41. Module 1 -- Polynomial Division Instructional Unit Polynomial and Rational Functions. DayOne. 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

42. Polynomial Representation polynomial division. polynomial division. polynomial division. polynomial division.polynomial division. polynomial division. polynomial division. polynomial division. http://www-users.aston.ac.uk/~blowkj/Internetworks/crc/

43. Polynomial Division polynomial division. x9+x5+x4+1. x3+1. The method is thatof long division which we first arrange in the normal way http://www-users.aston.ac.uk/~blowkj/Internetworks/crc/tsld002.htm

44. Lecture 23: Polynomial Division Lecture 23 polynomial division. List of Lectures Math 1100 Index Assignment. Assignments during third test period. These are 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=8 12. Section 5.3 ; page 421; problems 2n+1, for n = 0,...,25 and 6n+1, for n=9 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.

45. NRICH Mathematics Enrichment Club (1890.html) Maclaurin series for tan(x), and polynomial division By Anonymous onFriday, January 26, 2001 1246 pm Hi there, I was fiddling http://www.nrich.maths.org.uk/askedNRICH/edited/1890.html

Asked NRICH NRICH Prime NRICH Club ... Get Printable Page April 03 Magazine Site Update News Events Problems Solutions ... Games Archive Problems Solutions Articles Inspirations ... Interactivities Web board Ask NRICH Asked NRICH NRICH Club Register Tough Nuts About Help! ... Where is NRICH? Associated Projects Maths Thesaurus MOTIVATE EuroMaths Millennium Maths ... Project Display maths using fonts images Help Back Issues Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News Maclaurin series for tan(x), and polynomial division By Anonymous on Friday, January 26, 2001 - 12:46 pm Hi there, I was fiddling with the Maclaurin Series, or the Power Series, which ever you like to call it. .... and on and on Thanks for your help in advance. By Kerwin Hui (Kwkh2) on Friday, January 26, 2001 - 02:04 pm Upon dividing the two series, you get

46. NRICH Mathematics Enrichment Club (2388.html) Remainder Theorem and polynomial division By Anonymous on Tuesday, May 1,2001 0921 pm Here's a web page introducing polynomial division Brad. http://www.nrich.maths.org.uk/askedNRICH/edited/2388.html

Asked NRICH NRICH Prime NRICH Club ... Get Printable Page April 03 Magazine Site Update News Events Problems Solutions ... Games Archive Problems Solutions Articles Inspirations ... Interactivities Web board Ask NRICH Asked NRICH NRICH Club Register Tough Nuts About Help! ... Where is NRICH? Associated Projects Maths Thesaurus MOTIVATE EuroMaths Millennium Maths ... Project Display maths using fonts images Help Back Issues Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News Remainder Theorem and polynomial division By Anonymous on Tuesday, May 1, 2001 - 09:21 pm What does this question mean? Can someone please explain for me. Thank you By Kerwin Hui (Kwkh2) on Tuesday, May 1, 2001 - 10:58 pm This means that you have to prove Kerwin By Anonymous on Thursday, May 3, 2001 - 12:04 am Thanks.

47. The Great CRC Mystery: LISTING ONE The Great CRC Mystery LISTING ONE. polynomial division Binary Trace.(Reconstructed by hand from 10year-old listings. This is Turbo http://www.ciphersbyritter.com/ARTS/CRCLIST1.HTM

The Great CRC Mystery: LISTING ONE Polynomial Division Binary Trace Terry Ritter , his current address , and his top page Last updated:

48. Untitled Fast parallel polynomial division via reduction to triangular Toeplitzmatrix inversion and to polynomial inversion modulo a power. http://www.dm.unipi.it/pages/bini/public_html/pub77-89.html

Su alcune condizioni di monotonia per matrici a blocchi. Calcolo 14, 133-141, 1977. Su alcune questioni di complessita' computazionale numerica. Boll. U.M.I. 15-A, 327-351, 1978 (with M. Capovani). Lower bounds of the complexity of linear algebras. Information Processing Letters 46-47, 1979 (with M. Capovani). Approximate solution for the bilinear form computational problem. SIAM J. Comput., 692-697, 1980 (with G. Lotti, F. Romani). Relation between exact and approximate bilinear algorithms. Applications. Calcolo 17, 87-97, 1980. Stability of fast algorithms for matrix multiplications. Numerische Mathematik, 36, 63-72, 1980 (with G. Lotti). Reply to the paper "The numerical instability of Bini's algorithm". Information Processing Letters, 14, 144-145, 1982. Spectral and computational properties of symmetric band Toeplitz matrices. Linear Algebra Appl. 52, 99-126, 1983 (with M. Capovani). Fast parallel and sequential computations and spectral properties concerning band Toeplitz matrices. Calcolo 20,177-189, 1983 (with M. Capovani). On commutativity and approximation. Theoretical Computer Science, 28, 135-150, 1984.

49. Deconv (Signal Processing Toolbox) q,r = deconv(b,a) deconvolves vector a out of vector b , using long division....... Deconvolution and polynomial division. Syntax q,r = deconv(b,a). http://www.csb.yale.edu/userguides/datamanip/matlab/help/toolbox/signal/deconv.h

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.

50. Deconv (MATLAB Function Reference) q,r = deconv(v,u) deconvolves vector u out of vector v , using long division....... Deconvolution and polynomial division. Syntax q,r = deconv(v,u). http://www.csb.yale.edu/userguides/datamanip/matlab/help/techdoc/ref/deconv.html

MATLAB Function Reference Go to function: Search Help Desk deconv Examples See Also 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 convmtx , and filter in the Signal Processing Toolbox, and: conv Convolution and polynomial multiplication residue Convert between partial fraction expansion and polynomial coefficients Previous Help Desk Next

51. Solutions To Integration Using A Power Substitution so that. and. . Substitute into the original problem, replacing all formsof , getting. (Use polynomial division.). . (Use polynomial division.). . http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/powersubsoldirectory/PowerSu

Next: About this document ... SOLUTIONS TO INTEGRATION USING A POWER SUBSTITUTION SOLUTION 1 Integrate . Use the power substitution so that and Substitute into the original problem, replacing all forms of , getting (Use polynomial division.) Click HERE to return to the list of problems. SOLUTION 2 Integrate . Use the power substitution so that and Substitute into the original problem, replacing all forms of , getting (Use polynomial division.) Click HERE to return to the list of problems. SOLUTION 3 Integrate . Use the power substitution so that and Substitute into the original problem, replacing all forms of , getting (Use polynomial division.) Click HERE to return to the list of problems. SOLUTION 4 Integrate . Use the power substitution so that and Substitute into the original problem, replacing all forms of , getting Click HERE to return to the list of problems. SOLUTION 5 Integrate . Use the power substitution so that and Substitute into the original problem, replacing all forms of , getting Click HERE to return to the list of problems.

52. Solutions To Integration By Partial Fractions SOLUTION 4 Integrate . Because the degree of the numerator is not less thanthe degree of the denominator, we must first do polynomial division. http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/partialfracsoldirectory/Part

Next: About this document ... SOLUTIONS TO INTEGRATION BY PARTIAL FRACTIONS SOLUTION 1 Integrate . Factor and decompose into partial fractions, getting (After getting a common denominator, adding fractions, and equating numerators, it follows that let let (Recall that Click HERE to return to the list of problems. SOLUTION 2 Integrate . Factor and decompose into partial fractions, getting (After getting a common denominator, adding fractions, and equating numerators, it follows that let let Click HERE to return to the list of problems. SOLUTION 3 Integrate . Factor and decompose into partial fractions, getting (After getting a common denominator, adding fractions, and equating numerators, it follows that let let Click HERE to return to the list of problems. SOLUTION 4 Integrate . Because the degree of the numerator is not less than the degree of the denominator, we must first do polynomial division. Then factor and decompose into partial fractions, getting (After getting a common denominator, adding fractions, and equating numerators, it follows that let let (Recall that Click HERE to return to the list of problems.

53. Members.lycos.co.uk/rfam/9x50g/polydiv.txt To be able to open this file in Xchange, rename the extension to .CTF. TITLEpolynomial division AUTHOR Roy FA Maclean EMAIL rfam@lycosmail.com WEB http://members.lycos.co.uk/rfam/9x50g/polydiv.txt

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

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 The MathWorks, Inc. Trademarks Privacy Policy

55. Deconv (Signal Processing Toolbox) . q,r = deconv(b,a) deconvolves vector a out......deconv Deconvolution and polynomial division. Syntax q,r = deconv(b,a). http://www.mathworks.com/access/helpdesk_r12p1/help/toolbox/signal/deconv.shtml

Signal Processing Toolbox deconv 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 The deconv function is part of the standard MATLAB language. Example The convolution of a and b is c conv(a,b) c = Use deconv to divide b back out. [q,r] deconv(c,a) q = r = Algorithm This function calls filter to compute the deconvolution as the impulse response of an IIR filter. See Also conv filter residuez decimate demod The MathWorks, Inc. Trademarks Privacy Policy

56. Polynomial Long Division, Answer 1 Exercise 1. Use long polynomial division to rewrite. Answer. The answeris Solution. Divide the leading term of the numerator polynomial http://www.math.utep.edu/sosmath/algebra/factor/fac01a1/fac01a1.html

Exercise 1. Use long polynomial division to rewrite Answer. The answer is: Solution. Divide the leading term of the numerator polynomial by the leading term of the divisor: Multiply "back": , and subtract: Divide the leading term of the bottom polynomial by the leading term of the divisor: Multiply back: , and subtract: You're done! The answer is: [Back] [Exercises] [Next] [Algebra] ... S.O.S MATHematics home page Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard Helmut Knaust Fri Jun 6 13:11:33 MDT 1997 Contact us Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA users online during the last hour

57. Real Roots Of Polynomial Functions Again, consider our basic definition of polynomial division Dividend f(x). Divisorh(x). This we will call the remainder theorem for polynomial division. http://id.mind.net/~zona/mmts/functionInstitute/polynomialFunctions/roots/realRo

Real Roots of Polynomial Functions Roots Polynomial Functions Function Institute Contents ... Home Definition of terms and symbols when dividing polynomials: Dividend: f(x) Divisor: h(x) Quotient: q(x) Remainder: r(x) If any of these are constants, for example if r(x) is constant, as in: r(x) = 5 or: r(x) = a then variable, rather than function, notation may be used for that value, as in: r = 5 or: r = a When f(x) is divided by h(x), the result is the value of q(x) plus r(x), as in: f(x)/h(x) = q(x) + r(x) This can also be written as: f(x) = h(x)q(x) + r(x) The remainder, r(x), will either be equal to 0, or it will be less in degree than the degree of the divisor, h(x). If h(x) has a degree of 1, then the degree of the remainder must be 0. That is, the remainder must be a constant, as in: r(x) = cx = c Under these conditions variable notation is fine, as in: r = c Therefore, if f(x) is divided by the linear polynomial (x - c), the remainder is a constant, r. Again, consider our basic definition of polynomial division: Dividend: f(x) Divisor: h(x) Quotient: q(x) Remainder: r(x) f(x) = h(x)q(x) + r(x) Make the divisor, h(x), equal to the zero degree polynomial (x - c). This will create a remainder, r, that is a constant.

58. Synthetic Division Consider dividing f(x) = 4x 3 3x 2 + x - 4 by (x - 2). Standard polynomial divisionwould look like this 4x 2 + 5x + 11 - x - 2 )4x 3 http://id.mind.net/~zona/mmts/functionInstitute/polynomialFunctions/roots/synthe

Synthetic Division Polynomial Functions Function Institute Contents Index ... Home Consider this polynomial function: f(x) = 4x + x - 4 Suppose that we evaluate it at an input of x = 2, like this: f(2) = 4(2 f(2) = 32 - 12 = 2 - 4 f(2) = 18 In this process we raised the input to a power, as in 2 Let us see that there is a way to evaluate this polynomial function using only multiplication and addition. Start with the original polynomial and factor out an x. So, this: + x - 4 Becomes: x(4x Factor out another x from the parenthesized expression: x(x(4x - 3) + 1) - 4 Now, imagine that you evaluate f(x) at x = 2. Begin with the inner most expression. Place a 2 for the input value of x, as in: x(x(4(2) - 3) + 1) - 4 Now you would multiply 2 (the input) by 4 (the original coefficient of x ) and then add -3 (the original coefficient of x ). This would evaluate to 5. The expression now looks like: x(x(5) + 1) - 4 Place a 2 for the next input value of x, as in: x(2(5) + 1) - 4 Now you would multiply 2 (the input) by 5 and then add 1 (the original coefficient of x). This would evaluate to 11. The expression now looks like: x(11) - 4 Place a 2 for the last input value of x, as in:

59. Verilog Exercises XORed). Figure 3.1 Figure 2.16 from Data Networks by Bertsekas and Gallagerto remind you how the polynomial division is computed. If http://www.cse.ucsc.edu/~karplus/222/w00/verilog-hw/verilog-hw.html

Verilog Exercises 1. Full adder Write two Verilog modules for a full adder. One should be a behavioral description using continuous assignment: The other should be a gate-level description, using built-in gates, transistors, or user-defined primitives. Build a testing module that exhaustively tests the full-adder modules, making sure that they agree for all possible inputs. If you use a static adder design, you do not need to worry about clocking discipline, but if you use a dynamic design, you need to test the clocking. 2. 64-bit adder Write two Verilog descriptions for a 64-bit adder. One should be a behavioral description using the standard addition operator a+b ; the other should be a structural description of a ripple-carry adder built using one of the full-adder modules from Problem Build a testing module to run through a standard set of adder tests (that you found for a previous homework assignment). Deliberately introduce an error (mis-wiring the modules, shorting two signals together, setting one input of one module always high or always low, or substituting a faulty module for one of the correct modules), and show how the test catches the error. 3. Sequential logic

60. Cyber Calc I They also discuss polynomial division, homework guidelines, math study skills,and links to other internet resources (including self tests). http://mrs.umn.edu/~mcquarrb/CyberCalcI/CCI.html

Cyber Calculus I Instructor Contact Course Information The Math Room Resources ... tests Instructor Contact My main webpage Questions? Comments? email me My current Schedule , which is the most up to date listing of where I am. Course Information Calculus Survival Guide . The things you need to know about calculus before your first lecture. Common Mathematical Errors . The mathematical errors that can hinder your study of calculus. The Course Guide A Study Guide and some Review Notes for Chapter 1 Links to Mathematica resources on my teaching page. The groups for presentation problems for Fall 2002. Homework related material (solutions, sample solutions from lectures). Test related material (practice questions, solutions, review). The Handout on Curves and Surfaces in 3D is available. The Math Room Fall 2002 Schedule 7-10pm Monday, Tuesday, Wednesday, Thursday. Resources and Other Fun Things The History of Calculus is an interesting and fun read. The basic motivation for calculus were mainly four problems: 1) determining a relationship between acceleration, velocity, position; 2) finding the tangent line to a curve (and defining what the heck a tangent is!); 3) finding maximum and minimum values of a function; and 4) finding lengths of curves. An excellent resource for algebra is the PurpleMath page at http://www.purplemath.com/index.htm

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