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Topic: Polynomial long division

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	Polynomial long division -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		This method is entirely reminiscent of the (The operation of division in which the sequence of steps are indicated in detail) long division learned in elementary arithmetic classes -- in fact, that technique is simply a special case of polynomial long division, where x=10.
		In fact, in synthetic division, we disregard the coefficient of the leading power of x in the divisor, since it is assumed to be 1 (other cases will be taken care of later).
		However, it is possible to extend the idea to division by polynomials of nth degree, as long as the dividend has a higher power than its divisor.
	www.absoluteastronomy.com /encyclopedia/p/po/polynomial_long_division.htm (3283 words)

	Polynomial - Open Encyclopedia (Site not responding. Last check: 2007-10-07)
		Because of their simple structure polynomials are very easy to evaluate and are used extensively in numerical analysis for polynomial interpolation or to numerically integrate more complex functions.
		As there is generally no closed formula to calculate the roots of a polynomial of degree 5 and higher root-finding algorithm are studied in numerical analysis to approximate the roots.
		Depending on the degree of the polynomial to be considered, simply checking if the polynomial has linear factors can eliminate several cases, and then resorting to checking divisibility of some other irreducible polynomials, however Eisenstein's criterion can be used to more efficiently determine irreducibility.
	open-encyclopedia.com /Polynomial (1761 words)

	Encyclopedia: Polynomial long division
		In arithmetic, long division is an algorithm for division of two real numbers.
		In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on.
		Algorithms The polynomial remainder theorem in algebra is an application of polynomial long division.
	www.nationmaster.com /encyclopedia/Polynomial-long-division (3859 words)

	Learn more about Polynomial in the online encyclopedia. (Site not responding. Last check: 2007-10-07)
		Note that the polynomials of degree ≤ n are precisely those functions whose (n+1)st derivative is identically zero.
		The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Weierstrass approximation theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial.
		In order to determine function values of polynomials for given values of the variable x, one does not apply the polynomial as a formula directly, but uses the much more efficient Horner scheme instead.
	www.onlineencyclopedia.org /p/po/polynomial.html (1534 words)

	Long division -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		In (The branch of pure mathematics dealing with the theory of numerical calculations) arithmetic, long division is an algorithm for ((botany) taxonomic unit of plants corresponding to a phylum) division of two (Any rational or irrational number) real numbers.
		However, the procedure requires various numbers to be divided by the (The number by which a dividend is divided) divisor: this is simple with single-digit divisors, but becomes harder with larger ones.
		A more generalized version of this method is used for (Click link for more info and facts about dividing polynomials) dividing polynomials (sometimes using a shorthand version called (Click link for more info and facts about synthetic division) synthetic division).
	www.absoluteastronomy.com /encyclopedia/L/Lo/Long_division.htm (375 words)

	eZine - ARTICLES (Site not responding. Last check: 2007-10-07)
		Long division has been especially targeted for de-emphasis, or even elimination from the school curriculum(2).
		The standard arithmetic algorithms, and the long division algorithm in particular, did not appear at all, or were drastically abridged, in all of the elementary school curricula on the government's list.
		The student's experience of carrying long division past the decimal point is his or her first experience with infinite processes of any kind converging.
	www.homeschooltimes.com /ezine/092-01/03.htm (6877 words)

	Long Division (Site not responding. Last check: 2007-10-07)
		There is a long standing consensus among those most knowledgeable in mathematics that standard algorithms(1)of arithmetic should be taught to school children.
		The long division algorithm is the essential tool in establishing that any rational number has a repeating block of digits in its decimal representation.
		There are at least two separate areas where the insights gained from understanding the long division algorithm are crucial, first calculus, and second, applications of polynomials in advanced areas of mathematics and related fields.
	math.stanford.edu /ftp/milgram/long-division/longdivsiondone.htm (6312 words)

	Algebraic Long Division
		Algebraic long division is very similar to traditional long division (which you may have come across earlier in your education).
		If the polynomial/ expression that you are dividing has a term in x missing, add such a term by placing a zero in front of it.
		With algebraic long division, practise makes perfect- the best way to learn how to do them properly is to do loads of examples until you get them right every time.
	www.mathsrevision.net /alevel/pages.php?page=1 (434 words)

	Glencoe Mathematics - Online Study Tools
		The binomial is not a factor of the polynomial.
		The polynomial is of degree greater than 2.
		The polynomial is of degree less than or equal to 2.
	www.glencoe.com /sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-825083-8&chapter=12&lesson=5&headerFile=4&state=nc (57 words)

	PreCalculus
		A rational expression is improper if the degree of the numerator polynomial is greater than or equal to the degree of the denominator polynomial.
		Polynomial long division can be used to write an improper rational expression as the sum of a polynomial and a proper rational expression.
		All of the factors of the each of the polynomials must be present in the factors of the LCM.
	home.att.net /~srschmitt/precalc/precalc-05-01-00.html (355 words)

	Karl's Calculus Tutor - Notes and Basic Algebra Concepts
		With division, you can still do the same thing, but you must be aware that the results only count in the cases when the divisor is not zero.
		A rule of thumb when multiplying polynomials is that the degree of the product is always the sum of the degrees of the two polynomials you multiplied.
		When a 1st degree polynomial can be factored out of a higher degree polynomial, the 1st degree polynomial is said to represent a root of the higher degree polynomial.
	www.karlscalculus.org /notes.html (4161 words)

	spacekitten.org: synthetic division
		I'm sure everyone has, at one point in their life, asked themselves the question, "Is there any way to make polynomial long division even better than it already is?" Wonder no longer, for here I present to you the secret to synthetic division.
		It can be used on polynomial equations of any degree as long as the divisor is of the form x + c where c is a constant.
		Synthetic division is also a fast way to evaluate expressions if you happen to be unable to calculate, say, 185 in your head.
	www.spacekitten.org /2002/07/synthetic-division.html (382 words)

	Synthetic Division
		Now, this method of evaluating the output value of a polynomial can be captured in a type of short-hand notation by positioning the input value and the coefficients of the polynomial in a certain form.
		Notice that the remainder from the standard polynomial long division is the last value calculated in the synthetic division.
		Notice that the coefficients of the quotient from the standard polynomial long division are aligned before the remainder on the last row, under the line, of the synthetic division.
	id.mind.net /~zona/mmts/functionInstitute/polynomialFunctions/roots/syntheticDivision.html (593 words)

	The Remainder and Factor Theorems - Chapter 6 Review: Section 5
		Polynomial long division can be used to divide one polynomial by another when the dividend has a higher degree than the divisor.
		Treat the polynomial furthest down the page like it were the dividend and start again at step 3.
		The numbers under the line are the coefficients of the quotient polynomial, in descending order of degrees.
	webpages.charter.net /thejacowskis/chapter6/section5.html (518 words)

	Polynomial Division and Factoring
		Notice how we used some common divisibility rules (i.e., "even numbers are divisible by 2", "numbers ending in 5 are divisible by 5", etc.) in order to narrow our search to primes that are likely to be factors.
		Polynomial division is very similar to ordinary integer division.
		With integer division, we automatically put the most significant digits on the left and insert 0's where appropriate; with polynomials we must explicitly line up the terms, from most significant (i.e., highest powers) to least significant, inserting 0 terms where necessary to keep like terms aligned.
	campus.northpark.edu /math/PreCalculus/Algebraic/Polynomial/Factoring (2897 words)

	StudyWorks! Online : Dividing Polynomials (Site not responding. Last check: 2007-10-07)
		You can also use long division to solve this type of problem.
		Polynomial long division works just like arithmetic long division except that you line up your columns according to degree.
		Set up a "division box" with the divisor outside at the left and dividend inside the box.
	www.studyworksonline.com /cda/content/article/0,,NAV14-105_SAR1865,00.shtml (334 words)

	Untitled
		The methods of long division and synthetic division are quite similar.
		With long division, the divisor is a monomial and dividend is a polynomial; Long division with polynomial is very similar to long division with whole numbers by using four basic steps: estimate, multiply, subtract and bring down the next term;
		With synthetic division, the divisor is a number(a zero of the polynomial) and the dividend are coefficients of the polynomial.
	www.gomath.com /Questions/p13.html (461 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		Introduction There is a long standing consensus among those most knowledgable in mathematics that standard algorithms of arithmetic should be taught to school children.
		Long division has been especially targeted for de-emphasis, or even elimination from the school curriculum. We hope to make clear why this tendency should be reversed by explaining the importance of the long division algorithm on conceptual grounds.
		Place value and Division Before beginning the discussion of long division a teacher has to be sure that students understand place value.
	math.stanford.edu /ftp/milgram/long-division-try-again.doc (2239 words)

	Polynomial Division and Factoring: Solutions
		Here are some solutions to the Exercises to accompany the section Polynomial Division and Factoring.
		Use polynomial long-division to compute the quotient, q, and remainder, r, after dividing p by d for each of the following pairs of polynomials.
		Repeat this Exercise as often as necessary until you are confident in your ability to divide polynomials.
	campus.northpark.edu /math/PreCalculus/Algebraic/Polynomial/Factoring/Exercises/solutions.html (991 words)

	T3 answers
		where q(x) is a polynomial and where the degree of r is less than the degree of d.
		Some of you thought it was necessary to expand the polynomial.
		Note: There is another trick for reducing the polynomial using long division instead of synthetic division.
	www.lsus.edu /sc/math/rmabry/math121/t3-2001fall-answers/t3-2001fall-answers.html (1065 words)

	Polynomial long division
		In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of lower degree, a generalized version of the familiar arithmatic technique called long division.
		The problem is written down like a regular (non-alegbraic) long division problem:
		This method is entirely reminiscent of the long division learned in elementary arithmatic classes -- in fact, that technique is simply a special case of polynomial long division, where x=10.
	www.sciencedaily.com /encyclopedia/polynomial_long_division (709 words)

	Synthetic Division Lesson - I (Site not responding. Last check: 2007-10-07)
		Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor (and
		In the long division, I divided by the factor
		It is always true that, when you use synthetic division, your answer (in the bottom row) will be of one degree less than what you'd started with, because you have divided out a linear factor.
	www.purplemath.com /modules/synthdiv.htm (762 words)

	GiNaC: normal.cpp Source File
		!b.info(info_flags::rational_polynomial))) 00378 throw(std::invalid_argument("quo: arguments must be polynomials over the rationals")); 00379 00380 // Polynomial long division 00381 ex r = a.
		!b.info(info_flags::rational_polynomial))) 00432 throw(std::invalid_argument("rem: arguments must be polynomials over the rationals")); 00433 00434 // Polynomial long division 00435 ex r = a.
		info(info_flags::rational_polynomial))) 00497 throw(std::invalid_argument("prem: arguments must be polynomials over the rationals")); 00498 00499 // Polynomial long division 00500 ex r = a.
	www.ginac.de /reference/normal_8cpp-source.html (2979 words)

	College Algebra Tutorial on Long Division
		In this tutorial we revisit something that you may not have seen since grade school: long division.
		In this tutorial we are dividing polynomials, but it follows the same steps and thought process as when you apply it numbers.
		Note that the "scratch work" that you see at the right of the long division shows you how that step is filled in.
	www.wtamu.edu /academic/anns/mps/math/mathlab/col_algebra/col_alg_tut36_longdiv.htm (946 words)

	Polynomial Long Division
		are polynomials, and the degree of d(x) is less than or equal to the degree of f(x), then there exist unique polynomials q(x) and r(x), so that
		Other ways of checking include graphing both sides (if you have a graphing calculator), or plugging in a few numbers on both sides (this is not always 100% foolproof).
		of the polynomial on the last line by the leading term x of the divisor to obtain -2x, and add this term to the
	www.sosmath.com /algebra/factor/fac01/fac01.html (629 words)

	Mathematics Weblog :: Polynomial Division
		A long time ago UK students used to learn how to do polynomial long division before they were 16.
		They don’t actually need to do long division as examples are usually simple enough to allow one to guess factors; for example
		Nevertheless, it is useful to be able to do long division and, given the time constraints, I usually use Synthetic Division.
	sixthform.info /maths/index.php?p=84&more=1&c=1 (449 words)

	PreCalculus (Site not responding. Last check: 2007-10-07)
		It cannot have both a horizontal and an oblique asymptote.
		Using polynomial long division, it can be converted into the sum of a linear polynomial and a proper rational expression.
		is the quotient obtained by long division of
	home.att.net /~srschmitt/precalc/precalc-12-03-00.html (152 words)

	Math Forum - Ask Dr. Math
		Here's an example of when you'd use polynomial long division.
		The steps are quite similar to the steps you do when dividing numbers, but that may not be helpful until you already know how to do it.
		Let's say we have the polynomial x^5 - 2x^2 + 4, and we want to divide it by x-4.
	mathforum.org /library/drmath/view/52855.html (259 words)

	Useless math - Physics Help and Math Help - Physics Forums
		Sometimes if you have a polynomial you can spot one factor real easy, but the others are hard to find.
		also, i believe, that some kids cant complete squares easily, so they need to use polynomial long division to get the +d-d part as a -d remainder.
		Polynomial division is good for curve sketching too.
	www.physicsforums.com /showthread.php?t=32000 (1245 words)

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