Higher-Degree Polynomials
We have devoted chapters to quadratic, cubic, and quartic polynomials. This pattern cannot continue through all degrees, and not just because the resulting book would be infinitely long. It turns out that results of the sort we have obtained do not exist for polynomials in degree greater than four....
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| Vydáno v: | Beyond the Quadratic Formula s. 179 - 216 |
|---|---|
| Hlavní autor: | |
| Médium: | Kapitola |
| Jazyk: | angličtina |
| Vydáno: |
Washington
The Mathematical Association of America
2013
Mathematical Association of America American Mathematical Society |
| Vydání: | 1 |
| Témata: | |
| ISBN: | 0883857839, 9780883857830 |
| On-line přístup: | Získat plný text |
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| Abstract | We have devoted chapters to quadratic, cubic, and quartic polynomials. This pattern cannot continue through all degrees, and not just because the resulting book would be infinitely long. It turns out that results of the sort we have obtained do not exist for polynomials in degree greater than four. Therefore, we will content ourselves with a survey of some central results about higher-degree polynomials, combining proof sketches (or no proofs at all) with historical discussions. The chapter ends with a proof of the fundamental theorem of algebra.
Quintic Polynomials
We have obtained the quadratic formula, Cardano's formula, and Euler's formula for solutions of polynomial equations up through degree four. What about quintic polynomials? By Theorem 1.14, quintics have real roots. It is natural to seek a formula for a root like our other formulas, one involving the polynomial's coefficients and the operations of sum, product, quotient, and nth root extraction. If we had such an expression, we could solve quintic equations by radicals, the word "radical" referring to an nth root. Through the seventeenth and eighteenth centuries, progress was made in simplifying the problem of solving quintic equations, but no solution by radicals was found. Mathematicians began to suspect that a solution may not exist, and this was shown to be true in the nineteenth century. |
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| AbstractList | We have devoted chapters to quadratic, cubic, and quartic polynomials. This pattern cannot continue through all degrees, and not just because the resulting book would be infinitely long. It turns out that results of the sort we have obtained do not exist for polynomials in degree greater than four. Therefore, we will content ourselves with a survey of some central results about higher-degree polynomials, combining proof sketches (or no proofs at all) with historical discussions. The chapter ends with a proof of the fundamental theorem of algebra.
We have obtained the quardic formula, Cardano’ formula, and Euler’s formula for solutions We have devoted chapters to quadratic, cubic, and quartic polynomials. This pattern cannot continue through all degrees, and not just because the resulting book would be infinitely long. It turns out that results of the sort we have obtained do not exist for polynomials in degree greater than four. Therefore, we will content ourselves with a survey of some central results about higher-degree polynomials, combining proof sketches (or no proofs at all) with historical discussions. The chapter ends with a proof of the fundamental theorem of algebra. Quintic Polynomials We have obtained the quadratic formula, Cardano's formula, and Euler's formula for solutions of polynomial equations up through degree four. What about quintic polynomials? By Theorem 1.14, quintics have real roots. It is natural to seek a formula for a root like our other formulas, one involving the polynomial's coefficients and the operations of sum, product, quotient, and nth root extraction. If we had such an expression, we could solve quintic equations by radicals, the word "radical" referring to an nth root. Through the seventeenth and eighteenth centuries, progress was made in simplifying the problem of solving quintic equations, but no solution by radicals was found. Mathematicians began to suspect that a solution may not exist, and this was shown to be true in the nineteenth century. |
| Author | Ron Irving |
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2013 2013 the Mathematical Association of America, Inc. |
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| Keywords | Abel, Niels McMullen, Curt polynomial equation ring Tschirnhausen, Ehrenfried Walter von resolvent polynomial general notion Viète root of a polynomial field definition ring definition mathematical induction, principle of induction Jerrard, George Lagrange, Joseph Klein, Felix d'Alembert, Jean-le-Rond Bézout, Etienne symmetric polynomials definition change of variables Euler, Leonhard fundamental theorem of algebra Galois, Évariste Laplace, Pierre-Simon Liouville, Joseph Girard, Albert quintic polynomials polynomial factorization existence Bring, Erland Galois Theory Hermite, Charles polynomial factorization quintic polynomials Bring-Jerrard form Fefferman, Charles fundamental theorem of algebra Eisenstein, Gotthold field Ruffini, Paolo symmetric polynomials polynomial discriminant Doyle, Peter Gauss, Carl Friedrich Tschirnhausen Argand, Jean-Robert Waring, Edward Newton, Isaac |
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| PublicationTitle | Beyond the Quadratic Formula |
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| Snippet | We have devoted chapters to quadratic, cubic, and quartic polynomials. This pattern cannot continue through all degrees, and not just because the resulting... |
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| StartPage | 179 |
| SubjectTerms | Algebra Axiomatic set theory Coefficients Complex numbers Degrees of polynomials Discrete mathematics Discriminants Factorization Formal logic Fundamental theorems History of mathematics Integers Logic Logical topics Mathematical expressions Mathematical logic Mathematical objects Mathematical procedures Mathematical set theory Mathematical theorems Mathematics Number theory Numbers Philosophy Polynomials Pure mathematics Rational numbers Real numbers |
| Title | Higher-Degree Polynomials |
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