61. DMTCS SERIES An Introduction to Polynomials Construction and representation of polynomials; Complexity and cost; polynomial division; Polynomial factorization; http://www.cs.auckland.ac.nz/CDMTCS/docs/mignotte.html

M. Mignotte, D. Stefanescu. Polynomials. An Algorithmic Approach, Springer-Verlag, Singapore, 1999. Approx. 320pp. ISBN: 981-4021-51-2. US$49 softcover. This textbook gives a well-balanced presentation of the classic procedures of polynomial algebra which are computationally relevant and some algorithms developed during the last decade. The first chapter discusses the constrcution and the representation of polynomials. The second chapter focuses on the computational aspects of the analytical theory of polynomials. Polynomials with coefficients in a finaite field are then described in chapetr three, and the final chapter is devoted to factorization of polynomials with integral coefficients. The book is primarily aimed at graduate students taking courses in Polynomial Algebra, with a prerequisite knowledge of set theory, usual fields and basic algebra. Fully worked out examples, hints and references complement the main text, and details concerning the implementation of algorithms as well as indicators of their efficiency are provided. The book is also useful as a supplementary text for courses in scientific computing, analysis of algorithms, computational polynomial factorization, and computational geometry of polynomials. Contents: 1. An Introduction to Polynomials: Construction and representation of polynomials; Complexity and cost; Polynomial division; Polynomial factorization; Polynomial roots. Eliminations. Resultants; Symmetric functions; Polynomial interpolation; Irreducinility criteria. 2. Complex Polynomials: Polynomial size; Geometry of polynomials; Stable polynomials; Polynomial roots inside the unit disk; Bounds for the roots; Applications to integer polynomials; Separation of roots. 3. Polynomials with Coefficients in a Finite Field: Finite fields; Cyclotomic polynomials; Fast Fourier transform; Number of irreducible polynomials over a finite field; Constrcution of irreducible polynomials over a finite field; Roots of polynomials over finite fields; Squarefree polynomials; Berlekamp's algorithm; Niederreiter's algorithm. 4. Integer Polynomials: Kronecker's factorization method; The berlekamp-Zassenhaus algorithm; The LLL factorization algorithm. Bibliography; Notation; List of Algorithms; Index.

62. MultiPoly.html Maple has implemented the multivariate polynomial division algorithm. Both commands perform the polynomial division in the multivariate case. http://mark.math.helsinki.fi/Symbolinen laskenta/Worksheets/Lecture12/MultiPoly1

Multivariate polynomials in Maple Monomial orderings and manipulation of polynomials Maple provides built-in functions for manipulating multivariate polynomials. Let us consider, for example, the polynomial Polynomial:=expand((x+2*y)*(2*x+y)*(x^7+y^7)*x^4*y^2-5*x*y^3+y*x^5+x^6-5*x^3*y-5*y^2*x^2+y^3*x^3-5*y^4); You can sort the monomials of this poolynomial in the pure lexicographic order (plex) as follows: sort(Polynomial,plex); The plex ordering is not used by default. By default Maple uses the total degree ordering in which the monomials with highest total degree come first and ties are broken by lexicographical ordering. In this particular example, both orderings are the same: sort(Polynomial); Terms containing the same exponent of a variable can be collected into a single term by the collect command. Warning, premature end of input collect(Polynomial,y); collect(Polynomial,x); You can extract the coefficients of multivariate polynomials by the coeff command. That, however, treats a multivariate polynomial just like a univariate one. coeff(Polynomial,x^3);

63. /TeXnik/equnarray/PolDiv A polynomial division writing in mathmode is possible with \underline{}, \hspace{} and use of this makes polynomial division very easy! For example http://www.tug.org/TeXnik/mainFAQ.cgi?file=equnarray/PolDiv

64. Algebra BIA Brain Summary Synthetic Division Technique used to divide polynomials of a particular form polynomial division (jump) – Intimately tied with rational functions http://braintrax.umr.edu/NewBTWebSite/Teachers/algebraBIAtreedescription.htm

Algebra Brain Summary Malcolm E. Hays 28 October 2002 This is a list of all the thoughts located in the Algebra Brain. Each thought is followed by a statement indicating the content associated by that thought. Thought followed by (jump) are jump thoughts, which are located on the left hand side of the central thought in the Brain matrix. Jump thoughts often take the user to entirely different sections of the Brain or provide reference information about the central thought. This compilation of thoughts only includes those thoughts listed under Content in the Algebra Brain. Algebra Glossary (jump) Several definitions of common terms used throughout Algebra Algebra Alphabet A list of how each letter of the alphabet is used throughout Algebra Fundamentals of Algebra (jump) – The basics needed to fully understand what Algebra is all about Course Content (jump) The 5 basic concepts covered in all Algebra courses, from 8th grade to 13th grade Algebraic Expressions The expression is one of the basic units used in Algebra, is part of an algebraic function Basic Rules of Algebra Covers many of the fundamental properties exhibited by different operators in Algebra Properties of Equality What equality really means Properties of Exponents How to manipulate exponents to simplify algebraic expressions Exponential Functions (jump) – Basic definition of the exponential function Properties of Fractions How to manipulate fractions to simplify algebraic expressions Rational Functions (jump)

65. Polydiv.red Module Polydiv; % Enhanced Polynomial Division % FJ % polydiv.red module polydiv; % enhanced polynomial division % FJWright@Maths.QMW.ac.uk, 6 Nov 1995 % Defines (or redefines) the following polynomial http://centaur.maths.qmul.ac.uk/Computer_Algebra/REDUCE/OLD/polydiv/polydiv.red

< new_kord := t; updkorder x; p := reorder p; q := reorder q >> where kord!* = kord!*; % preserve environment u := if pseudo then pseudo!-qremf(p, q, x) else qremf(p, q); p := car u; q := cdr u; if new_kord then >; n := if domainp v or (var and not(mvar v eq var)) then else >; %% The following special-case code for n = and m < %% Compute the quotient q EFFICIENTLY. %% q0 = (q_0 ... q_k) without powers of v_n q0 := (car_u := car u) . q0; vv := cdr v; u := for each c in cdr u collect < c := multf(c, car_v); if vv then < c := subtrf(c, multf(car_u, car vv)); vv := cdr vv >>; c >>; k := k - 1 >>; if q0 then

66. Cyclic Redundancy Check Polynomials Tutorial Press the check symbol beside polynomial division. A window will appear with the generator sequence as a divisor in a long division sum. http://www.macs.hw.ac.uk/~pjbk/nets/crc/crctext.html

Cyclic Redundancy Check Polynomials Tutorial Cyclic redundancy check polynomials are the theory which lie behind the checksum algorithm used in most modern communication systems. A generator is chosen (using theory which will not be detailed here). This is a sequence of bits, of which the first and last are 1. This sequence is used with the bits of the message to calculate a check sequence which has 1 fewer bits than the generator. The check sequence is appended to the original message. At the receiver, the same calculation is performed on the message and check sequence combined. If the result is no transmission error is assumed to have occurred. Try it now using the tutorial applet above. Enter the message Enter the generator Pressing the Step button repeatedly will transfer bits from the message to the transmitted buffer. After all the message bits have been transferred, continue to press Step a further 5 times. You will observe that the bits have been added to the end of the transmitted buffer. Now press the Clear button and try the message and the sequence will be added.

67. Lecture 23: Polynomial Division Lecture 23 polynomial division. List of Lectures Math 1100 Index . Assignment. Assignments during third test period. These are the problems you should http://www.math.uncc.edu/~hbreiter/m1100/lectures/lect23.htm

Lecture 23: Polynomial Division List of Lectures Math 1100 Index Assignment Assignments during third test period. These are the problems you should work before April 2: Section 3.6; page 318; problems 6n+1, for n = 0,...,14 and number 87. Review; page 326; problems 6n+1, for n = 0,...,12. Section 4.1; page 339; problems 4n+1, for n = 0,...,10. These are the problems you should work before April 9: Section 4.2; page 348; problems 6n+1, for n = 0,...,12. Section 5.1; page 402; problems 1, 7, 15, 27, 29, 31, and 47. These are the problems you should work before April 16: Section 5.2 ; page 413; problems 2n+1, for n = 0,...,23; and 6n+1, for n=8Â 12. Section 5.3 ; page 421; problems 2n+1, for n = 0,...,25 and 6n+1, for n=9Â 14. These are the problems you should work before April 23: Section 5.4 ; page 431; problems 2n+1, for n = 0,...,22 and 4n+1 for n = 12...20. Section 5.5; page 442; problems 1-4, 7, 10, 13, 25-26, 35, 45, 49-50, 55, 57, 59, and 74. Today we talked about two important classes of problems, examples of which can be found by clicking here.

68. CenterSpace API Documentation - Polynomial Division Operator polynomial division Operator. Divides a polynomial by a scalar. public static Polynomial operator /( Polynomial p, double s http://www.centerspace.net/doc/NMath/Core/ref/CenterSpace.NMath.Core.Polynomial.

NMath Core Reference Guide Version 2.2 Polynomial Division Operator  Divides a polynomial by a scalar. public static  Polynomial  operator /( Polynomial p double s Parameters p A polynomial. s A scalar. Return Value A new polynomial containing the quotient. See Also Polynomial Class CenterSpace.NMath.Core Namespace

69. Journal Of Inequalities And Applications TWO polynomial division INEQUALITIES IN L p. P. GOETGHELUCK. Received 4 March 1997 and in revised form 6 August 1997. This paper is a first attempt to give http://www.hindawi.com/journals/jia/volume-2/S1025583498000186.html

Home About this Journal MS Tracking System Author Index ... Contents JIA 2:3 (1998) 285-296. DOI: 10.1155/S1025583498000186 TWO POLYNOMIAL DIVISION INEQUALITIES IN L p P. GOETGHELUCK Received 4 March 1997 and in revised form 6 August 1997 This paper is a first attempt to give numerical values for constants C p and C p , in classical estimates P C p n x P and P C p n x P where P is an algebraic polynomial of degree at most n n and denotes the p -metric on . The basic tools are Markov and Bernstein inequalities. Keywords and phrases: Polynomial inequalities; Schur inequality; Explicit constants. 1991 Mathematics Subject Classification: Primary: 41A17; Secondary: 26D05. The following files are available for this article: Full-Text PDF for all readers This page contains MathML. Click here for more information. Hindawi Publishing Corporation

70. Calendar/Syllabus 3.1, Polynomial functions, 112,14,16,17,21,36. 3.2, polynomial division, 1-8,29,30 (no synthetic division), Dividing polynomials(video)* Long division http://www.math.uri.edu/~pakula/111webs5/calendar_syllabus.htm

Home PrintSyllabus Click [ PrintSyllabus for a version of this page suitable for printing. The following calendar gives a timetable for the course. Your class may be slightly behind or ahead at any given time. Below the calendar is a list of sections in the textbook, with suggested problems and on-line material. Some sections will be assigned by your instructor for reading only and will not be discussed in class. Make sure you attend the class regularly to keep pace with the course. The listed problems may be done in class or homework. Your instructor will be more specific. You should attempt them all. Starred problems may be more challenging. (Note: In the problem lists, a notation like 3-9 means that all the problems 3,4,5,6,7,8,9 are assigned. A notation like" 3-9 odd" means that problems 3,5,7,9 are assigned.) COURSE CALENDAR AND SYLLABUS FOR MTH 111 SPRING 2005 Week of Events Text Jan 17 1st class 1/18 1.1, Intro to Calc., 1.2 Jan. 24 Jan. 31 Feb. 7 Feb.14 Ex I, 2/15 Feb.21 No Mon 2/21, Mon on 2/22 Feb. 28

71. MTH 111 Precalculus -- Summer 2004, Schedule July 12, 3.2 polynomial division 3.3 Polynomial zeros 3.2, polynomial division, 18,29,30 (no synthetic division), Dividing polynomials(video)* http://www.math.uri.edu/~thoma/teaching/mth111_summer2004/schedule.html

MTH111 Precalculus Calendar Summer(II) 2004 The following calendar gives a timetable for the course. Your class may be slightly behind or ahead at any given time. Below the calendar is a list of sections in the textbook, with suggested problems and on-line material. Some sections will be assigned by your instructor for reading only and will not be discussed in class. Make sure you attend the class regularly to keep pace with the course. The listed problems may be done in class or homework. Your instructor will be more specific. You should attempt them all. Starred problems may be more challenging. (Note: In the problem lists, a notation like 3-9 means that all the problems 3,4,5,6,7,8,9 are assigned. A notation like" 3-9 odd" means that problems 3,5,7,9 are assigned. DAY SECTIONS SUGGESTED PROBLEMS June 28 1.1 Functions 1.2 Linear functions 1.4 More on functions June 30 1.5 Symmetry/transformation 1.6 Variation 1.7 Distance, midpoint, circle

72. This Program Listing Is CTF Compliant And Xchange Compatible. To TITLE polynomial division AUTHOR Roy FA Maclean EMAIL rfam@lycos.co.uk WEB http//members.lycos.co.uk/rfam DATE 13Aug1996, 15Jan1998, http://members.lycos.co.uk/rfam/9x50g/polydiv.txt

73. This Program Listing Is CTF Compliant And Xchange Compatible. To TITLE polynomial division AUTHOR Roy FA Maclean EMAIL rfam@lycos.co.uk WEB http//members.lycos.co.uk/rfam DATE 13Aug1996 MAKE CASIO http://members.lycos.co.uk/rfam/wide/polydiv.txt

Goto 1 ; loop 1 stores first set of coefficients "COEFFSDEN" Lbl 2 ?->Q[D] Isz D D Goto 2 ; loop 2 stores second set of coefficients "QUOTIENT" Lbl 3 ; loop 3 displays the quotient coeffs (the 'C' value) G[F]%Q->C_ ; use fraction symbol for % F->D:0->E Lbl 4 G[D]-CQ[E]->G[D] Isz D:Isz E ; loop 4 uses the 'C' from loop 3 to E Goto 4 ; subtract from dividend Isz F A-F>=B=>Goto 3 "REM" Lbl 5 G[F]_ Isz F F Goto 1 "COEFFSDEN" Lbl 2 ?->Q[D] Isz D D Goto 2 "QUOTIENT" Lbl 3 G[F]%Q->C_ F->D:0->E Lbl 4 G[D]-CQ[E]->G[D] Isz D:Isz E E Goto 4 Isz F A-F>=B=>Goto 3 "REM" Lbl 5 G[F]_ Isz F F Goto 5 Goto

74. Deconv (MATLAB Functions) . q,r = deconv(v,u) deconvolves vector u out of vector v , using long......Deconvolution and polynomial division. Syntax. q,r = deconv(v,u). http://www.mathworks.com/access/helpdesk/help/techdoc/ref/deconv.html

MATLAB Function Reference deconv Deconvolution and polynomial division Syntax [q,r] = deconv(v,u) Description [q,r] = deconv(v,u) deconvolves vector u out of vector v , using long division. The quotient is returned in vector q and the remainder in vector r such that v conv(u,q)+r If u and v are vectors of polynomial coefficients, convolving them is equivalent to multiplying the two polynomials, and deconvolution is polynomial division. The result of dividing v by u is quotient q and remainder r Examples If u = [1 2 3 4] v = [10 20 30] the convolution is c = conv(u,v) c = Use deconvolution to recover u [q,r] = deconv(c,u) q = r = This gives a quotient equal to v and a zero remainder. Algorithm deconv uses the filter primitive. See Also conv residue decic Trademarks

75. 6.170 Spring 2005 that we should stop polynomial division once the dividend gets a degree lower This is certainly true, but only if we regard the zero polynomial as http://theory.csail.mit.edu/classes/6.170/archive/Old-2005-Spring/modules.php?na

76. TSP (libtsp) - MSdDeconvCof Given polynomials U(D) and V(D), the polynomial division U(D)/V(D) gives quotient Q(D) and remainder R(D), where U(D) = Q(D) V(D) + R(D). http://www.tsp.ece.mcgill.ca/MMSP/Documents/Software/Packages/libtsp/MS/MSdDecon

MSdDeconvCof Routine int MSdDeconvCof (const double u[], int Nu, const double v[], int Nv, double q[], double r[]) Purpose Polyomial divide (deconvolve the coefficients of two vectors) Description This routine divides one polynomial by another. Given polynomials U(D) and V(D), the polynomial division U(D)/V(D) gives quotient Q(D) and remainder R(D), where U(D) = Q(D) V(D) + R(D). Polynomial U(D) is represented as U(D) = u[0] + u[1] D + u[2] D^2 + ... + u[Nu-1] D^(Nu-1). The quotient Q(D) normally has Nu-Nv+1 terms. The remainder R(C) normally has Nv-1 terms. Parameters Number of quotient coefficients. This is normally Nu-Nv+1. If some of the higher order terms in v are zero, this may be as large as Nu. If the last non-zero term in v is v[k], then the number of quotient coefficients is returned as Nu-k. Input vector of coefficients (Nu values) Number of coefficients in u Input vector of coefficients (Nv values) Output vector of Nu coefficients representing the quotient of the polynomial division. The quotient vector can be the same as the vector u. If v[k] is the last non-zero coefficient of v, the last k coefficients of q will be zero. Output vector of Nu coefficient representing the remainder of the polynomial division. If r is NULL, the remainder is not calculated. The remainder satisfies R(D) = U(D) - Q(D) U(D). If v[k] is the last nonzero coefficient of v (k can be ascertained from the function return value which is Nu-k), the last Nu-k terms of R(D) will be zero.

77. Partial Fractions polynomial division. The first step in expanding the ration is dividing if the After polynomial division, we are left with a proper rational function. http://www-ee.eng.buffalo.edu/faculty/paololiu/edtech/roaldi/References/partial_

Partial Fraction Expansion When solving circuit analysis problems in the s-Domain with Laplace transforms, you are generally left with a ratio of polynomials of s. To convert these back into functions in the time domain, the ratio may need to be converted into the addition of simpler polynomials which can then be looked up in a table to find their inverse. Polynomial Division The first step in expanding the ration is dividing if the ratio is improper. The ration is improper if the order (the largest power) of the numerator is greater than the order of the denominator. If this is the case then the numerator should be divided by the denominator in a way which is very similar to standard long division. + 2s² + 2) / (s² - 2s + 1) + 2s² + 2) / (s² - 2s + 1) = 5s + 12 + (19s - 10) / (s² - 2s + 1) The two terms which are not part of a fraction now can be looked up in a table to find the inverse Laplace transform. The rest of this reference section will deal with how to reduce the remaining fractional part to something that can be converted easily. Simple Poles After polynomial division, we are left with a proper rational function. At this point, our objective is to convert this function into the sum of simple fractions. For example we have

78. Academics-Penn State Erie-Learning Resource Center Tape 10 polynomial division, Synthetic Division, Real Zeros of Polynomial Functions. Tape 11 - Complex Zeros and the Fundamental Theorem of Algebra, http://pennstatebehrend.psu.edu/academic/lrc/videos.htm

79. Deconv (Signal Processing Toolbox) . q,r = deconv(b,a) deconvolves vector a out of vector b , using long......Deconvolution and polynomial division. Syntax. q,r = deconv(b,a). http://odysseus.nat.uni-magdeburg.de/matlab/toolbox/signal/deconv.html

Signal Processing Toolbox Go to function: Search Help Desk deconv Examples See Also Deconvolution and polynomial division. Syntax [q,r] = deconv(b,a) Description [q,r] = deconv(b,a) deconvolves vector a out of vector b , using long division. The result (quotient) is returned in vector q and the remainder in vector r such that b = conv(q,a) + r If a and b are vectors of polynomial coefficients, convolving them is equivalent to polynomial multiplication, and deconvolution is equivalent to polynomial division. The result of dividing b by a is quotient q and remainder r deconv is part of the standard MATLAB environment. Example The convolution of a = [1 2 3] and b = [4 5 6] is c = conv(a,b) c = Use deconv to divide b back out: [q,r] = deconv(c,a) q = r = Algorithm This function is an M-file in the MATLAB environment that uses the filter primitive. Deconvolution is the impulse response of an IIR filter. See Also conv Convolution and polynomial multiplication. filter Filter data with a recursive (IIR) or nonrecursive (FIR) filter. residuez z -transform partial fraction expansion.

80. Creating Polynomial Division Logical Devices Patent Freshpatents.com offers information on a variety of new patent applications, updated each week check out Creating polynomial division logical devices http://www.freshpatents.com/Creating-polynomial-division-logical-devices-dt20041

Monitor Patents - Creating polynomial division logical devices Title/Abstract Agents Agents by City Apps by Location FreshPatents Search A FREE service from FreshPatents... PATENT KEYWORD MONITOR 3 steps to Unlock the Secrets of new technologies before competitors, co-workers or other inventors! Sign up (takes 30 seconds). Fill in the keywords to be monitored. 3. Each week you receive an email with patent applications related to your keywords. Start now! Browse Patent Applications: Prev Next USPTO Class 706 Creating polynomial division logical devices A method of creating a logical device performing polynomial division includes using a hardware description language to build code directly describing synthesizable logic for performing the polynomial division. The logic is then implemented on a target device. The code receives as inputs a parameter identifying a polynomial and a parameter identifying a number of data bits for which the polynomial division is performed. For a given n-degree polynomial, performing the polynomial division includes calculating a next n-term remainder for a data unit having d terms. Agent: Agilent Technologies, Inc. Legal Department, Dl429

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z

Page 4     61-80 of 88    Back | 1  | 2  | 3  | 4  | 5  | Next 20