21. 5 Left Or Right Polynomial Division 5 Left or right polynomial division. The operator nc_divide computes the one sided quotient and remainder of two polynomials nc_divide( p1 , p2 ); http://www.uni-koeln.de/REDUCE/3.6/doc/ncpoly/node5.html

TOP Next: 6 Left or right polynomial reduction Up: NCPOLY: Computation in noncommutative polynomial ideals Previous: Top: REDUCE Online Documentation 5 Left or right polynomial division The operator computes the one sided quotient and remainder of two polynomials: The result is a list with quotient and remainder. The division is performed as a pseudodivision, multiplying by coefficients if necessary. The result is defined by the relation for direction left and for direction right where is an expression that does not contain any of the ideal variables, and the leading term of is lower than the leading term of according to the actual term order. TOP Next: 6 Left or right polynomial reduction Up: NCPOLY: Computation in noncommutative polynomial ideals Previous: Top: REDUCE Online Documentation REDUCE WWW Pages maintained by Strotmann@RRz.Uni-Koeln.DE at

22. Deconvolution Or Descending Polynomial Division Deconvolution or Descending polynomial division. http://www.omatrix.com/manual/deconv.htm

23. Deconvolution Or Descending Polynomial Division Deconvolution and descending polynomial division are equivalent operations. The descending polynomial b is the quotient resulting from dividing the http://www.omatrix.com/manual/deconv_frame2.htm

This Section Description Deconvolution and descending polynomial division are equivalent operations. The descending polynomial b is the quotient resulting from dividing the descending polynomial y by the descending polynomial corresponding a . The descending polynomial r is the remainder of the division; i.e., y x a x b x r where denoted multiplication of the corresponding polynomials. From another point of view y conv a b r where conv denotes convolution. Example You can perform the following polynomial division x x x by entering y = [1, 3, 2] a = [1, 1] [b, r] = deconv(y, a) You can print the quotient polynomial which is x by entering b to which O-Matrix will reply and you can print the remainder by entering r to which O-Matrix will reply

24. Polynomial Division This is polynomial division by A(Z). We have for the output Y(Z) polynomial division feedback filter Y(Z) = X(Z) / A(Z) subroutine polydiv( na, aa, http://sepwww.stanford.edu/sep/prof/pvi/zp/paper_html/node15.html

Next: Spectrum of a pole Up: DAMPED OSCILLATION Previous: Narrow-band filters Polynomial division Convolution with the coefficients b t of B Z A Z ) is a narrow-banded filtering operation. If the pole is chosen very close to the unit circle, the filter bandpass becomes very narrow, and the coefficients of B Z ) drop off very slowly. A method exists of narrow-band filtering that is much quicker than convolution with b t . This is polynomial division by A Z ). We have for the output Y Z Multiply both sides of ( ) by A Z For definiteness, let us suppose that the x t and y t vanish before t = 0. Now identify coefficients of successive powers of Z to get Let N a be the highest power of Z in A Z ). The k -th equation (where k N a ) is Solving for y k , we get Equation ( ) may be used to solve for y k once are known. Thus the solution is recursive . The value of N a is only 2, whereas N b is technically infinite and would in practice need to be approximated by a large value. So the feedback operation ( ) is much quicker than convolving with the filter B Z A Z ). A program for the task is given below. Data lengths such as

25. Polynomial Division polynomial division. polynomial division. Kolmogorov crossspectral factorization, therefore, provides a tool to factor the helical 1-D filter of length http://sepwww.stanford.edu/public/docs/sep109/paper_html/node36.html

Next: Synthetic examples Up: Implicit extrapolation theory Previous: Cross-spectral factorization Polynomial division Kolmogorov cross-spectral factorization, therefore, provides a tool to factor the helical 1-D filter of length 2 N x + 1 into minimum-phase causal (and maximum-phase anti-causal) filters of length N x +1. Deconvolution with minimum-phase filters is unconditionally stable. However, inverse-filtering with the entire filters would be an expensive operation. Fortunately, filter coefficients drop away rapidly from either end. In practice, small-valued coefficients can be safely discarded, without violating the minimum-phase requirement; so for a given grid-size, the cost of the matrix inversion scales linearly with the size of the grid. The unitary form of equation ( ) can be maintained by factoring the right-hand-side matrix, in equation ( ), with Kolmogorov before applying it to Chapter and Appendix extend the concept of recursive inverse filtering to handle non-stationarity. There are pitfalls associated with this process, however; consequently, in this Chapter I limit the examples to the constant velocity case. Next: Synthetic examples Up: Implicit extrapolation theory Previous: Cross-spectral factorization Stanford Exploration Project

26. Activity Exchange - Activity Detail activity title, polynomial division. activity overview, This Computer Algebra System (CAS) activity encourages students to investigate polynomial division, http://education.ti.com/educationportal/activityexchange/activity_detail.do?cid=

27. Polynomial Division polynomial division. Review of Long Division. Example Use long division to calculate of the polynomial Q(x). Example Use synthetic division to find http://www.ltcconline.net/greenl/courses/103a/polynomials/polydiv.htm

Polynomial Division Review of Long Division Example Use long division to calculate and will write the steps for this process without using any numbers. Solution We see that we follow the steps: Write it in long division form. Determine what we need to multiply the quotient by to get the first term. Place that number on top of the long division sign. Multiply that number by the quotient and place the product below. Subtract Repeat the process until the degree of the difference is smaller than the degree of the quotient. Write as sum of the top numbers + remainder/quotient. P(x)/D(x) = Q(x) + R(x)/D(x) Below is a nonsintactical version of a computer program: do divide first term of remainder by first term of denominator and place above quotient line; multiply result by denominator and place product under the remainder; subtract product from remainder for new remainder; Write expression above the quotient line + remainder/denominator; Exercises + 5x + 7)/(x + 1) + x - 1)/(x Synthetic Division For the special case that the denominator is of the form x - r , we can use a shorthand version of polynomial division called synthetic division. Here is a step by step method for synthetic division for

28. 9.4 Polynomial Division, Factors, And Remanders This site shows how to divide polynomials by either long division or synthetic division. It also explains the factor theorem and the remainder theorem http://drago5.0.tripod.com/

setAdGroup('67.18.104.18'); var cm_role = "live" var cm_host = "tripod.lycos.com" var cm_taxid = "/memberembedded" Search: Lycos Tripod Free Games Share This Page Report Abuse Edit your Site ... Next 9.4 polynomial division, Factors, and Remanders Long Division Synthetic Division Theorems This site shows how to divide polynomials by either long division or synthetic division. It also explains the factor theorem and the remainder theorem Enter supporting content here

29. Worksheet11 Section 3.2 is about polynomial division, and what polynomial division can tell The division algorithm for polynomial long division says if $ f(x)$ and http://cs.jsu.edu/mcis/faculty/leathrum/ms105/Worksheet11/

30. Polynomial Division In Pari? To pariusers@list.cr.yp.to; Subject polynomial division in Pari? Or is it even possible to divide two dyadic polynomials? Regards and thanks for your http://pari.math.u-bordeaux.fr/archives/pari-users-0507/msg00002.html

Sascha Rissel on Mon, 18 Jul 2005 21:18:24 +0200 Date Prev Date Next Thread Prev Thread Next ... Thread Index polynomial division in Pari? To pari-users@list.cr.yp.to Subject : polynomial division in Pari? From s_rissel@web.de Date : Mon, 18 Jul 2005 21:16:02 +0200 Delivery-date : Mon, 18 Jul 2005 21:18:24 +0200 Mailing-list : contact pari-users-help@list.cr.yp.to; run by ezmlm Sender s_rissel@web.de User-agent : Mozilla Thunderbird 1.0.5 (Windows/20050711) Hello, how can I do some polynomial divisions in Pari? Or is it even possible to divide two dyadic polynomials? Regards and thanks for your help, Sascha. Follow-Ups Re: polynomial division in Pari? From: Prev by Date: Re: Class groups Next by Date: Re: polynomial division in Pari? Previous by thread: Re: Class groups Next by thread: Re: polynomial division in Pari? Index(es): Date Thread

31. Re: Polynomial Division In Pari? To pariusers@list.cr.yp.to; Subject Re polynomial division in Pari? Or is it even possible to divide two dyadic polynomials? http://pari.math.u-bordeaux.fr/archives/pari-users-0507/msg00004.html

Sascha Rissel on Tue, 19 Jul 2005 20:04:21 +0200 Date Prev Date Next Thread Prev Thread Next ... Thread Index Re: polynomial division in Pari? To pari-users@list.cr.yp.to Subject : Re: polynomial division in Pari? From s_rissel@web.de Date : Tue, 19 Jul 2005 19:37:03 +0200 Delivery-date : Tue, 19 Jul 2005 20:04:21 +0200 In-reply-to 20050719102928.GA19742@pari.math.u-bordeaux.fr Mailing-list : contact pari-users-help@list.cr.yp.to; run by ezmlm References 42DBFFF2.6010800@web.de 20050719102928.GA19742@pari.math.u-bordeaux.fr Sender s_rissel@web.de User-agent : Mozilla Thunderbird 1.0.5 (Windows/20050711) Follow-Ups Re: polynomial division in Pari? From: References polynomial division in Pari? From: Re: polynomial division in Pari? From: Prev by Date: Re: polynomial division in Pari? Next by Date: Re: polynomial division in Pari? Previous by thread: Re: polynomial division in Pari? Next by thread: Re: polynomial division in Pari? Index(es): Date Thread

32. Long Polynomial Division Set up the long division x2 + 5x + 9 = (x + 2)(x + 3) + 3. Dividend, x2 + 5x + 9. Divisor, x + 2. Quotient, x + 3. Remainder, 3. http://www.sci.wsu.edu/~kentler/Fall97_101/Chapter5/lpd_1.html

Problem: Divide x x + 2 Set up the long division x x x Choose x since x x = x(x + 2). Subtract x + 2x from x Result is x Choose since = 3(x + 2). Subtract 3x + 6 from 3x + 9. Result is is the remainder. Answer: x x + 2 x + 3 + x + 2 Notes x + 5x + 9 = (x + 2)(x + 3) + 3 Dividend x Divisor x + 2 Quotient x + 3 Remainder

33. Module 1 -- Polynomial Division Instructional Unit. Polynomial and Rational Functions. Day One. by. Behnaz Rouhani. Return to Behnaz Rouhani s Page. http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002/Rouhani/IU/module1.html

Instructional Unit Polynomial and Rational Functions Day One by Behnaz Rouhani Return to Behnaz Rouhani's Page

34. Detailed Information For Polynomial Division Program , Simple program to divide polynomials. Authors, Eliel Louzoun Mikael Sundstrom......polynomial division Program. Filename, polydiv.zip. http://www.hpcalc.org/details.php?id=357

35. Polynomial Division First Previous Next Last Index Home Text. Slide 18 of 26. http://www.cs.berkeley.edu/~kfall/EE122/lec06/sld018.htm

36. Polynomial Division polynomial division. 10011010000. 1101. 1. 1101. 1001. 1101. 1. 1000. 1101. 1011. 1101. 1. 1. 1. 0. 0. 1. 1100. 1101. 1000. 1101. 101 http://www.cs.berkeley.edu/~kfall/EE122/lec06/tsld018.htm

Polynomial Division Previous slide Next slide Back to first slide View graphic version

37. Fq_x polynomial division. abstract Fq_x, divideAndRemainder(Fq_x B) polynomial division Returns the Quotient and Remainder. abstract Fq_x, egcd(Fq_x B) http://dragongate-technologies.com/jSaluki/com/dragongate_technologies/saluki/Fq

Overview Package Class Use Tree Deprecated Index ... METHOD com.dragongate_technologies.saluki Class Fq_x com.dragongate_technologies.saluki.Fq_x Direct Known Subclasses: public abstract class extends java.lang.Object f q (x) Version: Author: Dragongate Technologies Ltd. Constructor Summary Fq [] val) (int n) Method Summary abstract add B) Polynomial Addition int degree Returns the degree of the polynomial. divide B) Polynomial Division abstract divideAndRemainder B) Polynomial Division Returns the Quotient and Remainder. abstract egcd B) Extended Polynomial GCD Fq eval Fq a) Evaluate the value of the polynomial when x is equal to a. abstract gcd B) Polynomial GCD boolean isZero Fq Returns the leading coefficient of the polynomial. mod B) Polynomial Division abstract mul B) Polynomial Multiplication abstract negate Returns the Additive Inverse. abstract Fq a) Add a to each coefficient. abstract Fq a) Multiply each coefficient by a. sub B) Polynomial Subtraction java.lang.String

38. F2m_x polynomial division Algorithm 3.1.1 from of A Course in Computational Algebraic Number Theory by Henri Cohen. Fq_x, egcd(Fq_x B) Extended Polynomial GCD http://dragongate-technologies.com/jSaluki/com/dragongate_technologies/saluki/F2

Overview Package Class Use Tree Deprecated Index ... METHOD DETAIL: FIELD CONSTR METHOD com.dragongate_technologies.saluki Class F2m_x com.dragongate_technologies.saluki.Fq_x com.dragongate_technologies.saluki.F2m_x public class extends f m (x) Version: Author: Dragongate Technologies Ltd. Field Summary static I Multiplicative Identity Element static O Additive Identity Element Constructor Summary Fq [] val) (int n) Method Summary add a) Polynomial Addition divideAndRemainder B) Polynomial Division Algorithm 3.1.1 from of "A Course in Computational Algebraic Number Theory" by Henri Cohen egcd B) Extended Polynomial GCD Algorithm 3.2.2 from of "A Course in Computational Algebraic Number Theory" by Henri Cohen gcd B) Extended Polynomial GCD Algorithm 3.2.1 from of "A Course in Computational Algebraic Number Theory" by Henri Cohen mul f) Polynomial Multiplication Algorithm from section 3.1.2 of "A Course in Computational Algebraic Number Theory" by Henri Cohen. negate Returns the Additive Inverse.

39. Donald Ramirez Mt This link contains exercises in polynomial long division and synthetic division which is used to divide a polynomial by a binomial of the form x a, http://www.msjc.edu/math/dramirez/Polynomial Division Exercises Home Page.htm

Donald Ramirez Mt. San Jacinto College Adjunct Mathematics Instructor Fall 2003 Polynomial Division Exercises This link contains exercises in polynomial long division and synthetic division which is used to divide a polynomial by a binomial of the form x a , where a is a given constant. For each long division exercise which involves a divisor of this form, there is a corresponding synthetic division exercise. Each of the following exercises is presented as a Microsoft PowerPoint slide show. Different slides are reached by clicking the up or down vertical slider arrows on the right side of your screen. After you are finished with an exercise, click "Back" to return to this page. The intent of these exercises is for the student to obtain as much of the solution as they can before viewing the solutions. Not doing so lessens the learning process. The first page of each slide show contains the problem to be worked. Each successive page contains a step and/or simplification in the solution process until the last slide which contains a verification of the solution. To see the slides more clearly, right-click anywhere on the first slide and select "Full Screen".

40. Matlab Manual Page: Deconv deconv. Purpose. Deconvolution and polynomial division. them is equivalent to multiplying the two polynomials, and deconvolution is polynomial division. http://www.utexas.edu/math/Matlab/Manual/deconv.html

deconv Purpose Deconvolution and polynomial division. Synopsis [q,r] = deconv(b,a) Description [q,r] = deconv(b,a) deconvolves vector a out of vector b , using long division. The 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 multiplying the two polynomials, and deconvolution is polynomial division. The result of dividing b by a is quotient q and remainder r Examples If a = [1 2 3 4] b = [10 20 30] the convolution is c = conv(a,b) c = Use deconvolution to divide a back out: [q,r] = deconv(c,a) q = r = gives a quotient equal to b and a zero remainder. Algorithm deconv uses the filter primitive. See Also conv residue convmtx , and f ilter in the Signal Processing Toolbox

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