61. Polynomials 3. polynomial division polynomial division is a lot like division of whole numberswhere you get a quotient and a remainder, eg 13 divided by 5 gives quotient http://www.math.uri.edu/~pakula/111web_f02/polynomials.htm

[UnitView] [Syllabus] Unit Guide for Polynomials Review adding, multiplying and factoring polynomials. Sections R3 and R4 are intended as a review of algebra-this should not be the first time you are seeing these things. Some special points: Multiplying polynomials: Be sure that you can multiply things like (3 x x x + x - x ). The "FOIL" method can't be used for this, so you should know how to perform this calculation by careful use of the distributive property. Special patterns: The formulas (A+B) = A + 2AB + B and (A-B)(A+B) = A - B appear very frequently and you should recognize these patterns. Factoring: Make sure you know how to factor perfect squares, difference of two squares, and situations where you must pull out common factors. You don't need to know how to factor the difference of two cubes. A very simple but common factoring situation that you should know is AB + A = A(B+1). Polynomial Functions. The main points to learn about polynomials are a. For large values of x, a polynomial behaves like the term of highest degree. This is reflected in the appearance of the graph in a very big viewing window. (The "Leading Term Test")

62. ThinkQuest : Library : Go Forth & Multiply: A Mathematics Adventure Long Division. This is a method used in arithmetic and polynomial division.First, we ll take a look at an example of division in arithmetic. http://library.thinkquest.org/C0110248/algebra/allongdiv.htm

Index Education Want to learn more about the world of mathematics? Then go forth, and enter the wildest math adventure you've ever been! Learn new math concepts and refresh your knowledge for those you've already known. Understand how the formulae you use were derived from. Or, you can take a step back into the past and read about how mathematics and its concepts originated. Go forth and multiply! Visit Site 2001 ThinkQuest Internet Challenge Awards Achievement Award Students Teow Lim Raffles Junior College, Singapore, Singapore Vee San Coaches Poh Kheng Pioneer Junior College, Singapore, Singapore Jee Wah Want to build a ThinkQuest site? The ThinkQuest site above is one of thousands of educational web sites built by students from around the world. Click here to learn how you can build a ThinkQuest site. Privacy Policy

63. 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.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

64. Cyclic Redundancy Check Polynomials Tutorial Press the check symbol beside polynomial division. A window will appearwith 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.

65. File Verification Using CRC CRC calculations are done using a technique with the formidable nameof polynomial division . A block of data, regardless os how http://www.dogma.net/markn/articles/crcman/crcman.htm

File Verification Using CRC by Mark Nelson Dr. Dobb's Journal May, 1992 This page contains my original text and figures for the article that appeared in the May 1992 DDJ. I haven't broken it up into pages, so loading the entire thing might take some time. File Verification Using the CRC by Mark Nelson Recently I have found myself thinking a lot about file verification. By file verification, I mean the process of determining whether a file on my computer has been modified unexpectedly. Whether it happened through hardware failure, program error, or malicious tampering, I like to know when a file has had its contents altered. Likewise, I would like a convenient way to check the integrity of a file to verify that it hasn't been changed. The problem of file integrity has been on my mind because of several nearly simultaneous incidents. First of all, I recently ran dozens of relatively untested programs through my home systems while I was judging the Dr. Dobb's Data Compression Contest. At least two of these programs caused inadvertent damage to the file systems on my computer, one under UNIX and one under MS-DOS. In both cases, I was able to spot a lot of the damage, but after I restored the data that looked bad, I was left feeling unsure about the rest of my system. Had other files been damaged in more subtle ways? I suddenly felt as though I couldn't trust my system. An even more alarming incident occurred a couple of weeks later. A programmer who supplies us with a product for resale called us up and casually mentioned that his office had been infested with the notorious "Stoned" virus. Had we by any chance noticed anything funny in oursystems? We see funny things on our systems on an hourly basis, sosuddenly we were once again in the position of not trusting any of the files on our computers. (Fortunately this turned out to be a false alarm).

66. Polynomial Expressions The polynomial division and remainder are done using the defined monomial orderin the base ring. Parentheses are used to enforce groupings in an expression. http://www.math.columbia.edu/online/Macaulay1-rel0994-html/node22.html

Next: Integer lists Up: Input Syntax Previous: Integer Expressions Polynomial expressions The syntax for polynomial expressions is similar to integer expressions. The legal operators for polynomials, in order of increasing precedence, are given in the following table. (polynomial division), (polynomial remainder) unary operators (both are exponentiation operators) The polynomial division and remainder are done using the defined monomial order in the base ring. Parentheses are used to enforce groupings in an expression. The following remarks describe certain aspects of polynomial expressions. Polynomials in Macaulay are output in an abbreviated notation: each polynomial is displayed as a number of monomials separated by ``+'' or ``-''. Each monomial is preceeded by its coefficient and each indeterminate is followed by its degree with no intervening blanks. Rational number coefficients are represented by x y . A polynomial which extends over a line boundary is displayed by using as the continuation character. If you wish to have polynomials output in Mathematica format, use the prmat command.

67. P04-Factor Remainder Theorems.html f(r) = R the remainder when (xr) is divided into f(x) is R It makes a connectionbetween the remainder of a polynomial division and evaluating a polynomial. http://www.mapleapps.com/powertools/precalc/html/P04-Factor Remainder Theorems.h

P04-Factor Remainder Theorems.mws 0. Code 1. The Remainder Theorem 2. Evaluating a Polynomial using Synthetic Division ... 4. The Factor Theorem Exposition and application of the factor and remainder theorems. [Directions : Execute the Code Resource section first. Although there will be no output immediately, these definitions are used later in this worksheet.] 0. Code restart; with(plots): Warning, the name changecoords has been redefined 1. The Remainder Theorem One version of the Remainder Theorem is this : For a polynomial f(x), the following are equivalent : - f(r) = R - the remainder when (x-r) is divided into f(x) is R It makes a connection between the remainder of a polynomial division and evaluating a polynomial. Let's look at some examples. Example 1.1 : Let's create a polynomial and pick a number r. Then we will evaluate both of the expressions above and note that the two are the same. r := 4; f(r); LongDiv( f(x), r); Example 1.2 : Here is another example. r := -1; f(r); LongDiv( f(x), r);

68. EE4253 Polynomial The first test division by (x+1) yields a zero remainder, and a factor is found.This polynomial is not prime. Answer polynomial division in Hardware. http://www.ee.unb.ca/tervo/ee4253/poly.htm

EE4253 Digital Communications Manipulation of Binary Messages as Polynomials Manipulation of long binary values requires some special techniques. The polynomials notation lends itself to computation in hardware using only shift registers and exclusive-OR (XOR) gates. Once the mathematics of polynomials has been defined, the concept of prime polynomials can be introduced. Introduction to Polynomials For example, can be written as x x x x x x and simplified as: x + x + x + 1 The order of a polynomial is the power of the highest non-zero coefficient. The above example shows a polynomial of "order 5". 2. Polynomials can be manipulated using the usual arithmetic rules, and these properites (closure, associative, commutative, etc) define a field Example 1: 110011 x 10 = 1100110 can be written as: (x + x + x + 1) (x) = x + x + x + x 3. Polynomials always use modulus 2 arithmetic . This is equivalent to the exclusive-OR operation, as shown below. Example 2: 11 x 11 can be computed as: 11 x 11 11 + 110 101 Note modulus 2 addition and as a polynomial: (x + 1)(x + 1) = x + x + x + 1 = x Note that x + x = when simplifying this result.

69. DBLP: Dario Bini ISSAC 1993 193200. 20, Dario Bini, Victor Y. Pan Improved Parallelpolynomial division. SIAM J. Comput. 22(3) 617-626 (1993). 1992. 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 Coauthor Index - Ask others: ACM DL ACM Guide CiteSeer CSB ... 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 Dario Bini, Luca Gemignani Victor Y. Pan

70. Synthetic Division Suggested Use Study algebra of polynomial division. Topics college algebra,polynomials, gcd, synthetic division, symbolic algebra. Number of Pages 1. http://www.mathwright.com/book_pgs/book055.html

Been away for a while? Check out our new building by clicking the picture on the right! This WorkBook requires Mathwright Library Player 2000 to read it. To download the book, press the button on the left. A self-extracting file will be downloaded. Either save it to disk and execute it later, or simply select "Open it" from the popup dialog. This places the book, along with its documentation, on the Start, Programs, Mathwright Library menu, so that you may read it whenever you like. Size: 131 KB Find similar WorkBooks in the Rooms below: Categories: Home Study Tools Math and Computers Subjects: Algebra College Algebra Precalculus Factorization ... Rational Functions Title: Synthetic Division Book Description: In this command-line WorkBook, students may explore synthetic division of polynomials with rational coefficients. There is a command called Synthetic that returns the quotient of one polynomial by another (with rational coefficients) together with the remainder part. There are also several programs that support exploration. These are: Divide(num,den) returns the same result that synthetic would. This result can then be used by another command (for example, to define a function and draw its graph). Pquotient and Premainder returns the results (quotient and remainder) from the Euclidean algorithm. Finally, GCD returns the greatest common divisor of two rational polynomials.

71. Matlab Manual Page: Deconv . q,r = deconv(b,a) deconvolves vector......deconv. Purpose. Deconvolution and polynomial division. Synopsis. q,r= deconv(b,a) 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

72. Page7 We put the polynomial division algorithms in different classes PolynomialDivisor(one variable), PolynomialPseudoDivisor (one or many variables, but wrt a http://mate.dm.uba.ar/~caniglia/Pablo/

Mathematical Objects This is the change set of this work The object of this project is to solve the Implicitation Problem, i.e. given an algebraic variety described by a rational parametric equation, find a system of polynomial equations which define it in implicit form. Description We created classes to model multivariate polynomials: MonomialLiteral, Monomial y Polynomial. MonomialLiteral models the literal part of the monomials, i.e., products of indeterminate powers. The Monomials know a coefficient and a literal. Finally, each Polynomial represets a polynomial and stores the collection of not null monomials (Monomials). We put the polynomial division algorithms in different classes: PolynomialDivisor (one variable), PolynomialPseudoDivisor (one or many variables, but w.r.t. a variable), MultiPolynomialDivisor (many variables with fixed monomial ordering). We also have objects which are rational functions; we created the RationalFunction class, this class knows a numerator and a denominator. Specification and Implementation Class Ideal You can create ideals from a generating set Class protocol instance creation generator: anIdealGenerator Also, we can ask an ideal for a generating set accessing generators The ideals allow arithmetic operations

73. ThinkQuest : Library : Seeing Is Believing polynomial Functions. Synthetic division ex. divide 3x3 4x + 3 by (x + 1), using synthetic division. 3x- 4x + 3 is equal to 3x+ 0x- 4x + 3 http://library.thinkquest.org/10030/8syndiv.htm

Index Education Seeing is Believing Need a primer on math, science, technology, education, or art, or just looking for a new Internet search engine? This catch-all site covers them all. Maybe you're doing your homework and need to quickly look up a basic term? Here you'll find a brief yet concise reference source for all these topics. And if you're still not sure what's here, use the search feature to scan the entire site for your topic. Visit Site 1997 ThinkQuest Internet Challenge Languages English Students Suranthe H Oakhill College, Sydney, Australia Peter Oakhill College, Castle Hill, Sydney, Australia Coaches Tina Oakhill College, Castle Hill, Sydney, Australia Tina Oakhill College, Castle Hill, Sydney, Australia Want to build a ThinkQuest site? The ThinkQuest site above is one of thousands of educational web sites built by students from around the world. Click here to learn how you can build a ThinkQuest site. Privacy Policy

74. Learn.co.uk - Learning Resources For The National Curriculum, Online Lessons, GC 2 + 3 2 leading coefficient of product = 3 1 x 3 3 constant term of product = 12 - 4 x 3 division of a polynomial expression by a binomial The binomial http://www.learn.co.uk/default.asp?WCI=Unit&WCU=27254

75. Polynomial Long Division - Wikipedia, The Free Encyclopedia polynomial long division. From Wikipedia, the free encyclopedia. A similar shortcutmethod exists for dividing by a quadratic or higher degree monic polynomial. http://en.wikipedia.org/wiki/Polynomial_long_division

Polynomial long division From Wikipedia, the free encyclopedia. Server will be down for maintenance on 2004-06-11 from about 18:00 to 18:30 UTC. In algebra polynomial long division is an algorithm similar to long division for dividing a polynomial into another polynomial of a larger degree. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones. For any polynomials f(x) and g(x) g(x) being of lesser degree than f(x) , there exist unique polynomials q(x) and r(x) such that Synthetic division will find the quotient q(x) and remainder r(x) given a dividend f(x) and divisor g(x) . The problem is written down like this: When the problem is written, all the terms with exponents less than the largest one must be written, even if their coefficients are zero. edit Example Find: The problem is written like this (note that the x term is included): 1. Divide the first term of the dividend by the first term of the divisor. Place the result above the bar ( x ÷ x = x 2. Multiply the divisor by the term you just wrote. Write the result under the first two terms of the dividend ( x * (x-3) = x 3. Subtract the second term of the result you just got from the second term of the dividend and write the result under both of them. This can be tricky at times, because of the sign. (

76. Polynomial Long Division - Wikipedia, The Free Encyclopedia polynomial long division. (Redirected from Synthetic division). A similar shortcutmethod exists for dividing by a quadratic or higher degree monic polynomial. http://en.wikipedia.org/wiki/Synthetic_division

Polynomial long division From Wikipedia, the free encyclopedia. (Redirected from Synthetic division Server will be down for maintenance on 2004-06-11 from about 18:00 to 18:30 UTC. In algebra polynomial long division is an algorithm similar to long division for dividing a polynomial into another polynomial of a larger degree. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones. For any polynomials f(x) and g(x) g(x) being of lesser degree than f(x) , there exist unique polynomials q(x) and r(x) such that Synthetic division will find the quotient q(x) and remainder r(x) given a dividend f(x) and divisor g(x) . The problem is written down like this: When the problem is written, all the terms with exponents less than the largest one must be written, even if their coefficients are zero. edit Example Find: The problem is written like this (note that the x term is included): 1. Divide the first term of the dividend by the first term of the divisor. Place the result above the bar ( x ÷ x = x 2. Multiply the divisor by the term you just wrote. Write the result under the first two terms of the dividend (

77. Mathwords: Polynomial Long Division V. W. X. Y. Z. A to Z. polynomial Long division. A method used to divide polynomials.polynomial long division is essentially the same as long division for numbers. http://www.mathwords.com/p/polynomial_long_division.htm

G o o g ... e www mathwords Polynomial Long Division A method used to divide polynomials . Polynomial long division is essentially the same as long division for numbers. This method can be used to write an improper rational expression as the sum of a polynomial and a proper rational expression See also Synthetic division this page updated 7-jun-04

78. Beginning Algebra Tutorial On Division Of Polynomials Divide a polynomial by a polynomial using long division. Introduction. Divide polynomialpolynomial. Using Long division. Step 1 Set up the long division. http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut30_di

(Back to the Beginning Algebra Homepage) Beginning Algebra Tutorial 30: Division of Polynomials Learning Objectives After completing this tutorial, you should be able to: Divide a polynomial by a monomial. Divide a polynomial by a polynomial using long division. Introduction In this tutorial we revisit something that you may not have seen since grade school: long division. I hope that your experiences with long division have been better than Nerwin's in this comic. In this tutorial we are dividing polynomials, but it follows the same steps and thought process as when you apply it numbers. Let's forge ahead. Tutorial Divide Polynomial Monomial Step 1: Use distributive property to write every term of the numerator over the monomial in the denominator. If you need a review on the distributive property, go to Tutorial 8: Properties of Real Numbers. Step 2: Simplify the fractions. If you need a review on simplifying fractions, go to Tutorial 3: Fractions. Example 1 : Divide Step 1: Use distributive property to write every term of the numerator over the monomial in the denominator AND Step 2: Simplify the fractions.

79. Division Of Polynomials Lesson Page, Math A. Steps for Dividing a polynomial by a Monomial. 1. Divide eachterm of the polynomial by the monomial. a) Divide numbers b) Subtract exponents. http://regentsprep.org/Regents/math/divpoly/Ldiv.htm

Lesson Page Math A Steps for Dividing a Polynomial by a Monomial Divide each term of the polynomial by the monomial. a) Divide numbers b) Subtract exponents Keep this in mind * The number of terms in the polynomial equals the number of terms in the answer Numbers do not cancel! A number divided by itself is Remember to write the appropriate sign in between the terms. Example: Schultzkie

80. Division Of A Polynomial By A Monomial division OF A polynomial BY A MONOMIAL. division, like multiplication,may be distributive. division OF A polynomial BY A polynomial. http://www.tpub.com/math1/10g.htm

Click here to make tpub.com your Home Page Division of a polynomial by a monomial tpub.com Updates Back Home Up ... Back Rationalizing denominators Up Introduction to Mathematics and Algebra Next Synthetic Division DIVISION OF A POLYNOMIAL BY A MONOMIAL Division, like multiplication, may be distributive. Consider, for example, the problem 2, which may be solved by adding the numbers within the parentheses and then dividing the total by 2. Thus, Now notice that the problem may also-be solved distributively. CAUTION: Do not confuse problems of the type just described with another type which is similar in appearance but not in final result. For example, in a problem such as 2 the beginner is tempted to divide 2 successively by 4, then 6, and then -2, as follows: Notice that we have canceled the "equals" sign, because 2 + 8 is obviously not equal to 1/2 + 2/6. - 1. The distributive method applies only in those cases in which several different numerators are to be used with the same denominator When literal numbers are present in an expression, the distributive method must be used, as in the following two problems: Quite often this division may be done mentally, and the intermediate steps need not be

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