (1) self -daul For quartic polyomials it is possible to fid type (1) self-dual solutios usig The remaiig equality we eed to satisfy is Which is solved usig r = - r 1, r = - r, s = - s 1, s = - s r 1 + r = s 1 + s r 1 = ab + cd, r = ad - bc, s 1 = ab - cd, s = ad + bc, From this we ca geerate a ifiite umber of iequivalet solutios. Type () self -daul We ca also look for type () self-dual solutios usig This the requires s 1 = - r 1, s = - r, s = - r, s = - r

5 r 1 + r + r + r = 0 ad r 1 + r + r + r = 0 the secod equatio gives The usig the secod equatio we get But we also have (r 1 + r )(r 1 - r 1 r + r ) + (r + r )(r - r r + r ) = 0 r 1 - r 1 r + r = r - r r + r (r 1 + r ) = (r + r ) Ad combiig the differet quadratics we also get (r 1 - r ) = (r - r ) It quickly follows that the two polyomials must be equal, so o type () self-dual solutios are possible for =. Type () self daul I this case we set =5 The the remaiig equatios to solve are r 1 = -s 1, r = -s, r = -s, r = -s, r 5 = -s 5 r 1 + r + r + r + r 5 = 0 ad r 1 + r + r + r + r 5 = 0 This has a ifiite umber of iequivalet solutios e.g. from this sequece for z ay positive iteger. r 1 = 1, r = z + z, r = -(z + z + 1), r = -(z + z + ), r 5 = z + 5z + =6 Type (1) self -daul For degree six polyomials cosider type (1) self-dual solutios usig The the remaiig equatios to solve are r = - r 1, r 5 = - r, r 6 = - r, s = - s 1, s 5 = - s, s 6 = - s r 1 + r + r = s 1 + s + s ad r 1 + r + r = s 1 + s + s Agai this has a ifiite umber of iequivalet solutios e.g. from this sequece. r 1 = 0, r = r = z + z + 1, s 1 = z + 1, s = z + z, s = z + z + 1

6 =8 Type (1) self -daul For =8 a brute force umerical search has produced some example solutios the smallest of which is { r i } = {-,-,-1,-5,5,1,,} { s i } = {-5,-1,-16,-,,16,1,5} Fial Remarks Solutios seem to be reasoably abudat up to degree 8 but there is o obvious patter that allows us to fid geeral solutios for arbitrarily high degree. As icreases the umber of variables icreases at twice the rate of the costraits, but the costraits are of icreasigly high degree. It is therefore a iterestig questio as to whether solutios exist for all. Although this problem has bee studied here for its ow iterest it may have applicatios to other problems where it is required to fid systems of umbers with small differeces that have may factors.

Properties and Tests of Zeros of Polynomial Functions

Properties and Tests of Zeros of Polynomial Functions Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by

More information

Recurrence Relations

Recurrence Relations Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The

More information

In algebra one spends much time finding common denominators and thus simplifying rational expressions. For example:

In algebra one spends much time finding common denominators and thus simplifying rational expressions. For example: 74 The Method of Partial Fractios I algebra oe speds much time fidig commo deomiators ad thus simplifyig ratioal epressios For eample: + + + 6 5 + = + = = + + + + + ( )( ) 5 It may the seem odd to be watig

More information

MAT 271 Project: Partial Fractions for certain rational functions

MAT 271 Project: Partial Fractions for certain rational functions MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,

More information

UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 116C. Problem Set 4. Benjamin Stahl. November 6, 2014

UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 116C. Problem Set 4. Benjamin Stahl. November 6, 2014 UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 6C Problem Set 4 Bejami Stahl November 6, 4 BOAS, P. 63, PROBLEM.-5 The Laguerre differetial equatio, x y + ( xy + py =, will be solved

More information

MATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006

MATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006 MATH 34 Summer 006 Elemetary Number Theory Solutios to Assigmet Due: Thursday July 7, 006 Departmet of Mathematical ad Statistical Scieces Uiversity of Alberta Questio [p 74 #6] Show that o iteger of the

More information

Chapter 4. Fourier Series

Chapter 4. Fourier Series Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,

More information

Unit 4: Polynomial and Rational Functions

Unit 4: Polynomial and Rational Functions 48 Uit 4: Polyomial ad Ratioal Fuctios Polyomial Fuctios A polyomial fuctio y px ( ) is a fuctio of the form p( x) ax + a x + a x +... + ax + ax+ a 1 1 1 0 where a, a 1,..., a, a1, a0are real costats ad

More information

1 Generating functions for balls in boxes

1 Generating functions for balls in boxes Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways

More information

RADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify

RADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) 16 1 2 c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL

More information

Sequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

Sequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet

More information

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as

More information

Math 475, Problem Set #12: Answers

Math 475, Problem Set #12: Answers Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe

More information

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1

September 2012 C1 Note. C1 Notes (Edexcel) Copyright   - For AS, A2 notes and IGCSE / GCSE worksheets 1 September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright

More information

SNAP Centre Workshop. Basic Algebraic Manipulation

SNAP Centre Workshop. Basic Algebraic Manipulation SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)

More information

Stochastic Matrices in a Finite Field

Stochastic Matrices in a Finite Field Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices

More information

ECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations

ECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows

More information

[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.

[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms. [ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural

More information

Inverse Matrix. A meaning that matrix B is an inverse of matrix A.

Inverse Matrix. A meaning that matrix B is an inverse of matrix A. Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix

More information

Polynomial Functions and Their Graphs

Polynomial Functions and Their Graphs Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively

More information

We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n

We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at

More information

Zeros of Polynomials

Zeros of Polynomials Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

More information

11. FINITE FIELDS. Example 1: The following tables define addition and multiplication for a field of order 4.

11. FINITE FIELDS. Example 1: The following tables define addition and multiplication for a field of order 4. 11. FINITE FIELDS 11.1. A Field With 4 Elemets Probably the oly fiite fields which you ll kow about at this stage are the fields of itegers modulo a prime p, deoted by Z p. But there are others. Now although

More information

Chapter 2. Periodic points of toral. automorphisms. 2.1 General introduction

Chapter 2. Periodic points of toral. automorphisms. 2.1 General introduction Chapter 2 Periodic poits of toral automorphisms 2.1 Geeral itroductio The automorphisms of the two-dimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad

More information

NICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =

NICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) = AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

TEACHER CERTIFICATION STUDY GUIDE

TEACHER CERTIFICATION STUDY GUIDE COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra

More information

End-of-Year Contest. ERHS Math Club. May 5, 2009

End-of-Year Contest. ERHS Math Club. May 5, 2009 Ed-of-Year Cotest ERHS Math Club May 5, 009 Problem 1: There are 9 cois. Oe is fake ad weighs a little less tha the others. Fid the fake coi by weighigs. Solutio: Separate the 9 cois ito 3 groups (A, B,

More information

x x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,

x x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0, Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative

More information

Recursive Algorithms. Recurrences. Recursive Algorithms Analysis

Recursive Algorithms. Recurrences. Recursive Algorithms Analysis Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects

More information

PAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION

PAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION Iteratioal Joural of Pure ad Applied Mathematics Volume 94 No. 204, 9-20 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/0.2732/ijpam.v94i.2 PAijpam.eu

More information

Hoggatt and King [lo] defined a complete sequence of natural numbers

Hoggatt and King [lo] defined a complete sequence of natural numbers REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies

More information

CALCULATION OF FIBONACCI VECTORS

CALCULATION OF FIBONACCI VECTORS CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College

More information

Lecture 23 Rearrangement Inequality

Lecture 23 Rearrangement Inequality Lecture 23 Rearragemet Iequality Holde Lee 6/4/ The Iequalities We start with a example Suppose there are four boxes cotaiig $0, $20, $50 ad $00 bills, respectively You may take 2 bills from oe box, 3

More information

x c the remainder is Pc ().

x c the remainder is Pc (). Algebra, Polyomial ad Ratioal Fuctios Page 1 K.Paulk Notes Chapter 3, Sectio 3.1 to 3.4 Summary Sectio Theorem Notes 3.1 Zeros of a Fuctio Set the fuctio to zero ad solve for x. The fuctio is zero at these

More information

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered

More information

PROBLEM SET 5 SOLUTIONS 126 = , 37 = , 15 = , 7 = 7 1.

PROBLEM SET 5 SOLUTIONS 126 = , 37 = , 15 = , 7 = 7 1. Math 7 Sprig 06 PROBLEM SET 5 SOLUTIONS Notatios. Give a real umber x, we will defie sequeces (a k ), (x k ), (p k ), (q k ) as i lecture.. (a) (5 pts) Fid the simple cotiued fractio represetatios of 6

More information

U8L1: Sec Equations of Lines in R 2

U8L1: Sec Equations of Lines in R 2 MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie

More information

REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.

REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains. The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values of the variable it cotais The relatioships betwee

More information

Linear Differential Equations of Higher Order Basic Theory: Initial-Value Problems d y d y dy

Linear Differential Equations of Higher Order Basic Theory: Initial-Value Problems d y d y dy Liear Differetial Equatios of Higher Order Basic Theory: Iitial-Value Problems d y d y dy Solve: a( ) + a ( )... a ( ) a0( ) y g( ) + + + = d d d ( ) Subject to: y( 0) = y0, y ( 0) = y,..., y ( 0) = y

More information

18th Bay Area Mathematical Olympiad. Problems and Solutions. February 23, 2016

18th Bay Area Mathematical Olympiad. Problems and Solutions. February 23, 2016 18th Bay Area Mathematical Olympiad February 3, 016 Problems ad Solutios BAMO-8 ad BAMO-1 are each 5-questio essay-proof exams, for middle- ad high-school studets, respectively. The problems i each exam

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors 5 Eigevalues ad Eigevectors 5.3 DIAGONALIZATION DIAGONALIZATION Example 1: Let. Fid a formula for A k, give that P 1 1 = 1 2 ad, where Solutio: The stadard formula for the iverse of a 2 2 matrix yields

More information

Chapter 7 COMBINATIONS AND PERMUTATIONS. where we have the specific formula for the binomial coefficients:

Chapter 7 COMBINATIONS AND PERMUTATIONS. where we have the specific formula for the binomial coefficients: Chapter 7 COMBINATIONS AND PERMUTATIONS We have see i the previous chapter that (a + b) ca be writte as 0 a % a & b%þ% a & b %þ% b where we have the specific formula for the biomial coefficiets: '!!(&)!

More information

Axioms of Measure Theory

Axioms of Measure Theory MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that

More information

Chapter 10: Power Series

Chapter 10: Power Series Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because

More information

Section 11.8: Power Series

Section 11.8: Power Series Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i

More information

Summary: Congruences. j=1. 1 Here we use the Mathematica syntax for the function. In Maple worksheets, the function

Summary: Congruences. j=1. 1 Here we use the Mathematica syntax for the function. In Maple worksheets, the function Summary: Cogrueces j whe divided by, ad determiig the additive order of a iteger mod. As described i the Prelab sectio, cogrueces ca be thought of i terms of coutig with rows, ad for some questios this

More information

Bertrand s Postulate

Bertrand s Postulate Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a

More information

REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.

REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains. the Further Mathematics etwork wwwfmetworkorguk V 07 The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values

More information

Induction: Solutions

Induction: Solutions Writig Proofs Misha Lavrov Iductio: Solutios Wester PA ARML Practice March 6, 206. Prove that a 2 2 chessboard with ay oe square removed ca always be covered by shaped tiles. Solutio : We iduct o. For

More information

CHAPTER 5. Theory and Solution Using Matrix Techniques

CHAPTER 5. Theory and Solution Using Matrix Techniques A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL

More information

Math 113, Calculus II Winter 2007 Final Exam Solutions

Math 113, Calculus II Winter 2007 Final Exam Solutions Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this

More information

4755 Mark Scheme June Question Answer Marks Guidance M1* Attempt to find M or 108M -1 M 108 M1 A1 [6] M1 A1

4755 Mark Scheme June Question Answer Marks Guidance M1* Attempt to find M or 108M -1 M 108 M1 A1 [6] M1 A1 4755 Mark Scheme Jue 05 * Attempt to fid M or 08M - M 08 8 4 * Divide by their determiat,, at some stage Correct determiat, (A0 for det M= 08 stated, all other OR 08 8 4 5 8 7 5 x, y,oe 8 7 4xy 8xy dep*

More information

Let us consider the following problem to warm up towards a more general statement.

Let us consider the following problem to warm up towards a more general statement. Lecture 4: Sequeces with repetitios, distributig idetical objects amog distict parties, the biomial theorem, ad some properties of biomial coefficiets Refereces: Relevat parts of chapter 15 of the Math

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

It is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.

It is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial. Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable

More information

Solutions to Final Exam

Solutions to Final Exam Solutios to Fial Exam 1. Three married couples are seated together at the couter at Moty s Blue Plate Dier, occupyig six cosecutive seats. How may arragemets are there with o wife sittig ext to her ow

More information

ANOTHER GENERALIZED FIBONACCI SEQUENCE 1. INTRODUCTION

ANOTHER GENERALIZED FIBONACCI SEQUENCE 1. INTRODUCTION ANOTHER GENERALIZED FIBONACCI SEQUENCE MARCELLUS E. WADDILL A N D LOUIS SACKS Wake Forest College, Wisto Salem, N. C., ad Uiversity of ittsburgh, ittsburgh, a. 1. INTRODUCTION Recet issues of umerous periodicals

More information

CSE 1400 Applied Discrete Mathematics Number Theory and Proofs

CSE 1400 Applied Discrete Mathematics Number Theory and Proofs CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of

More information

Lemma Let f(x) K[x] be a separable polynomial of degree n. Then the Galois group is a subgroup of S n, the permutations of the roots.

Lemma Let f(x) K[x] be a separable polynomial of degree n. Then the Galois group is a subgroup of S n, the permutations of the roots. 15 Cubics, Quartics ad Polygos It is iterestig to chase through the argumets of 14 ad see how this affects solvig polyomial equatios i specific examples We make a global assumptio that the characteristic

More information

Created by T. Madas SERIES. Created by T. Madas

Created by T. Madas SERIES. Created by T. Madas SERIES SUMMATIONS BY STANDARD RESULTS Questio (**) Use stadard results o summatios to fid the value of 48 ( r )( 3r ). 36 FP-B, 66638 Questio (**+) Fid, i fully simplified factorized form, a expressio

More information

(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?

(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m? MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle

More information

Definitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.

Definitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients. Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,

More information

Vector Spaces and Vector Subspaces. Remarks. Euclidean Space

Vector Spaces and Vector Subspaces. Remarks. Euclidean Space Vector Spaces ad Vector Subspaces Remarks Let be a iteger. A -dimesioal vector is a colum of umbers eclosed i brackets. The umbers are called the compoets of the vector. u u u u Euclidea Space I Euclidea

More information

MATH 304: MIDTERM EXAM SOLUTIONS

MATH 304: MIDTERM EXAM SOLUTIONS MATH 304: MIDTERM EXAM SOLUTIONS [The problems are each worth five poits, except for problem 8, which is worth 8 poits. Thus there are 43 possible poits.] 1. Use the Euclidea algorithm to fid the greatest

More information

PAPER : IIT-JAM 2010

PAPER : IIT-JAM 2010 MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure

More information

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled 1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how

More information

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3 MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special

More information

Solutions to Problem Set 8

Solutions to Problem Set 8 8.78 Solutios to Problem Set 8. We ow that ( ) ( + x) x. Now we plug i x, ω, ω ad add the three equatios. If 3 the we ll get a cotributio of + ω + ω + ω + ω 0, whereas if 3 we ll get a cotributio of +

More information

Addition: Property Name Property Description Examples. a+b = b+a. a+(b+c) = (a+b)+c

Addition: Property Name Property Description Examples. a+b = b+a. a+(b+c) = (a+b)+c Notes for March 31 Fields: A field is a set of umbers with two (biary) operatios (usually called additio [+] ad multiplicatio [ ]) such that the followig properties hold: Additio: Name Descriptio Commutativity

More information

Lesson 10: Limits and Continuity

Lesson 10: Limits and Continuity www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

More information

The Binomial Theorem

The Binomial Theorem The Biomial Theorem Robert Marti Itroductio The Biomial Theorem is used to expad biomials, that is, brackets cosistig of two distict terms The formula for the Biomial Theorem is as follows: (a + b ( k

More information

Math 61CM - Solutions to homework 3

Math 61CM - Solutions to homework 3 Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig

More information

Kinetics of Complex Reactions

Kinetics of Complex Reactions Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 wcolema@wellesley.edu Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet

More information

R is a scalar defined as follows:

R is a scalar defined as follows: Math 8. Notes o Dot Product, Cross Product, Plaes, Area, ad Volumes This lecture focuses primarily o the dot product ad its may applicatios, especially i the measuremet of agles ad scalar projectio ad

More information

Curve Sketching Handout #5 Topic Interpretation Rational Functions

Curve Sketching Handout #5 Topic Interpretation Rational Functions Curve Sketchig Hadout #5 Topic Iterpretatio Ratioal Fuctios A ratioal fuctio is a fuctio f that is a quotiet of two polyomials. I other words, p ( ) ( ) f is a ratioal fuctio if p ( ) ad q ( ) are polyomials

More information

Application of Jordan Canonical Form

Application of Jordan Canonical Form CHAPTER 6 Applicatio of Jorda Caoical Form Notatios R is the set of real umbers C is the set of complex umbers Q is the set of ratioal umbers Z is the set of itegers N is the set of o-egative itegers Z

More information

Linear Regression Demystified

Linear Regression Demystified Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces

More information

Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018

Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018 CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical

More information

+ au n+1 + bu n = 0.)

+ au n+1 + bu n = 0.) Lecture 6 Recurreces - kth order: u +k + a u +k +... a k u k 0 where a... a k are give costats, u 0... u k are startig coditios. (Simple case: u + au + + bu 0.) How to solve explicitly - first, write characteristic

More information

arxiv: v1 [math.fa] 3 Apr 2016

arxiv: v1 [math.fa] 3 Apr 2016 Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert

More information

THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART II Calculators are NOT permitted Time allowed: 2 hours

THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART II Calculators are NOT permitted Time allowed: 2 hours THE 06-07 KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART II Calculators are NOT permitted Time allowed: hours Let x, y, ad A all be positive itegers with x y a) Prove that there are

More information

ACO Comprehensive Exam 9 October 2007 Student code A. 1. Graph Theory

ACO Comprehensive Exam 9 October 2007 Student code A. 1. Graph Theory 1. Graph Theory Prove that there exist o simple plaar triagulatio T ad two distict adjacet vertices x, y V (T ) such that x ad y are the oly vertices of T of odd degree. Do ot use the Four-Color Theorem.

More information

APPENDIX F Complex Numbers

APPENDIX F Complex Numbers APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios

More information

Roger Apéry's proof that zeta(3) is irrational

Roger Apéry's proof that zeta(3) is irrational Cliff Bott cliffbott@hotmail.com 11 October 2011 Roger Apéry's proof that zeta(3) is irratioal Roger Apéry developed a method for searchig for cotiued fractio represetatios of umbers that have a form such

More information

Homework Set #3 - Solutions

Homework Set #3 - Solutions EE 15 - Applicatios of Covex Optimizatio i Sigal Processig ad Commuicatios Dr. Adre Tkaceko JPL Third Term 11-1 Homework Set #3 - Solutios 1. a) Note that x is closer to x tha to x l i the Euclidea orm

More information

LESSON 2: SIMPLIFYING RADICALS

LESSON 2: SIMPLIFYING RADICALS High School: Workig with Epressios LESSON : SIMPLIFYING RADICALS N.RN.. C N.RN.. B 5 5 C t t t t t E a b a a b N.RN.. 4 6 N.RN. 4. N.RN. 5. N.RN. 6. 7 8 N.RN. 7. A 7 N.RN. 8. 6 80 448 4 5 6 48 00 6 6 6

More information

Classroom. We investigate and further explore the problem of dividing x = n + m (m, n are coprime) sheep in

Classroom. We investigate and further explore the problem of dividing x = n + m (m, n are coprime) sheep in Classroom I this sectio of Resoace, we ivite readers to pose questios likely to be raised i a classroom situatio. We may suggest strategies for dealig with them, or ivite resposes, or both. Classroom is

More information

CALCULATING FIBONACCI VECTORS

CALCULATING FIBONACCI VECTORS THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet

More information

[ 47 ] then T ( m ) is true for all n a. 2. The greatest integer function : [ ] is defined by selling [ x]

[ 47 ] then T ( m ) is true for all n a. 2. The greatest integer function : [ ] is defined by selling [ x] [ 47 ] Number System 1. Itroductio Pricile : Let { T ( ) : N} be a set of statemets, oe for each atural umber. If (i), T ( a ) is true for some a N ad (ii) T ( k ) is true imlies T ( k 1) is true for all

More information

The Phi Power Series

The Phi Power Series The Phi Power Series I did this work i about 0 years while poderig the relatioship betwee the golde mea ad the Madelbrot set. I have fially decided to make it available from my blog at http://semresearch.wordpress.com/.

More information

Topic 9 - Taylor and MacLaurin Series

Topic 9 - Taylor and MacLaurin Series Topic 9 - Taylor ad MacLauri Series A. Taylors Theorem. The use o power series is very commo i uctioal aalysis i act may useul ad commoly used uctios ca be writte as a power series ad this remarkable result

More information

Chapter Vectors

Chapter Vectors Chapter 4. Vectors fter readig this chapter you should be able to:. defie a vector. add ad subtract vectors. fid liear combiatios of vectors ad their relatioship to a set of equatios 4. explai what it

More information

Z ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew

Z ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew Problem ( poits) Evaluate the itegrals Z p x 9 x We ca draw a right triagle labeled this way x p x 9 From this we ca read off x = sec, so = sec ta, ad p x 9 = R ta. Puttig those pieces ito the itegralrwe

More information

Proof of Fermat s Last Theorem by Algebra Identities and Linear Algebra

Proof of Fermat s Last Theorem by Algebra Identities and Linear Algebra Proof of Fermat s Last Theorem by Algebra Idetities ad Liear Algebra Javad Babaee Ragai Youg Researchers ad Elite Club, Qaemshahr Brach, Islamic Azad Uiversity, Qaemshahr, Ira Departmet of Civil Egieerig,

More information

Chapter 9 - CD companion 1. A Generic Implementation; The Common-Merge Amplifier. 1 τ is. ω ch. τ io

Chapter 9 - CD companion 1. A Generic Implementation; The Common-Merge Amplifier. 1 τ is. ω ch. τ io Chapter 9 - CD compaio CHAPTER NINE CD-9.2 CD-9.2. Stages With Voltage ad Curret Gai A Geeric Implemetatio; The Commo-Merge Amplifier The advaced method preseted i the text for approximatig cutoff frequecies

More information

Some remarks for codes and lattices over imaginary quadratic

Some remarks for codes and lattices over imaginary quadratic Some remarks for codes ad lattices over imagiary quadratic fields Toy Shaska Oaklad Uiversity, Rochester, MI, USA. Caleb Shor Wester New Eglad Uiversity, Sprigfield, MA, USA. shaska@oaklad.edu Abstract

More information

Machine Learning for Data Science (CS 4786)

Machine Learning for Data Science (CS 4786) Machie Learig for Data Sciece CS 4786) Lecture & 3: Pricipal Compoet Aalysis The text i black outlies high level ideas. The text i blue provides simple mathematical details to derive or get to the algorithm

More information

MA131 - Analysis 1. Workbook 10 Series IV

MA131 - Analysis 1. Workbook 10 Series IV MA131 - Aalysis 1 Workbook 10 Series IV Autum 2004 Cotets 4.19 Rearragemets of Series...................... 1 4.19 Rearragemets of Series If you take ay fiite set of umbers ad rearrage their order, their

More information