Polynomials:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
New York [u.a.]
Springer
1995
|
| Ausgabe: | Corr. 2. printing |
| Schriftenreihe: | Problem books in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 438 - 448 |
| Beschreibung: | XXII, 455 S. graph. Darst. |
| ISBN: | 3540969195 0387969195 |
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| 100 | 1 | |a Barbeau, Edward J. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Polynomials |c E. J. Barbeau |
| 250 | |a Corr. 2. printing | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1995 | |
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| 490 | 0 | |a Problem books in mathematics | |
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Datensatz im Suchindex
| _version_ | 1804125145478987776 |
|---|---|
| adam_text | E.J. BARBEAU POLYNOMIALS WITH 36 ILLUSTRATIONS SPRINGER CONTENTS PREFACE
VII ACKNOWLEDGMENT OF PROBLEM SOURCES XIII 1 FUNDAMENTALS 1 /L.L THE
ANATOMY OF A POLYNOMIAL OF A SINGLE VARIABLE 1 1.1.5 MULTIPLICATION BY
DETACHED COEFFICIENTS 1.1.19 EVEN AND ODD POLYNOMIALS E.I SQUARE OF A
POLYNOMIAL E.2 SETS WITH EQUAL POLYNOMIAL-VALUE SUMS E.3 POLYNOMIALS AS
GENERATING FUNCTIONS ^1.2 QUADRATIC POLYNOMIALS 9 1.2.1 QUADRATIC
FORMULA 1.2.4 THEORY OF THE QUADRATIC 1.2.14 CAUCHY-SCHWARZ INEQUALITY
1.2.17 ARITHMETIC-GEOMETRIC MEAN INEQUALITY 1.2.18 APPROXIMATION OF
QUADRATIC IRRATIONAL BY A RATIONAL E.4 GRAPHICAL SOLUTION OF THE
QUADRATIC E.5 POLYNOMIALS, SOME OF WHOSE VALUES ARE SQUARES J(L.3
COMPLEX NUMBERS 14 1.3.8 DE MOIVRE S THEOREM 1.3.10 SQUARE ROOT OF A
COMPLEX NUMBER 1.3.15 ; TCHEBYCHEF POLYNOMIALS E.6 COMMUTING POLYNOMIALS
1.4 EQUATIONS OF LOW DEGREE 18 1.4.4 CARDAN S METHOD FOR CUBIC 1.4.11
DESCARTES METHOD FOR QUARTIC 1.4.12 FERRARI S METHOD FOR QUARTIC 1.4.13
RECIPROCAL EQUATIONS E.7 THE RECIPROCAL EQUATION SUBSTITUTION 1.5
POLYNOMIALS OF SEVERAL VARIABLES 25 1.5.2 CRITERION FOR HOMOGENEITY
1.5.5 ELEMENTARY SYMMETRIC POLYNOMIALS OF 2 VARIABLES 1.5.8 ELEMENTARY
SYMMETRIC POLYNOMIALS OF 3 VARIABLES 1.5.9 ARITHMETIC-GEOMETRIC MEAN
INEQUALITY FOR 3 NUMBERS 1.5.10 POLYNOMIALS WITH N VARIABLES E.8
POLYNOMIALS IN EACH VARIABLE SEPARATELY XVIII CONTENTS E.9 THE RANGE OF
A POLYNOMIAL E.10 DIOPHANTINE EQUATIONS 1.6 BASIC NUMBER THEORY AND
MODULAR ARITHMETIC 31 1.6.1 EUCLIDEAN ALGORITHM 1.6.5 MODULAR ARITHMETIC
1.6.6 LINEAR CONGRUENCE E.LL LENGTH OF EUCLIDEAN ALGORITHM E.12 THE
CONGRUENCE AX = B (MOD M) E.13 POLYNOMIALS WITH PRIME VALUES E.14
POLYNOMIALS WHOSE POSITIVE VALUES ARE FIBONACCI NUMBERS 1.7 RINGS AND
FIELDS 38 1.7.6 Z M E.15 IRREDUCIBLE POLYNOMIALS OF LOW DEGREE MODULO P
1.8 PROBLEMS ON QUADRATICS 40 1.9 OTHER PROBLEMS 43 HINTS 46 2
EVALUATION, DIVISION, AND EXPANSION 50 2.1 HORNER S METHOD 50 2.1.8-9
USE OF HORNER S METHOD FOR TAYLOR EXPANSION E.16 NUMBER OF
MULTIPLICATIONS FOR C N E.17 A HORNER S APPROACH TO THE BINOMIAL
EXPANSION E.18 FACTORIAL POWERS AND SUMMATIONS 2.2 DIVISION OF
POLYNOMIALS 57 2.2.2 FACTOR THEOREM 2.2.4 NUMBER OF ZEROS CANNOT EXCEED
DEGREE OF POLYNOMIAL 2.2.7 LONG DIVISION OF POLYNOMIALS; QUOTIENT AND
REMAINDER 2.2.9 DIVISION THEOREM 2.2.12 FACTOR THEOREM FOR TWO VARIABLES
2.2.15 GAUSS THEOREM ON SYMMETRIC FUNCTIONS E.19 CHROMATIC POLYNOMIALS
E.20 THE GREATEST COMMON DIVISOR OF TWO POLYNOMIALS E.21 THE REMAINDER
FOR SPECIAL POLYNOMIAL DIVISORS 2.3 THE DERIVATIVE 66 2.3.4 DEFINITION
OF DERIVATIVE 2.3.5 PROPERTIES OF THE DERIVATIVE 2.3.9 TAYLOR S THEOREM
2.3.15 MULTIPLICITY OF ZEROS E.22 HIGHER ORDER DERIVATIVES OF THE
COMPOSITION OF TWO FUNCTIONS E.23 PARTIAL DERIVATIVES E.24 HOMOGENEOUS
POLYNOMIALS CONTENTS XIX E.25 CAUCHY-RIEMANN CONDITIONS E.26 THE
LEGENDRE EQUATION 2.4 GRAPHING POLYNOMIALS 74 2.4.6. SYMMETRY OF CUBIC
GRAPH E.27 INTERSECTION OF GRAPH OF POLYNOMIAL WITH LINES E.28 ROLLE S
THEOREM 2.5 PROBLEMS 78 HINTS 81 3 FACTORS AND ZEROS 83 3.1 IRREDUCIBLE
POLYNOMIALS 83 3.1.3 IRREDUCIBILITY OF LINEAR POLYNOMIALS 3.1.10
IRREDUCIBILITY OVER Q RELATED TO IRREDUCIBILITY OVER Z 3.1.12 EISENSTEIN
CRITERION 3.2 STRATEGIES FOR FACTORING POLYNOMIALS OVER Z 87 3.2.6
UNDETERMINED COEFFICIENTS E.29 T 2 - T + A AS A DIVISOR OF T N +1 + B
E.30 THE SEQUENCE U N (T) 3.3 FINDING INTEGER AND RATIONAL ROOTS:
NEWTON S METHOD OF DIVISORS 93 3.3.5 NEWTON S METHOD OF DIVISORS E.31
RATIONAL ROOTS OF NT 2 + (N + L)T - (N + 2) 3.4 LOCATING INTEGER ROOTS:
MODULAR ARITHMETIC 98 3.4.7 CHINESE REMAINDER THEOREM E.32 LITTLE FERMAT
THEOREM E.33 HENSEL S LEMMA 3.5 ROOTS OF UNITY 104 3.5.1 ROOTS OF UNITY
3.5.7 PRIMITIVE ROOTS OF UNITY 3.5.9 CYCLOTOMIC POLYNOMIALS 3.5.18
QUADRATIC RESIDUE 3.5.19 SICHERMAN DICE E.34 DEGREE OF THE CYCLOTOMIC
POLYNOMIALS E.35 IRREDUCIBILITY OF THE CYCLOTOMIC POLYNOMIALS E.36
COEFFICIENTS OF THE CYCLOTOMIC POLYNOMIALS E.37 LITTLE FERMAT THEOREM
GENERALIZED 3.6 RATIONAL FUNCTIONS 110 3.6.4-6 PARTIAL FRACTIONS E.38
PRINCIPAL PARTS AND RESIDUES 3.7 PROBLEMS ON FACTORIZATION 114 3.8 OTHER
PROBLEMS 117 HINTS 120 XX CONTENTS 4 EQUATIONS 125 4.1 SIMULTANEOUS
EQUATIONS IN TWO OR THREE UNKNOWNS 125 4.1.2 TWO LINEAR HOMOGENEOUS
EQUATIONS IN THREE UNKNOWNS 4.1.7 USE OF SYMMETRIC FUNCTIONS OF ZEROS
4.2 SURD EQUATIONS 128 4.3 SOLVING SPECIAL POLYNOMIAL EQUATIONS 130
4.3.6 SURD CONJUGATE 4.3.6-10 FIELD EXTENSIONS E.39 SOLVING BY RADICALS
E.40 CONSTRUCTIONS USING RULER AND COMPASSES 4.4 THE FUNDAMENTAL THEOREM
OF ALGEBRA: INTERSECTING CURVES 142 4.5 THE FUNDAMENTAL THEOREM:
FUNCTIONS OF A COMPLEX VARIABLE 146 4.6 CONSEQUENCES OF THE FUNDAMENTAL
THEOREM 150 4.6.1 DECOMPOSITION INTO LINEAR FACTORS 4.6.2 C AS AN
ALGEBRAICALLY CLOSED FIELD 4.6.3 FACTORIZATION OVER R 4.6.7 UNIQUENESS
OF FACTORIZATION 4.6.8 UNIQUENESS OF POLYNOMIAL OF DEGREE N TAKING N + 1
ASSIGNED VALUES 4.6.10 A CRITERION FOR IRREDUCIBILITY OVER Z E.41 ZEROS
OF THE DERIVATIVE: GAUSS-LUCAS THEOREM 4.7 PROBLEMS ON EQUATIONS IN ONE
VARIABLE 154 4.8 PROBLEMS ON SYSTEMS OF EQUATIONS 155 4.9 OTHER PROBLEMS
159 HINTS 160 5 APPROXIMATION AND LOCATION OF ZEROS 163 5.1
APPROXIMATION OF ROOTS 163 5.1.1 THE METHOD OF BISECTION 5.1.3 LINEAR
INTERPOLATION 5.1.4 HORNER S METHOD 5.1.5 NEWTON S METHOD 5.1.9
SUCCESSIVE APPROXIMATION TO A FIXPOINT E.42 CONVERGENCE OF NEWTON
APPROXIMATIONS E.43 NEWTON S METHOD ACCORDING TO NEWTON E.44 NEWTON S
METHOD AND HENSEL S LEMMA E.45 CONTINUED FRACTIONS: LAGRANGE S METHOD OF
APPROXIMATION CONTENTS XXI E.46 CONTINUED FRACTIONS: ANOTHER APPROACH
FOR QUADRATICS 5.2 TESTS FOR REAL ZEROS 174 5.2.7 DESCARTES RULE OF
SIGNS 5.2.12 A BOUND ON THE REAL ZEROS 5.2.15 ROLLE S THEOREM 5.2.17-20
THEOREM OF FOURIER-BUDAN E.47 PROVING THE FOURIER-BUDAN THEOREM E.48
STURM S THEOREM E.49 OSCILLATING POPULATIONS 5.3 LOCATION OF COMPLEX
ROOTS 183 5.3.3 CAUCHY S ESTIMATE 5.3.8 SCHUR-COHN CRITERION 5.3.9
STABLE POLYNOMIALS 5.3.10 ROUTH-HURWITZ CRITERION FOR A CUBIC 5.3.11
NYQUIST DIAGRAM 5.3.13 ROUCHE S THEOREM E.50 RECURSION RELATIONS 5.4
PROBLEMS 190 HINTS 194 6 SYMMETRIC FUNCTIONS OF THE ZEROS 198 6.1
INTERPRETING THE COEFFICIENTS OF A POLYNOMIAL 198 6.1.9 CONDITION FOR
REAL CUBIC TO HAVE REAL ZEROS 6.1.12 THE ZEROS OF A QUARTIC EXPRESSED IN
TERMS OF THOSE OF ITS RESOLVENT SEXTIC 6.2 THE DISCRIMINANT 201 6.2.5
DISCRIMINANT OF A CUBIC E.51 THE DISCRIMINANT OF T N - 1 6.3 SUMS OF THE
POWERS OF THE ROOTS 202 6.3.6 THE RECURSION FORMULA E.52 SERIES APPROACH
FOR SUM OF POWERS OF ZEROS E.53 A RECURSION RELATION E.54 SUM OF THE
FIRST N FCTH POWERS 6.4 PROBLEMS 205 HINTS 208 7 APPROXIMATIONS AND
INEQUALITIES 210 7.1 INTERPOLATION AND EXTRAPOLATION 210 7.1.5 LAGRANGE
POLYNOMIAL 7.1.7-13 FINITE DIFFERENCES XXII CONTENTS 7.2 7.3 7.4 7.5
7.1.16 E.55 E.56 ,E.57 E.58 FACTORIAL POWERS BUILDING UP A POLYNOMIAL
PROPAGATION OF ERROR SUMMING BY DIFFERENCES THE ABSOLUTE VALUE FUNCTION
APPROXIMATION ON AN INTERVAL 7.2.1 7.2.2 7.2.5 E.59 E.60 LEAST SQUARES
ALTERNATION BERNSTEIN POLYNOMIALS TAYLOR APPROXIMATION COMPARISON OF
METHODS FOR APPROXIMATING SQUARE ROOTS INEQUALITIES 7.3.5-6 7.3.8 7.3.10
E.61 GENERALIZATIONS OF THE AGM INEQUALITY BERNOULLI INEQUALITY NEWTON S
INEQUALITIES AGM INEQUALITY FOR FIVE VARIABLES PROBLEMS ON INEQUALITIES
OTHER PROBLEMS HINTS 218 225 230 231 233 8 MISCELLANEOUS PROBLEMS 236
E.62 E.63 E.64 E.65 E.66 E.67 E.68 E.69 HINTS ZEROS OF*- 1 [(L + *) B
-L-* N ] TWO TRIGONOMETRIC PRODUCTS POLYNOMIALS ALL OF WHOSE DERIVATIVES
HAVE INTEGER ZEROS POLYNOMIALS WITH EQUALLY SPACED ZEROS COMPOSITION OF
POLYNOMIALS OF SEVERAL VARIABLES THE MANDELBROT SET SUMS OF TWO SQUARES
QUATERNIONS ANSWERS TO EXERCISES AND SOLUTIONS TO PROBLEMS NOTES ON
EXPLORATIONS GLOSSARY FURTHER READING INDEX 246 250 403 427 430 449
|
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| author | Barbeau, Edward J. |
| author_facet | Barbeau, Edward J. |
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| author_sort | Barbeau, Edward J. |
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| callnumber-sort | QA 3241 B264 41995 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 230 SK 400 SK 420 |
| classification_tum | MAT 266f |
| ctrlnum | (OCoLC)34283304 (DE-599)BVBBV010666024 |
| dewey-full | 512.9/42 512.9/4220 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.9/42 512.9/42 20 |
| dewey-search | 512.9/42 512.9/42 20 |
| dewey-sort | 3512.9 242 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Corr. 2. printing |
| format | Book |
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| genre | (DE-588)4143389-0 Aufgabensammlung gnd-content |
| genre_facet | Aufgabensammlung |
| id | DE-604.BV010666024 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:56:54Z |
| institution | BVB |
| isbn | 3540969195 0387969195 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007119818 |
| oclc_num | 34283304 |
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| owner | DE-20 DE-91 DE-BY-TUM DE-521 DE-11 |
| owner_facet | DE-20 DE-91 DE-BY-TUM DE-521 DE-11 |
| physical | XXII, 455 S. graph. Darst. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | Springer |
| record_format | marc |
| series2 | Problem books in mathematics |
| spelling | Barbeau, Edward J. Verfasser aut Polynomials E. J. Barbeau Corr. 2. printing New York [u.a.] Springer 1995 XXII, 455 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Problem books in mathematics Literaturverz. S. 438 - 448 Polynômes Polynomials Polynom (DE-588)4046711-9 gnd rswk-swf (DE-588)4143389-0 Aufgabensammlung gnd-content Polynom (DE-588)4046711-9 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007119818&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Barbeau, Edward J. Polynomials Polynômes Polynomials Polynom (DE-588)4046711-9 gnd |
| subject_GND | (DE-588)4046711-9 (DE-588)4143389-0 |
| title | Polynomials |
| title_auth | Polynomials |
| title_exact_search | Polynomials |
| title_full | Polynomials E. J. Barbeau |
| title_fullStr | Polynomials E. J. Barbeau |
| title_full_unstemmed | Polynomials E. J. Barbeau |
| title_short | Polynomials |
| title_sort | polynomials |
| topic | Polynômes Polynomials Polynom (DE-588)4046711-9 gnd |
| topic_facet | Polynômes Polynomials Polynom Aufgabensammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007119818&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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