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1. The Fundamental Theorem of Algebra (Undergraduate Texts in Mathematics) by Benjamin Fine, Gerhard Rosenberger, 1997-06-20
2. Constructive aspects of the fundamental theorem of algebra;: Proceedings of a symposium conducted at the IBM Research Laboratory, Zurich-Ruschlikon, Switzerland, June 5-7, 1967,
3. Constructive Aspects of the Fundamental Theorem of Algebra by Bruno & Peter Henrici. Eds. Dejon, 1969
4. Constructive Aspects of the Fundamental Theorem of Algebra by Bruno and Peter Henrici. Eds. Dejon, 1969
5. Descent: An optimization point of view on different fields [An article from: European Journal of Operational Research] by J. Brinkhuis, 2007-08-16  1. The Fundamental Theorem Of Algebra
The fundamental theorem of algebra. The multiplicity of roots. Let s factor the polynomial . We can pull out a term The fundamental theorem of algebra.
http://www.sosmath.com/algebra/factor/fac04/fac04.html
##### The multiplicity of roots.
Let's factor the polynomial . We can "pull out" a term Can we do anything else? No, we're done, we have factored the polynomial completely; indeed we have found the four linear (=degree 1) polynomials, which make up f x It just happens that the linear factor x shows up three times. What are the roots of f x )? There are two distinct roots: x =0 and x =-1. It is convenient to say in this situation that the root x =0 has multiplicity 3 , since the term x x -0) shows up three times in the factorization of f x ). Of course, the other root x =-1 is said to have multiplicity 1. We will from now on always count roots according to their multiplicity. So we will say that the polynomial has FOUR roots. Here is another example: How many roots does the polynomial have? The root x =1 has multiplicity 2, the root has multiplicity 3, and the root x =-2 has multiplicity 4. All in all, the polynomial has 9 real roots!
A degree 2 polynomial is called a quadratic polynomial. In factoring quadratic polynomials, we naturally encounter three different cases: Case 1A: Consider the quadratic polynomial depicted below.

2. Fundamental Theorem Of Algebra
fundamental theorem of algebra. This is a very powerful algebraic tool.2.4 It says that given any polynomial. we can always rewrite it as. where the points are the polynomial roots, and they may be
http://ccrma-www.stanford.edu/~jos/complex/Fundamental_Theorem_Algebra.html
Complex Basics
Complex Roots

Complex Numbers
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##### Fundamental Theorem of Algebra
This is a very powerful algebraic tool. It says that given any polynomial
we can always rewrite it as
where the points are the polynomial roots, and they may be real or complex. Complex Basics
Complex Roots

Complex Numbers
Contents ... Mathematics of the Discrete Fourier Transform (DFT) '', by Julius O. Smith III W3K Publishing ISBN
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by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

3. Fundamental Theorem Of Algebra
fundamental theorem of algebra. Gauss' Proof of the fundamental theorem of algebra. Translated by Ernest Fandreyer, M.S., Ed.D. Professor Emeritus. Fitchburg State College. Department of Mathematics. Fitchburg, MA 01420 USA
http://libraserv1.fsc.edu/proof/gauss.htm
 Fundamental Theorem of Algebra Gauss' Proof of the Fundamental Theorem of Algebra Translated by: Ernest Fandreyer, M.S., Ed.D. Professor Emeritus Professor of Mathmatics at Fitchburg State College from 1968 to 1998 Fitchburg State College Department of Mathematics Fitchburg, MA 01420 USA Fundamental Theorem of Algebra - pdf format Note: You must have Adobe Acrobat Reader installed to view this pdf document. If you do not have Acrobat Reader, you can download it free from the Adobe website: Download Acrobat Reader Back to FSC Library

4. Fundamental Theorem Of Algebra
The applet on this page is designed for experimenting with the fundamental theorem of algebra, which state that all polynomials with complex coefficients (and
http://www.math.gatech.edu/~carlen/applets/archived/ClassFiles/FundThmAlg.html
 The applet on this page is designed for experimenting with the fundamental theorem of algebra, which state that all polynomials with complex coefficients (and hence real as a special case) have a complete set of roots in the complex plane. The applet is designed to impart a geometric understanding of why this is true. It graphs the image in the complex plane, through the entered polynomial, of the circle of radius r. For small r, this is approximately a small circle around the constant term. For very large r, this is approximately a large circle that wraps n times around the origin, where n is the degree of the polynomial. For topological reasons, at some r value in between, the image must pass through the origin. When it does, a root is found. This applet lets you vary the radius and search out these roots. The real and imaginary parts of the polynomial must be entered separately in the function entering panels at the bottom of the applet in this version. There are instructions for how to enter other functions into these applets, but probably you should just try to enter things in and experiment always use * for multiplication, and ^ for powers, and make reasonable guesses about function names, and you may not need the instructions. Also, when you click to go to the radius entering panel, click again after you get there. For reason unbeknownst to me, the canvas on which the radius and such is reported erases itself after being drawn in. But a second click brings it back. The second click makes the exact same graphics calls, so this shouldn't happen. In any case, a second click cures it. If you know how to solve this the source is available on-line please let me know.

5. Fundamental Theorem Of Algebra
THE fundamental theorem of algebra. Our object is to prove that every nonconstant polynomial f(z) in one variable z over the complex numbers C has a root, i.e. that there is a complex number r in C
http://www.math.lsa.umich.edu/~hochster/419/fund.html
##### THE FUNDAMENTAL THEOREM OF ALGEBRA
Our object is to prove that every nonconstant polynomial f(z) in one variable z over the complex numbers C has a root, i.e. that there is a complex number r in C such that f(r) = 0. Suppose that The key point: one can get the absolute value of a nonconstant COMPLEX polynomial at a point where it does not vanish to decrease by moving along a line segment in a suitably chosen direction. We first review some relevant facts from calculus. Properties of real numbers and continuous functions Fact 1. Every sequence of real numbers has a monotone (nondecreasing or nonincreasing) subsequence. Proof. If the sequence has some term which occurs infinitely many times this is clear. Otherwise, we may choose a subsequence in which all the terms are distinct and work with that. Hence, assume that all terms are distinct. Call an element "good" if it is bigger than all the terms that follow it. If there are infinitely many good terms we are done: they will form a decreasing subsequence. If there are only finitely many pick any term beyond the last of them. It is not good, so pick a term after it that is bigger. That is not good, so pick a term after it that is bigger. Continuing in this way (officially, by mathematical induction) we get a strictly increasing subsequence. QED Fact 2. A bounded monotone sequence of real numbers converges.

6. Fund Theorem Of Algebra
The fundamental theorem of algebra. The fundamental theorem of algebra (FTA) states Every polynomial equation of degree n with complex
http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Fund_theorem_of_algebra.html
##### The fundamental theorem of algebra
Algebra index History Topics Index
The Fundamental Theorem of Algebra (FTA) states Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers. In fact there are many equivalent formulations: for example that every real polynomial can be expressed as the product of real linear and real quadratic factors. Early studies of equations by al-Khwarizmi (c 800) only allowed positive real roots and the FTA was not relevant. Cardan was the first to realise that one could work with quantities more general than the real numbers. This discovery was made in the course of studying a formula which gave the roots of a cubic equation. The formula when applied to the equation x x + 4 gave an answer involving -121 yet Cardan knew that the equation had x = 4 as a solution. He was able to manipulate with his 'complex numbers' to obtain the right answer yet he in no way understood his own mathematics. Bombelli , in his Algebra , published in 1572, was to produce a proper set of rules for manipulating these 'complex numbers'. Descartes in 1637 says that one can 'imagine' for every equation of degree n n roots but these imagined roots do not correspond to any real quantity.

 7. Fundamental Theorem Of Algebra From MathWorld fundamental theorem of algebra from MathWorld Every polynomial equation having complex coefficients and degree \geq 1 has at least one complex root. This theorem was first proven by Gauss. Ithttp://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/FundamentalTheorem

8. Fundamental Theorem Of Algebra -- From MathWorld
fundamental theorem of algebra. 101103, 1996. Krantz, S. G. The fundamental theorem of algebra. Â§1.1.7 and 3.1.4 in Handbook of Complex Variables.
http://mathworld.wolfram.com/FundamentalTheoremofAlgebra.html
 INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Algebra Polynomials Fundamental Theorem of Algebra Every polynomial equation having complex coefficients and degree has at least one complex root . This theorem was first proven by Gauss It is equivalent to the statement that a polynomial P z ) of degree n has n values (some of them possibly degenerate) for which Such values are called polynomial roots . An example of a polynomial with a single root of multiplicity is which has z = 1 as a root of multiplicity 2. Degenerate Frivolous Theorem of Arithmetic Polynomial Polynomial Factorization ... search Courant, R. and Robbins, H. "The Fundamental Theorem of Algebra." §2.5.4 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 101-103, 1996. Krantz, S. G. "The Fundamental Theorem of Algebra." §1.1.7 and 3.1.4 in

9. Gauss
One of the alltime greats, Gauss began to show his mathematical brilliance at the early age of seven. He is usually credited with the first proof of The fundamental theorem of algebra.
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Gauss.html
##### Born: 30 April 1777 in Brunswick, Duchy of Brunswick (now Germany) Died:
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At the age of seven, Carl Friedrich Gauss In 1788 Gauss began his education at the Gymnasium binomial theorem and the arithmetic- geometric mean, as well as the law of quadratic reciprocity and the prime number theorem. Kaestner , whom Gauss often ridiculed. His only known friend amongst the students was Farkas Bolyai . They met in 1799 and corresponded with each other for many years. ruler and compasses This was the most major advance in this field since the time of Greek mathematics and was published as Section VII of Gauss's famous work, Disquisitiones Arithmeticae Gauss returned to Brunswick where he received a degree in 1799. After the Duke of Brunswick had agreed to continue Gauss's stipend, he requested that Gauss submit a doctoral dissertation to the University of Helmstedt. He already knew Pfaff , who was chosen to be his advisor. Gauss's dissertation was a discussion of the

10. Fundamental Theorem Of Algebra
who lives on the other side of the mountain?) The fundamental theorem of algebra is a theorem about equation solving But the fundamental theorem of algebra states even more
http://www.cut-the-knot.com/do_you_know/fundamental2.shtml
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##### Statement and Significance
We already discussed the history of the development of the concept of a number. Here I would like to undertake a more formal approach. Thus, in the beginning there was counting. But soon enough people got concerned with equation solving. (If I saw 13 winters and my tribe's law allows a maiden to marry after her 15th winter, how many winters should I wait before being allowed to marry the gorgeous hunter who lives on the other side of the mountain?) The Fundamental Theorem of Algebra is a theorem about equation solving. It states that every polynomial equation over the field of complex numbers of degree higher than 1 has a complex solution. Polynomial equations are in the form P(x) = a n x n + a n-1 x n-1 + ... + a x + a where a n is assumed non-zero (for why to mention it otherwise?), in which case n is called the degree of the polynomial P and of the equation above. a i 's are known coefficients while x is an unknown number. A number a is a solution to the equation P(x) = if substituting a for x renders it identity : P(a) = 0. Coefficients are assumed to belong to a specific set of numbers where we also seek a solution. The polynomial form is very general but often studying P(x) = Q(x) is more convenient.

11. Fundamental Theorem Of Algebra
fundamental theorem of algebra. The fundamental theorem of algebra establishes this reason and is the topic of the discussion below.
http://www.cut-the-knot.org/do_you_know/fundamental.shtml
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##### Fundamental Theorem of Algebra
Complex numbers are in a sense perfect while there is little doubt that perfect numbers are complex. Starting from the tail, perfect numbers have been studied by the Ancients ( Elements, IX.36 ). Euler (1707-1783) established the form of even perfect numbers. [Conway and Guy, p137] say this: Are there any other perfect numbers? ... All we know about the odd ones is that they must have at least 300 decimal digits and many factors. There probably aren't any! Every one would agree it's rather a complex matter to write down a number in excess of 300 digits. Allowing for a pun, if there are odd perfect numbers they may legitimately be called complex. What about complex numbers in the customary sense? There is at least one good reason to judge them perfect. The Fundamental Theorem of Algebra establishes this reason and is the topic of the discussion below. In the beginning there was counting which gave rise to the natural numbers (or integers ): 1,2,3, and so on. In the space of a few thousand years, the number system kept getting expanded to include fractions, irrational numbers, negative numbers and zero, and eventually complex numbers. Even a cursory glance at the terminology would suggest (except for fractions) the reluctance with which the new numbers have been admitted into the family.

12. Fundamental Theorem Of Algebra
fundamental theorem of algebra. Statement and Significance. This is indeed so. But the fundamental theorem of algebra states even more.
http://www.cut-the-knot.org/do_you_know/fundamental2.shtml
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##### Statement and Significance
We already discussed the history of the development of the concept of a number. Here I would like to undertake a more formal approach. Thus, in the beginning there was counting. But soon enough people got concerned with equation solving. (If I saw 13 winters and my tribe's law allows a maiden to marry after her 15th winter, how many winters should I wait before being allowed to marry the gorgeous hunter who lives on the other side of the mountain?) The Fundamental Theorem of Algebra is a theorem about equation solving. It states that every polynomial equation over the field of complex numbers of degree higher than 1 has a complex solution. Polynomial equations are in the form P(x) = a n x n + a n-1 x n-1 + ... + a x + a where a n is assumed non-zero (for why to mention it otherwise?), in which case n is called the degree of the polynomial P and of the equation above. a i 's are known coefficients while x is an unknown number. A number a is a solution to the equation P(x) = if substituting a for x renders it identity : P(a) = 0. Coefficients are assumed to belong to a specific set of numbers where we also seek a solution. The polynomial form is very general but often studying P(x) = Q(x) is more convenient.

 13. Fundamental Theorem Of Algebra fundamental theorem of algebra. The fundamental theorem of algebra (FTA) states. Every polynomial of degree n with complex coefficients has n roots in the complex numbers.http://www.und.nodak.edu/dept/math/history/fundalg.htm

14. Fundamental Theorem Of Algebra - Wikipedia, The Free Encyclopedia
http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra
##### Fundamental theorem of algebra
The fundamental theorem of algebra (now considered something of a misnomer by many mathematicians) states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity. More formally, if (where the coefficients a a n can be real or complex numbers), then there exist ( not necessarily distinct) complex numbers z z n such that This shows that the field of complex numbers , unlike the field of real numbers , is algebraically closed n a and the sum of all the roots equals - a n The theorem had been conjectured in the 17th century but could not be proved since the complex numbers had not yet been firmly grounded. The first rigorous proof was given by Carl Friedrich Gauss in 1799. (An almost complete proof had been given earlier by d'Alembert .) Gauss produced several different proofs throughout his lifetime. All proofs of the fundamental theorem necessarily involve some analysis , or more precisely, the concept of continuity of real or complex polynomials. The main difficulty in the proof is to show that every non-constant polynomial has at least one zero. We mention approaches via complex analysis topology , and algebra
• Find a closed disk D of radius r p z p z r p z D is therefore achieved at some point z in the interior of D p z m p z ) is a holomorphic function in the entire complex plane. Applying

15. Fundamental Theorem Of Algebra - Wikipedia, The Free Encyclopedia
fundamental theorem of algebra. (Redirected from fundamental theorem of algebra). The fundamental theorem of algebra (now considered
http://en.wikipedia.org/wiki/Fundamental_Theorem_of_Algebra
##### Fundamental theorem of algebra
(Redirected from Fundamental Theorem of Algebra The fundamental theorem of algebra (now considered something of a misnomer by many mathematicians) states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity. More formally, if (where the coefficients a a n can be real or complex numbers), then there exist ( not necessarily distinct) complex numbers z z n such that This shows that the field of complex numbers , unlike the field of real numbers , is algebraically closed n a and the sum of all the roots equals - a n The theorem had been conjectured in the 17th century but could not be proved since the complex numbers had not yet been firmly grounded. The first rigorous proof was given by Carl Friedrich Gauss in 1799. (An almost complete proof had been given earlier by d'Alembert .) Gauss produced several different proofs throughout his lifetime. All proofs of the fundamental theorem necessarily involve some analysis , or more precisely, the concept of continuity of real or complex polynomials. The main difficulty in the proof is to show that every non-constant polynomial has at least one zero. We mention approaches via

16. Fundamental Theorem Of Algebra
This theorem is valid only within the complex normal field. When we expand the number system to e.g. logic sci.physics Subject Re Extending the fundamental theorem of algebra Date 26 Mar 1998 185531
http://www.math.niu.edu/~rusin/known-math/98/polynom_quater
 From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math,sci.logic,sci.physics Subject: Re: Extending the Fundamental Theorem of Algebra Date: 26 Mar 1998 18:55:31 GMT In article

17. Solving Polynomial Equations (C)
Theorem 2680( fundamental theorem of algebra) If p(z) is a polynomial whose coefficients are complex numbers and p(z) is
http://www.uwm.edu/~ericskey/TANOTES/Algebra/node12.html
Next: Factoring (C) Up: No Title Previous: Hidden Quadratic Equations (C)
##### Solving Polynomial Equations (C)
We haved seen that polynomial equations can be solved if they are quadratic equations or hidden quadratic equations. There are general formulae for solving cubic and quartic equations, but they are frequently so cumbersome as to be useless. Our stategy will be to try to factor polynomial equation to solve them. The thing to keep in mind is Theorem 2680 (Fundamental Theorem of Algebra) If p z ) is a polynomial whose coefficients are complex numbers and p z ) is not constant, then p z ) = has a solution. Hence polynomial equations have solutions, and the only question is whether or not we are clever enough to find them. Unless we say otherwise, we want to find all solutions, even the complex-valued ones.

Eric S Key

18. Complex Numbers: The Fundamental Theorem Of Algebra
Dave s Short Course on The fundamental theorem of algebra. As remarked before, in the 16th century Cardano noted that the sum of
http://www.clarku.edu/~djoyce/complex/fta.html
##### Dave's Short Course on The Fundamental Theorem of Algebra
As remarked before, in the 16th century Cardano noted that the sum of the three solutions to a cubic equation x bx cx d b , the negation of the coefficient of x . By the 17th century the theory of equations had developed so far as to allow Girard (1595-1632) to state a principle of algebra, what we call now "the fundamental theorem of algebra". His formulation, which he didn't prove, also gives a general relation between the n solutions to an n th degree equation and its n coefficients. An n th degree equation can be written in modern notation as x n a x n a n x a n x a n where the coefficients a a n a n , and a n are all constants. Girard said that an n th degree equation admits of n solutions, if you allow all roots and count roots with multiplicity. So, for example, the equation x x x + 1 = has the two solutions 1 and 1. Girard wasn't particularly clear what form his solutions were to have, just that there be n of them: x x x n , and x n Girard gave the relation between the n roots x x x n , and x n and the n coefficients a a n a n , and a n that extends Cardano's remark. First, the sum of the roots

19. Fundamental Theorem Of Algebra
fundamental theorem of algebra. The fundamental theorem of algebra (now considered something of a misnomer by many mathematicians
http://www.fact-index.com/f/fu/fundamental_theorem_of_algebra_1.html
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##### Fundamental theorem of algebra
The fundamental theorem of algebra (now considered something of a misnomer by many mathematicians) states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity. More formally, if (where the coefficients a a n can be real or complex numbers), then there exist (not necessarily distinct) complex numbers z z n such that This shows that the field of complex numbers , unlike the field of real numbers , is algebraically closed n a and the sum of all the roots equals - a n The theorem had been conjectured in the 17th century but could not be proved since the complex numbers had not yet been firmly grounded. The first rigorous proof was given by Carl Friedrich Gauss in the early 19th century. (An almost complete proof had been given earlier by d'Alembert .) Gauss produced several different proofs throughout his lifetime. All proofs of the fundamental theorem necessarily involve some analysis , or more precisely, the concept of continuity of real or complex polynomials. The main difficulty in the proof is to show that every non-constant polynomial has at least one zero. We mention approaches via

20. The Fundamental Theorem Of Algebra
Bibliography for the fundamental theorem of algebra short. Another topological proof of the fundamental theorem of algebra.