41. The Fundamental Theorem Of Algebra. How to think of a proof of the fundamental theorem of algebra. Prerequisites.A familiarity with polynomials and with basic real analysis. Statement. http://www.dpmms.cam.ac.uk/~wtg10/ftalg.html

How to think of a proof of the fundamental theorem of algebra Prerequisites A familiarity with polynomials and with basic real analysis. Statement Every polynomial (with arbitrary complex coefficients) has a root in the complex plane. (Hence, by the factor theorem, the number of roots of a polynomial, up to multiplicity, equals its degree.) Preamble How to come up with a proof. If you have heard of the impossibility of solving the quintic by radicals, or if you have simply tried and failed to solve such equations, then you will understand that it is unlikely that algebra alone will help us to find a solution of an arbitrary polynomial equation. In fact, what does it mean to solve a polynomial equation? When we `solve' quadratics, what we actually do is reduce the problem to solving quadratics of the particularly simple form x =C. In other words, our achievement is relative: if it is possible to find square roots, then it is possible to solve arbitrary quadratic equations. But is it possible to find square roots? Algebra cannot help us here. (What it can do is tell us that the existence of square roots does not lead to a contradiction of the field axioms. We simply "adjoin" square roots to the rational numbers and go ahead and do calculations with them - just as we adjoin i to the reals without worrying about its existence. See my

42. Theorem EQ-3: On Ternary Boolean Algebra next up previous Next theorem EQ4 Group Theory Up Summary ofOtter Outputs Previous theorem EQ-2 Robbins algebra, theorem http://www-fp.mcs.anl.gov/~lusk/papers/contest/node19.html

Next: Theorem EQ-4: Group Theory Up: Summary of Otter Outputs Previous: Theorem EQ-2: Robbins Algebra, Theorem EQ-3: On Ternary Boolean Algebra Karen D. Toonen

43. Theorem EQ-2: Robbins Algebra, Boolean next up previous Next theorem EQ3 On Ternary Up Summary of Otter Outputs Previoustheorem EQ-1 The Commutator theorem EQ-2 Robbins algebra, 2 2 Boolean. http://www-fp.mcs.anl.gov/~lusk/papers/contest/node18.html

Next: Theorem EQ-3: On Ternary Up: Summary of Otter Outputs Previous: Theorem EQ-1: The Commutator Theorem EQ-2: Robbins Algebra, 2#2 Boolean Karen D. Toonen

44. Fundamental Theorem Of Algebra Fundamental theorem of algebra Gauss' Proof of the Fundamental theorem ofalgebra Translated by Ernest Fandreyer, MS, Ed.D. Professor Emeritus 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

45. Fundamental Theorem Of Algebra Fundamental theorem of algebra. Keywords FTA, Fundamental, theorem, algebra, Constructive,Real, Complex, Polynomial. The README file of the contribution http://coq.inria.fr/contribs/fta.html

Fundamental Theorem of Algebra A constructive proof of the Fundamental Theorem of Algebra (every non-trivial polynomial equation P(z)=0 always has a solution in the complex plane) Download (archive compatible with Coq V7.4) Authors: Herman Geuvers Freek Wiedijk Jan Zwanenburg Randy Pollack Henk Barendregt Luis Cruz-Filipe Institution: Nijmegen university Keywords: FTA, Fundamental, Theorem, Algebra, Constructive, Real, Complex, Polynomial The README file of the contribution: This page was automatically generated from this description file

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

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. There are many other equivalent versions of this, for example that every real polynomial can be expressed as the product of real linear and real quadratic factors. Early work with equations only considered positive real roots so the FTA was not relevant. Cardan realized that one could work with numbers outside of the reals while studying a formula for the roots of a cubic equation. While solving x = 15x + 4 using the formula he got an answer involving the square root of -121. He manipulated this to obtain the correct answer, x = 4, even though he did not understand exactly what he was doing with these "complex numbers." In 1572 Bombelli created rules for these "complex numbers." In 1637 Descartes said that one can "imagine" for every equation of degree n n roots, but these imagined roots do not correspond to any real quantity. Albert Girard , a Flemish mathematiciam, was the first to claim that there are always n solutions to a polynomial of degree n in 1629 in . He does not say that the solutions are of the form a + b i , a, b real. Many mathematicians accepted Girard's claim that a polynomial equation must have

47. Fundamental Theorem Of Algebra Fundamental theorem of algebra. The fundamental theorem of algebra (FTA)states Every polynomial of degree n with complex coefficients http://www.und.edu/instruct/lgeller/fundalg.html

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. There are many other equivalent versions of this, for example that every real polynomial can be expressed as the product of real linear and real quadratic factors. Early work with equations only considered positive real roots so the FTA was not relevant. Cardan realized that one could work with numbers outside of the reals while studying a formula for the roots of a cubic equation. While solving x = 15x + 4 using the formula he got an answer involving the square root of -121. He manipulated this to obtain the correct answer, x = 4, even though he did not understand exactly what he was doing with these "complex numbers." In 1572 Bombelli created rules for these "complex numbers." In 1637 Descartes said that one can "imagine" for every equation of degree n n roots, but these imagined roots do not correspond to any real quantity. Albert Girard , a Flemish mathematiciam, was the first to claim that there are always n solutions to a polynomial of degree n in 1629 in . He does not say that the solutions are of the form a + b i , a, b real. Many mathematicians accepted Girard's claim that a polynomial equation must have

48. The Binomial Theorem And Other Algebra The Binomial theorem and other algebra. At its simplest, the binomial theoremgives an expansion of (1 + x) n for any positive integer n. We have http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node15.html

Next: Sequences Up: Introduction. Previous: Absolute Value Contents Index The Binomial Theorem and other Algebra At its simplest, the binomial theorem gives an expansion of (1 + x n for any positive integer n . We have x n nx x x k x n Recall in particular a few simple cases: x x x x x x x x x x x x x x x There is a more general form: a b n a n na n-1 b a n-2 b a n-k b k b n with corresponding special cases. Formally this result is only valid for any positive integer n ; in fact it holds appropriately for more general exponents as we shall see in Chapter Another simple algebraic formula that can be useful concerns powers of differences: a b a b a b a b a b a ab b a b a b a a b ab b and in general, we have a n b n a b a n-1 a n-2 b a n-3 b a b n b n-1 Note that we made use of this result when discussing the function after Ex And of course you remember the usual ``completing the square'' trick: ax bx c a x x c a x c Next: Sequences Up: Introduction. Previous: Absolute Value Contents Index Ian Craw 2002-01-07

49. The Fundamental Theorem Of Algebra The Fundamental theorem of algebra. theorem 6.12 (Fundamental theorem of algebra)Every polynomial with complex coefficients has a root. Ran Levi 200003-13. http://www.maths.abdn.ac.uk/~ran/mx4509/mx4509-notes/node18.html

Next: Calculation of the Fundamental Up: Some First Applications Previous: The Degree of a S The Fundamental Theorem of Algebra We now present one of the most classical theorems in the history of mathematics, which admits several proofs, some of which the reader might already be familiar with. Theorem 6.12 (Fundamental Theorem of Algebra) Every polynomial with complex coefficients has a root. Ran Levi

50. Fundamental Theorem Of Linear Algebra Fundamental theorem of Linear algebra. Inner Products and Orthogonality.Thetheorem. An Example. Up to Linear algebra Part II http://www.ma.iup.edu/projects/CalcDEMma/linalg2/linalg218.html

Fundamental Theorem of Linear Algebra Inner Products and Orthogonality TheTheorem An Example Up to Linear Algebra Part II

51. Leaving Cert. Higher Level Maths - Algebra - The Factor Theorem You are here Home / Category Index / algebra / The Factor theorem.The Factor theorem By David Spollen. How to use this applet http://www.netsoc.tcd.ie/~jgilbert/maths_site/applets/algebra/the_factor_theorem

Search for: in Entire website Algebra Complex Numbers Matrices Sequences and series Differentiation Integration Circle Vectors Linear Transformations Line Geometry Trigonometry Probability Further Calculus and Series Website Home Algebra The Factor Theorem No Title ... Integration You are here: Home Category Index Algebra / The Factor Theorem The Factor Theorem - By David Spollen How to use this applet: The Factor Theorem: -Enter the co-efficients (a, b, c, d) of any given cubic polynomial in the boxes provided for them. -Enter the value of x in its box provided. -Click on the "Enter x" button. -An equation should appear containing your values and a report as to whether your x expression is a factor of your polynomial or not. Notes on the maths used in the applet: Proof of the factor theorem: Some other key pointers: Last updated: Thursday. May 03 2001

52. NRICH | Secondary Topics | Algebra | Polynomials | Binomial Theorem Top Level + 3D algebra + Applications + Equations Linear + Logical + Operations+ Percentages - Polynomials - Binomial theorem Binomial ( February 2001 http://www.nrich.maths.org.uk/topic_tree/Algebra/Polynomials/Binomial_Theorem/

NRICH Prime NRICH Club ... Get Printable Page March 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 Click on the folders to browse problem topics from the secondary site. You can then go directly to each of the problems. Top Level Algebra Applications Equations Euclid s Algorithm Exponentials Inequalities Linear Logical Operations Percentages Polynomials Binomial Theorem Binomial ( February 2001 ) Cubics Equations Factorisation Inequalities Quadratics Proofs Relations Sequences Series proofs Analysis Calculus Combinatorics Complex Numbers Geometry Geometry-Cartesian Geometry-Coordinate Geometry-Coordinate Geometry-Euclidean Graph Theory Groups Investigation Investigations Logic Measures Mechanics Number Pre-calculus Probability Programs Properties of Shapes Pythagoras Sequences Statistics Symmetry Trigonometry Unclassified algebra number

53. Citations: Theorem Proving And Algebra - Goguen (ResearchIndex) Retrieving documents Joseph Goguen. theorem Proving and algebra. MIT, to appear1996. Joseph Goguen. theorem Proving and algebra. MIT, to appear 1996. http://citeseer.nj.nec.com/context/38836/0

39 citations found. Retrieving documents... Joseph Goguen. Theorem Proving and Algebra . MIT, to appear 1996. Home/Search Document Not in Database Summary Related Articles Check This paper is cited in the following contexts: First 50 documents JAIST RESEARCH REPORT IS-RR-96-0024S 1 Logical Semantics.. - Razvan Diaconescu And (Correct) ....underlying logics. The following table shows the correspondence between specification programming paradigms and logics as they appear in the actual version of CafeOBJ, also pointing to some basic references. ABBREVIATION LOGIC SPEC PGM PARADIGM BASIC REF. MSA many sorted algebraic specification algebra OSA order sorted algebraic specification [13, 22, 15] algebra with subtypes HSA hidden sorted behavioural specification [16] algebra HOSA hidden order sorted behavioural specification [16, 1] algebra with subtypes RWL rewriting logic concurrent [27] algebraic specification OSRWL . ....between specification programming paradigms and logics as they appear in the actual version of CafeOBJ, also pointing to some basic references. ABBREVIATION LOGIC SPEC PGM PARADIGM BASIC REF.

54. Citations: Automated Theorem Proving In Support Of Computer Algebra: Symbolic De are several systems that study application like this AGLM99, FSS98, Fis00, DHHW98All of them use conventional theorem provers or computer algebra systems in http://citeseer.nj.nec.com/context/1301627/273622

4 citations found. Retrieving documents... A. Adams, H. Gottliebsen, S. Linton, and U. Martin. Automated theorem proving in support of computer algebra: symbolic definite integration as a case study . In Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation, Vancouver, Canada. ACM Press, 1999. Home/Search Document Details and Download Summary Related Articles ... Check This paper is cited in the following contexts: Towards an Knowledge-Centered Infrastructure for Web-Based.. - Kohlhase (2001) (Correct) ....base is that for a helpful answer, we have to determine, which cases t our situation, or at least which cases can be eliminated, a task which requires proving (in) consistency of the case descriptions with the situation at hand. There are several systems that study application like this [ , FSS98, Fis00, DHHW98] All of them use conventional theorem provers or computer algebra systems in the background, and could bene t from MathWeb, as well as contribute to it, if used as mathematical services. Finally, the user query can often contain variables (wildcards) itself.

55. No Match For Fundamental Theorem Of Algebra No match for Fundamental theorem of algebra. Sorry, the term Fundamentaltheorem of algebra is not in the dictionary. Check the spelling http://wombat.doc.ic.ac.uk/foldoc/foldoc.cgi?Fundamental Theorem of Algebra

56. PinkMonkey.com Algebra Study Guide 4.1 Theorem CHAPTER 4 QUADRATIC EQUATIONS. 4.1 theorem. If a, b Ï R and ab= 0 then either a = 0 or b = 0. Corollary 1 If a . b = 0 but a http://www.pinkmonkey.com/studyguides/subjects/algebra/chap4/a0404101.asp

Support the Monkey! Tell All your Friends and Teachers Get Cash for Giving Your Opinions! Free College Cash 4U Advertise Here See What's New on the Message Boards today! Favorite Link of the Week. Your house from the sky! ... Tell a Friend Win a $1000 or more Scholarship to college! Place your Banner Ads or Text Links on PinkMonkey for $0.50 CPM or less! Pay by credit card. Same day setup. Please Take our User Survey CHAPTER 4 : QUADRATIC EQUATIONS If a, b R and ab = then either a = or b = Corollary 1 : then b = Corollary 2 : 0, b then c = Corollary 3 : then either a = or b = c Example Solution : Example Solution : For what values of x is the expression x (3x - 6 ) = ? i.e. x (3x - 6 ) = either x = or 3x -6 = i.e. x = or 3x = 6 x = 2 ( x + 3 ) ( 3x + 1 ) - ( x - 2 ) ( x + 3 ) next page Index 4.1 Theorem 4.2 Definition 4.3 Methods of Solving Quadratic Equations Chapter 5 1012 PinkMonkey users are on the site and studying right now. Search: All Products Books Popular Music Classical Music Video DVD Electronics Software Outdoor Living Cell Phones Keywords:

57. Fundamental Theorem Of Algebra Fundamental theorem Of algebra. Start your search on Fundamental theoremOf algebra. Other educational search engines Ask Jeeves http://www.virtualology.com/virtualpubliclibrary/hallofeducation/Mathematics/Fun

You are in: Virtual Public Library Hall of Education Mathematics Fundamental Theorem Of Algebra Fundamental Theorem Of Algebra Start your search on Fundamental Theorem Of Algebra Other educational search engines: Ask Jeeves for Kids Britannica.com CyberSleuth Kids Education World ... Yahooligans Unauthorized Site: This site and its contents are not affiliated, connected, associated with or authorized by the individual, family, friends, or trademarked entities utilizing any part or the subject’s entire name. Any official or affiliated sites that are related to this subject will be hyper linked below upon submission and Virtualology's review. 2000 by Virtualology TM . All rights reserved. Virtualology TM Search: About Us e-mail us Editing Sponsor Editing Sponsors review and select all student work for publishing. To learn more about our editing sponsor program click here. Published Students Primary Secondary College Graduate ... Technical

58. Theorem 4.1.8: Algebra On Series theorem 4.1.8 algebra on Series. Let and be two absolutely convergent series.Then The sum of the two series is again absolutely convergent. http://www.shu.edu/projects/reals/numser/proofs/sumalgbr.html

Theorem 4.1.8: Algebra on Series Let and be two absolutely convergent series. Then: The sum of the two series is again absolutely convergent. Its limit is the sum of the limit of the two series. The difference of the two series is again absolutely convergent. Its limit is the difference of the limit of the two series. The product of the two series is again absolutely convergent. Its limit is the product of the limit of the two series ( Cauchy Product Context Proof: The proof of the first statement is a simple application of the triangle inequality. Let A n n and B n n Assume that A n converges to A and B n converges to B . Then S n + b + b n + b n n n A + B Therefore the sequence n is bounded above by A + B . The sequence is also monotone increasing so that it must converge to some limit. To find the limit, note that + b ) + (a + b ) + ... (a n + b n + a + ... + a n + b + ... + b n if n is big enough. Therefore the sequence n converges to A + B , as required. The proof for the difference of sums is similar. The proof for the Cauchy product, on the other hand, is much more complicated and will be given in the statement on the Cauchy Product Interactive Real Analysis , ver. 1.9.3

59. Theorem 6.5.7: Algebra With Derivatives theorem 6.5.7 algebra with Derivatives. Addition Rule If f and gare differentiable at x = c then f(x) + g(x) is differentiable http://www.shu.edu/projects/reals/cont/proofs/diffalg.html

Theorem 6.5.7: Algebra with Derivatives Addition Rule : If f and g are differentiable at x = c then f(x) + g(x) is differentiable at x = c and (f(x) + g(x)) = f'(x) + g'(x) Product Rule : If f and g are differentiable at x = c then f(x) g(x) is differentiable at x = c and (f(x) g(x)) = f'(x) g(x) + f(x) g'(x) Quotient Rule : If f and g are differentiable at x = c , and g(c) # then f(x) / g(x) is differentiable at x = c , and ( f(x) / g(x) ) = Chain Rule : If g is differentiable at x = c , and f is differentiable at x = g(c) then f(g(x)) is differentiable at x = c , and f(g(x)) = f'(g(x)) g'(x) Context Proof: These proofs, except for the chain rule, consist of adding and subtracting the same terms and rearranging the result. We will prove the product and chain rule, and leave the others as an exercise. Product rule: = f(c) g'(c) + g(c) f'(c) Chain Rule: A 'quick and dirty' proof would go as follows: Since g is differentiable, g is also continuous at x = c. Therefore, as x approaches c we know that g(x) approaches g(c). The first factor, by a simple substitution, converges to f'(u), where u = g(c). The second factor converges to g'(c). Hence, by our rule on product of limits we see that the final limit is going to be f'(u) g'(c) = f'(g(c)) g'(c), as required. But this 'simple substitution' may not be mathematically precise. Here is a better proof of the chain rule. Define the function h(t) as follows, for a fixed s = g(c):

60. Gauss’s 1799 Proof Of The Fundamental Theorem Of Algebra EDITORIAL. Teach Gauss’s 1799 Proof Of the FundamentalTheorem of algebra. From Spring 2002 21st Century issue. http://www.21stcenturysciencetech.com/articles/Spring02/Gauss_02.html

EDITORIAL From Spring 2002 21st Century issue. An Induced Mental Block A New Curriculum We have all heard the frequent laments among our co-thinkers and professional colleagues at the sadly reduced state of science and mathematics education in our nation. As in all such matters, after the righteous indignation and hand-wringing, is over, one must ask oneself the realistic question: Are you part of the problem, or part of the solution? If you are not sure, we have a proposal for you. To introduce it, I ask you to perform the following experiment. STEP 1: As a suitable subject, locate any person who has attended high school within the last 50 or so years. You may include yourself. Now, politely ask that person, if he or she would please construct for you a square root. Among the technically educated, it is very common, next, to see the diagonal of the square appear, often with the label 2 attached. As this has nothing whatsoever to do with the solution, I have found it most effective to point out in such cases, that the problem is really much simpler than that. No knowledge of the Pythagorean Theorem, nor any higher mathematics, is required. An Induced Mental Block What is the problem? No student of the classical method of education, which has been around for at least the past 2,500 years, could ever have any problem with this simple exercise. The mental block which arises here is the perfectly lawful result of the absurd and prevalent modern-day teaching that number can exist independent of any physically determining principle. This is the ivory-tower view of mathematics, which holds sway from grade school to university, and reaches up like a hand from the grave, even into the peer review process governing what can be reported as the results of experimental physics.

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