21. Polynomial Division 3 ? ( ? ) ( polynomial division Version 3 ). ? ( ? ) ( polynomial division Version 1 ). http://lpl.hkcampus.net/~lpl-wwk/Casio/Polynomial Division 3.htm

¦h¶µ¦¡°£ªk ( ²Ä¤Tª© ) ( Polynomial Division : Version 3 ) ª`·N FX-50F ¤W¨Ï¥Î¡C ( Cubic Polynomial ) ³Q¤@­Ó¤G¦¸¦h¶µ¦¡ ( Quadratic Polynomial ) °£©Ò±oªº°Ó¦¡ ( Quotient ) ¤Î¾l¼Æ¦¡ ( Remainder ) ¡C FX-50F/3600PV/3800P/3900PV ( LRN mode ) FX-3900PV ( EDIT mode ) Kout 5 Kin Kout 6 Kout 1 Kin ¡V Kout 5 Kin MR Kout 1 HLT Kout 6 Kout 2 HLT Kin ¡V MR Kout 2 Kin ¡V Kout 3 HLT Kout 4 MODE . 28 steps ¦pªG¨Ï¥Î LRN LRN ¼Ò¦¡«e¥ý«ö 1 Kin 5 ¡C ¨Ò¡G­pºâ ¡V 3x x ¡V ªº°Ó¦¡¤Î¾l¼Æ¦¡¡C Kin 1 3 +/- Kin 2 3 Kin 3 1 Kin 4 ( ¤T¦¸¦h¶µ¦¡ ( ³Q°£¦¡ ) ªº«Y¼Æ coefficient )¡A¦A«ö 1 Kin 5 2 +/- Kin 6 2 Min ( °£¦¡ªº«Y¼Æ ) ¡A¦A«ö RUN ( °Ó¦¡ªº«Y¼Æ ) ¦A«ö RUN ¡A¦A«ö RUN ¡V ( ¾l¼Æ¦¡ªº«Y¼Æ ) ¥ç§Y¬O»¡°Ó¦¡¬O ¡A¾l¼Æ¦¡¬O x ¡V ¡C FX-3800P FX-3900PV ¤W¨Ï¥Î¡C FX-3800P/3900PV ( LRN mode ) ENT 1 Kin 1 ENT 1 Kin 2 ENT 1 Kin 3 ENT 1 Kin 4 ENT 1 Kin 5 Kin ENT 1 Kin 6 Kout 1 Kin ¡V Kout 5 Kin ENT 1 Kin 5 Kout 1 HLT Kout 6 Kout 2 HLT Kin ¡V Kout 5 Kout 2 Kin ¡V Kout 3 HLT Kout 4 MODE . 39 steps FX-3900PV ( EDIT mode, MODE ) ENT Kin 1 ENT Kin 2 ENT Kin 3 ENT Kin 4 ENT Kin 5 Kin ENT Kin 6 Kout 1 Kin ¡V Kout 5 Kin ENT Kin 5 Kout 1 HLT Kout 6 Kout 2 HLT Kin ¡V Kout 5 Kout 2 Kin ¡V Kout 3 HLT Kout 4 MODE .

22. An Optimum Real-Time Systolic Polynomial Division AIgorithm IPSJ JOURNAL Abstract Vol.27 No.03 011. An Optimum Real-Time Systolic PolynomialDivision AIgorithm. UMEO HIROSHI ?1 , SUGIOKA TOSHIYUKI ?2. http://www.ipsj.or.jp/members/Journal/Eng/2703/article011.html

Last Update¡§Thu May 24 14:41:06 2001 IPSJ JOURNAL Abstract Vol.27 No.03 - 011 An Optimum Real-Time Systolic Polynomial Division AIgorithm UMEO HIROSHI SUGIOKA TOSHIYUKI Department of Electronic Engineering, FacuIty of Engineerlng, Osaka EIectro-Communication University Kamitani EIectronic Industry ¢¬Index Vol.27 No.03 IPSJ Journal Contents Web Members Service Menu Comments are welcome. Mail to address editt@ips j.or.jp , please.

23. Long Polynomial Division Problem Divide, x 4 2x 3 + 8x - 14 x 2 - 3. Set up the long division. Noticethe 0's put in as place holders for missing powers of x. x 2 =, x 4 x 2, . http://www.sci.wsu.edu/~kentler/Fall97_101/Chapter5/lpd_3.html

Problem: Divide x x Set up the long division. Notice the 's put in as place holders for missing powers of x. x x x Choose x since x matches x x = x (x Subtract x from x Result is x Choose since matches -2x = -2x(x Subtract -2x + 6x from Result is x Choose since matches 3x = 3(x Subtract 3x - 9 from Result is is the remainder. Answer: x x x x Notes x + 8x - 14 = (x - 3)(x Dividend x Divisor x Quotient x Remainder

24. Long Polynomial Division Problem Divide, x 2 + 5x + 9 x + 2. Set up the long division. x =,x 2 x, . Choose x since x · x matches x 2 . x 2 + 2x = x(x + 2). 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

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

26. 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

27. Software: Polynomial Division polynomial division. Home » Software Information Name, polynomial division(Download). Type, Software. Calculator, TI85. Version, 1.1. http://endeavor.ticalc.org/objects/96.html

Polynomial Division Home Name Polynomial Division (Download) Type Software Calculator TI-85 Version Added Friday, May 28, 1999 Description Does polynominal divisoin. Located In Folder(s) Home Education Mathematics Polynomials Contributer(s) Ben Author David Strauss

28. Software: Polynomial Division polynomial division. Home » Software Information Name, polynomial division(Download). Type, Software. Calculator, TI85. Version, 1.0. http://endeavor.ticalc.org/objects/340.html

Polynomial Division Home Name Polynomial Division (Download) Type Software Calculator TI-85 Version Added Saturday, March 04, 2000 Located In Folder(s) Home Education Mathematics Polynomials Contributer(s) Wally Hecht Author David Strauss

29. 4.3 Polynomial Division 4.3 polynomial division Topics polynomial division is a lot like integer division! Polynomialdivision is a lot like integer division! Integer division http://www.austin.cc.tx.us/~powens/ ColAlg/Html/04-3/04-3.html

30. Polynomial Division polynomial division. Review of Long Division. Example Use long division to calculate.495/12. and will write the steps for this process without using any numbers. 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

31. Polynomial Division polynomial division. I. Return Midterm II. II. Homework. II. Reviewof Long Division. The class will use long division to find. 495/12. http://www.ltcconline.net/greenl/courses/103a/premid3/polydiv.htm

Polynomial Division I. Return Midterm II II. Homework II. Review of Long Division The class will use long division to find and will write how to divide without using any numbers. We see that we follow the steps: 1) Write it in long division form. 2) Determine what we need to multiply the quotient by to get the first term. 3) Place that number on top of the long division sign. 4) Multiply that number by the quotient and place the product below. 5) Subtract 6) Repeat the process until the degree of the difference is smaller than the degree of the quotient. 7) Write as sum of the top numbers + remainder/quotient. P(x)/D(x) = Q(x) + R(x)/D(x) (A nonsintactical version of a computer program) do 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; We will do the following examples: A) (3x + 5x + 7)/(x + 1) B) (2x + x - 1)/(x III. 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 P(x)/(x - r):

32. IOLTW: Abstract: Fast Configurable Polynomial Division For Error Control Coding Fast Configurable polynomial division for Error Control Coding Applications. Itis based on a pipeline structure for the polynomial division. http://www.computer.org/proceedings/ioltw/1290/12900158abs.htm

Fast Configurable Polynomial Division for Error Control Coding Applications Fabrice Monteiro, Abbas Dandache, and Bernard Lepley University of Metz Abstract: The motivation for this paper is the need for high levels of reability in modern telecommunication systems requiring very high data transmission rates. The search for technologicaly independent solutions, easy to implement on low cost and popular devices such as FPGA is an important issue. In this paper, we present a method to improve effectively the speed performance of the polynomial division performed in most error detecting and error correcting circuits. It is based on a pipeline structure for the polynomial division. Furthermore, the proposed solution is fully configurable, both from the static and the dynamic points of view. At synthesis stage, the parallelism level (size of the pipeline structure) and the maximal size of the polynomial divisor must both be chosen. Afterwards, the actual divisor can be chosen and changed while the circuit is running. The architecture proved to be very effective, as data rates up to 2.5 Gbits/s have been reached. Proceedings of the Seventh International On-Line Testing Workshop

33. Polynomials:  Dividing Lesson You do the same thing with polynomial division. Since the remainderis –7 and since the divisor is 3x + 1, then turn this into http://www.purplemath.com/modules/polydiv.htm

Dividing Polynomials Lessons Home There are two cases for dividing polynomials: either the "division" is really just simplification and reduction of a fraction, or else you need to do long division Simplification / Reduction Simplify This is just a simplification problem, because there is only one term in the polynomial you're dividing by. And, in this case, there is a common factor in the numerator (top) and denominator (bottom), so it's easy to reduce this fraction. There are two ways of proceeding. I can split the division into two fractions, each with only one term on top, and then reduce: ...or I can factor out the common factor from the top and bottom, and then cancel off: Either way, the answer is the same: x Simplify Again, I can solve this in either of two ways: by splitting up the sum and simplifying each fraction separately: ...or by taking the common factor out front and canceling it off: Either way, the answer is the same:

34. Synthetic Division Lesson Synthetic division is a shorthand, or shortcut, method of polynomial division inthe special case of dividing by a linear factor (and ONLY works in this case). http://www.purplemath.com/modules/synthdiv.htm

Synthetic Division Lessons Home Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor (and ONLY works in this case). It is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials. If you are given, say, y x , you can factor this as y x x Then you can find the zeroes of y by setting each factor equal to zero and solving. You will find that x and x are the two zeroes of y You can, however, also work backwards from the zeroes to find the originating polynomial. For instance, if you are given that x and x are zeroes of a quadratic, then you know that x , so x is a factor, and x , so x is a factor. Therefore, you know that the quadratic must be of the form y a x x (The extra number " a " in that last sentence is in there because you don't know, when working backwards from the zeroes, which quadratic you're working toward. For any non-zero value of " a ", your quadratic will still have the same zeroes. This is a technical point; as long as you see the relationship between the zeroes and the factors, that's all you really need to know for this lesson.)

35. Homework: Partial Fractions: Polynomial Division [MATH*2080 Online] open this document in MS Word format.Write the following integrand in a format suitablefor the Partial Fractions technique, by performing polynomial division http://www.uoguelph.ca/~seanscot/Web2080/Homework/PartialFractionExamples/PolyDi

Write the following integrand in a format suitable for the Partial Fractions technique, by performing polynomial division: First step is to multiply out the bottom: Now do polynomial division. The whole trick is to always divide the highest-degree term of the divisor (here x ) into the highest degree term of the dividend (initially x Another Way: You can "avoid" polynomial division by algebraically manipulating the top as follows: As before, multiply out the bottom so you know what it looks like in "standard form": Now the trick is to continually work the top into this polynomial plus a lower-degree one. If you start with a higher-powered top like we have, you first take out as high a power of x as necessary to get the same power as the bottom: x -2x = x(x Now add and subtract from that cubic to make it the same as the bottom: = x(x - x² + x² + x - x Rearrange... = x(x - x² + x - 1 + x² - x - 1) = x(x - x² + x - 1) + x(x² - x - 1) This new top can now be manipulated in a similar way, first multiply it out to see the x x(x² - x - 1) = x - x² - x This actually isn't too far away from our bottom, it's simple now:

36. Polynomial- And Binary-Division Polynomial and Binary-Division with equnarray A polynomial division writing inmathmode is possible with \underline{}, \hspace{} and use of eqnarray. http://www.educat.hu-berlin.de/~voss/lyx/equnarray/PolDiv.phtml

Polynomial- and Binary-Division blue: latex preamble red: text in tex(red) with equnarray A polynomial division writing in mathmode is possible with and use of eqnarray. open LyX-mathbox with alt-m-d hit ctrl-enter to produce an eqnarray-environment (two lines with three columns) type in your first line, put the equal-sign in the middle-box start second line with , lyx puts by default the closing parenthesis. for it's the same; try a value for the space and so on ... With alt-M-n you'll get numbering for all lines (eqnarray!), with alt-M-N you can toogle between hide and show for numbering for some lines. A binary division looks a bit different to the polynomial division, but it's possible with eqnarray, too. You can download a LyX-Samplefile which shows all the other important math stuff, too. Or show the dvi-view with package polynom this makes polynomial division very easy! For example gives a nice dvi-view . For more information look at the doc, which comes with the package, which is available at CTAN or part of your local tex-installation! back voss@perce.de

37. PowerPoint Presentation - 9.4 Polynomial Division http://www2.guhsd.net/Algebra2/Alg2Show/Chapter09/9_5FindingRationalZeros.htm

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38. PowerPoint Presentation - 9.4 Polynomial Division http://www2.guhsd.net/Algebra2/Alg2Show/Chapter09/9_5FindingRationalZeros_files/

39. DBLP: Dario Bini 20, Dario Bini, Victor Y. Pan Improved Parallel polynomial division. 19, Dario Bini,Victor Y. Pan Improved Parallel polynomial division and Its Extensions. http://www.informatik.uni-trier.de/~ley/db/indices/a-tree/b/Bini:Dario.html

Dario Bini List of publications from the DBLP Bibliography Server FAQ Ask others: ACM CiteSeer CSB Google ... Home Page (Link generated by HomePageSearch EE Dario Bini, Gianna M. Del Corso Giovanni Manzini Luciano Margara : Inversion of Circulant Matrices over Z m ICALP 1998 EE Dario Bini, Victor Y. Pan : Computing Matrix Eigenvalues and Polynomial Zeros Where the Output is Real. SIAM J. Comput. 27 Dario Bini, Luca Gemignani : Erratum: Fast Parallel Computation of the Polynomial Remainder Sequence via Bezout and Hankel Matrices. SIAM J. Comput. 25 Dario Bini, Luca Gemignani : Fast Parallel Computation of the Polynomial Remainder Sequence Via Bezout and Hankel Matrices. SIAM J. Comput. 24 EE Dario Bini, Victor Y. Pan : Parallel Computations with Toeplitz-like and Hankel-like Matrices. ISSAC 1993 Dario Bini, Victor Y. Pan : Improved Parallel Polynomial Division. SIAM J. Comput. 22 Dario Bini, Victor Y. Pan : Improved Parallel Polynomial Division and Its Extensions. FOCS 1992 Dario Bini, Luca Gemignani : On the Complexity of Polynomial Zeros. SIAM J. Comput. 21

40. DBLP: Victor Y. Pan 46, Dario Bini, Victor Y. Pan Improved Parallel polynomial division. 44, Dario Bini,Victor Y. Pan Improved Parallel polynomial division and Its Extensions. http://www.informatik.uni-trier.de/~ley/db/indices/a-tree/p/Pan:Victor_Y=.html

Victor Y. Pan List of publications from the DBLP Bibliography Server FAQ Ask others: ACM CiteSeer CSB Google ... EE Victor Y. Pan: Randomized Acceleration of Fundamental Matrix Computations. STACS 2002 EE Ioannis Z. Emiris , Victor Y. Pan: Symbolic and Numeric Methods for Exploiting Structure in Constructing Resultant Matrices. JSC 33 EE Victor Y. Pan: Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding. JSC 33 EE Victor Y. Pan: Univariate polynomials: nearly optimal algorithms for factorization and rootfinding. ISSAC 2001 EE Victor Y. Pan, Yanqiang Yu : Certification of Numerical Computation of the Sign of the Determinant of a Matrix. Algorithmica 30 EE Keqin Li , Victor Y. Pan: Parallel Matrix Multiplication on a Linear Array with a Reconfigurable Pipelined Bus System. IEEE Transactions on Computers 50 Victor Y. Pan: Computation of Approximate Polynomial GCDs and an Extension. Information and Computation 167 EE Victor Y. Pan: Matrix structure, polynomial arithmetic, and erasure-resilient encoding/decoding. ISSAC 2000 EE Victor Y. Pan: A Homotopic Residual Correction Process.

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