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Topic: Binomial theorem

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	Binomial theorem - Wikipedia, the free encyclopedia
		This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century.
		It was, however, known to the Chinese mathematician Yang Hui in the 13th century, the earlier Persian mathematician Omar Khayyám in the 11th century, and the even earlier Indian mathematician Pingala in the 3rd century BC.
		The binomial theorem is mentioned in the Gilbert and Sullivan song I am the Very Model of a Modern Major General.
	en.wikipedia.org /wiki/Binomial_theorem (546 words)

	Binomial type - Wikipedia, the free encyclopedia
		Every sequence of binomial type may be expressed in terms of the Bell polynomials.
		The set of all polynomial sequences of binomial type is a group in which the group operation is "umbral composition" of polynomial sequences.
		of coefficients of the first-degree terms in a polynomial sequence of binomial type may be termed the cumulants of the polynomial sequence.
	en.wikipedia.org /wiki/Binomial_type (1095 words)

	the Binomial theorem
		Theorem: The elements of the Pascal triangle, indeed, are the C(n, m).
		Theorem: The coefficients of the binomial expansion of (x + y)^n, indeed, are given by the rows of the Pascal triangle.
		The statement of a theorem is defined as a implies b is equivalent to not a or b.
	www.rism.com /Trig/binomial.htm (789 words)

	theorem articles on Encyclopedia.com (Site not responding. Last check: 2007-11-03)
		theorem THEOREM [theorem] in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance.
		He originated Taylor's theorem, a formula important in differential calculus, which relates a function to its derivatives by means of a power series.
		The binomial theorem, or binomial formula, gives the expansion of the n th power of a binomial (x + y) for n= 1, 2, 3,ââ¦â, as follows: where the ellipsis (â¦) indicates a continuation of terms
	www.encyclopedia.com /articles/12792.html (406 words)

	PlanetMath: binomial theorem
		The binomial theorem is a formula for the expansion of
		This is version 12 of binomial theorem, born on 2001-10-16, modified 2005-02-22.
		It might be worth pointing out that this theorem also holds if a and b belong to an commutative rig (the spelling is right; I really mean "rig", not "ring" here) since we only use some basic algebraic properties of real or complex numbers in the proof.
	planetmath.org /encyclopedia/BinomialTheorem.html (118 words)

	BINOMIAL - LoveToKnow Article on BINOMIAL (Site not responding. Last check: 2007-11-03)
		The binonfial theorem is a celebrated theorem, originally due to Sir Isaac Newton, by which any power of a binomial can be expressed as a series.
		The original form of the theorem was first given in a letter, dated the i3th of June 1676, from Sir Isaac Newton to Henry Oldenburg for communication to Wilhelm G. Leibnitz, although Newton had discovered it some years previously.
		The binomial theorem was thus discovered as a development of John Walliss investigations in the method of interpolation.
	www.1911ency.org /B/BI/BINOMIAL.htm (735 words)

	Binomial theorem - Topics in precalculus
		The solution to the problem of the binomial coefficients without actually multiplying out, is called the binomial theorem.
		We found the binomial coefficients to be 1 5 10 10 5 1.
		In the binomial, x is "a", and −1 is "b".
	www.themathpage.com /aPreCalc/binomial-theorem.htm (873 words)

	Fermat's last theorem - an elementary proof by Nico de Jong (1992) (Site not responding. Last check: 2007-11-03)
		We shall demonstrate this with the aid of the general properties of the binomial theorem.
		In wording his theorem Fermat involved the case of w = 2 as well, by stating : "It is impossible to separate a cube.
		The rules of the proof are exclusively directed by those of the binomial theorem in concert with the fundamental theorem of arithmetic.
	www.geocities.com /elementaryfermat (5445 words)

	PlanetMath: binomial coefficient
		Properties 5 and 6 are the binomial theorem applied to
		Although the standard mathematical notation for the binomial coefficients is
		This is version 24 of binomial coefficient, born on 2001-10-17, modified 2005-07-27.
	planetmath.org /encyclopedia/BinomialCoefficient.html (208 words)

	The Binomial Theorem for rational exponents (Site not responding. Last check: 2007-11-03)
		The full question is this: Isaac Newton generalized the Binomial Theorem to rational exponents.
		It was this kind of observation that led Newton to postulate the Binomial Theorem for rational exponents.
		You need to know some calculus to study the Binomial Theorem for rational exponents and to determine for what values of x it is true.
	mathcentral.uregina.ca /qq/database/QQ.09.98/evans2.html (241 words)

	Binomial series: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-03)
		(the binomial series generalizes the purely algebraic binomial theorem binomial theorem quick summary:
		The binomial series generalizes the binomial formula binomial theorem quick summary:
		In that case the summation reduces to the binomial formula.
	www.absoluteastronomy.com /encyclopedia/b/bi/binomial_series.htm (811 words)

	Binomial theorem - Search Results - MSN Encarta
		Binomial theorem - Search Results - MSN Encarta
		About binomial theorems I'm teeming with a lot of news,
		Binomial, algebraic expression that consists of exactly two terms separated by + or -, such as x + y or ab - cd.
	encarta.msn.com /encnet/refpages/search.aspx?q=Binomial+theorem (138 words)

	The Binomial Theorem
		The Binomial Theorem is an important theorem, useful across a wide range of mathematics.
		The theorem enables you to calculate all the terms of this expansion in your head.
		the Binomial Theorem generalises to the situation where n is a number other than a positive integer, but this is outside the scope of this course.
	www.maths.abdn.ac.uk /~igc/tch/eg1006/notes/node17.html (314 words)

	The Binomial Theorem: Formulas
		The Binomial Theorem is a quick way (okay, a less slow way) of expanding (multiplying out) a binomial that has been raised to some (generally inconveniently large) power.
		As you might imagine, drawing Pascal's Triangle every time you have to expand a binomial would be a rather long process, especially if the binomial has a large exponent on it.
		The biggest source of errors in the Binomial Theorem (other than forgetting the Theorem) is the simplification process.
	www.purplemath.com /modules/binomial.htm (648 words)

	Fermat's Last Theorem is Solved
		A new theorem determining the irrationality of a number using its infinite series expansion is presented.
		Perhaps geometry and the binomial series are complementary; the former contained in the domain of real numbers the latter in the domain of absolutely converging infinite sums.
		The binomial series used in the present proof, equation (2), was invented by Newton (1642-1772) about 1676.
	www.coolissues.com /mathematics/Fermat/fermat.htm (791 words)

	Binomial Theorem (Site not responding. Last check: 2007-11-03)
		The sum of all the binomial coefficients from 0 to n, or the sum of the entries in the n
		When x and y are 1 and -1, the alternating sum of binomial coefficients is 0.
		The binomial theorem generalizes to the multinomial theorem when the original expression has more than two variables, although there isn't a triangle of numbers to help us picture it.
	www.mathreference.com /cmb,bint.html (247 words)

	Exponents (Site not responding. Last check: 2007-11-03)
		Any power of a binomial can be obtained from the Binomial Theorem.
		There are many binomial expansion applications in physics.
		The binomial expansion is a useful example of a series.
	hyperphysics.phy-astr.gsu.edu /hbase/alg3.html (105 words)

	L12.html
		Assume the theorem is true for n=k, and show that it is also true for n=k+1.
		Therefore by the inductive asumption the binomial theorem is true.
		Therefore Fermat's theorem is true for all a>=0 and all primes p.
	www.math.sfu.ca /~gfee/Math342/L121.html (1449 words)

	PHYS208 Binomial Theorem (Site not responding. Last check: 2007-11-03)
		For situations involving distribution of a net charge over an extended region, the calculated electric field dependence may be checked in the limit where the point of evaluation is far from the charge distribution.
		When a finite amount of charge is the source, the far-field behavior of the electric field should behave as a point charge of that amount.
		The binomial theorem is especially useful in converting negative or fractional exponents into ordinary polynomial expressions from which the leading-order dependence may be determined.
	www.physics.udel.edu /~watson/phys208/binomial-example.html (160 words)

	Binomial Trees
		that binomial trees only come in sizes that are a power of two.
		It is because the number of nodes at a given depth in the tree is determined by the binomial coefficient.
		The binomial tree of order h+1 is composed of two binomial trees of height h, one attached under the root of the other.
	www.brpreiss.com /books/opus4/html/page371.html (452 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-03)
		Date: 7/14/96 at 18:54:39 From: Doctor Anthony Subject: Re: Binomial Theorem by Induction To prove the binomial theorem by induction we use the fact that nCr + nC(r+1) = (n+1)C(r+1) We can see the binomial expansion of (1+x)^n is true for n = 1.
		Also, the binomial theorem results in (x+y)^n, which I try to prove for (x+y)^n+1.
		I will introduce a slightly different notation for the binomial coefficient: Let "n choose r," or nCr, be written as C(n,r).
	mathforum.org /library/drmath/view/54228.html (560 words)

	A proof of the binomial theorem - Topics in precalculus
		If we multiply those with a binomial, we will have 8 terms; and finally those multiplied with a binomial will produce 16 terms.
		What the binomial theorem does is to tell how many terms there are of each kind.
		The binomial theorem states that in the expansion of (a + b)
	www.themathpage.com /aPreCalc/proof-binomial-theorem.htm (1366 words)

	Re: Binomial Theorem -- Negative Exponents (Site not responding. Last check: 2007-11-03)
		I have a basic idea > of what the binomial theorem is, but I don't see how expanding > out (2) gives alternating positive and negative terms.
		Now to demonstrate the binomial theorem for -2 requires knowing the derivative of x^-2.
		So if he's trying to show that's -2/x^3 using the binomial theorem, he is making circular argument and demonstrating his mathematical incompetence.
	www.talkabouteducation.com /group/alt.math/messages/30514.html (340 words)

	The Binomial Theorem (by MathsRevision.net)
		The Binomial Theorem states that, where n is a positive integer:
		This means use the Binomial theorem to expand the terms in the brackets, but only go as high as x
		So to find the answer we substitute 4 for a in the Binomial theorem and 2x for b:
	www.mathsrevision.net /alevel/pure/binomial.php (349 words)

	A Study in… Moriarty’s Binomial Theorem
		This paper is a study of the supposed Moriarty’s Binomial Theorem but… it is actually a holmesian pastiche.
		This James was the nephew of the ex-viceroy Rufus James, at the moment tutor of a family in Yorkshire.
		This young man had supposedly taken steps forward in modern techniques linked to coefficients of Newton’s binomial theorem: his results are about the combination system in the case: t=3, k=4.
	soalinux.comune.fi.it /holmes/inglese/ing_teorema.htm (2198 words)

	Continuous Binomial Theorem
		Before we tackle the continuous case, you should be familiar with the traditional binomial theorem, and the associated binomial coefficients, written (n:k), and pronounced n choose k.
		Use the binomial coefficient (t:k) to represent this expression.
		This is the continuous binomial theorem in its full generality.
	www.mathreference.com /cx-pow,cbt.html (357 words)

	College Algebra Tutorial on the Binomial Theorem
		Use the Binomial Theorem to expand a binomial raised to a power.
		This theorem gives us a formula that enables us to find the expansion of a binomial raised to a power, without having to multiply the whole thing out.
		This formula is going to lead us into Binomial Theorem which gives us a shortcut way of expanding a binomial.
	www.wtamu.edu /academic/anns/mps/math/mathlab/col_algebra/col_alg_tut54_bi_theor.htm (2196 words)

	Proof of the binomial theorem (Site not responding. Last check: 2007-11-03)
		According to the Polynomial Factor Theorem has the polynom on the right hand side the zero -1 with multiplicity n.
		And then, using the Fundamental Theorem of Algebra, the polynom must be of degree n.
		You also rely on the fundamental theorem of algebra, a very non-trivial result...
	www.physicsforums.com /showthread.php?threadid=69245 (574 words)

	Leaving Cert. Higher Level Maths - Sequences And Series - Pascal's Triangle And The Binomial Theorem
		where n is the power of the binomial and r is the number of the term.
		where n is the power of the binomial and r is the term.
		Calculating the full binomial expansion is just a matter of generating each term in the expansion, one at a time.
	www.netsoc.tcd.ie /~jgilbert/maths_site/applets/sequences_and_series/pascal_s_triangle_and_the_binomial_theorem.html (155 words)

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