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# Topic: Polynomial

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 Polynomial - Wikipedia, the free encyclopedia Because of their simple structure, polynomials are easy to evaluate, and are used extensively in numerical analysis for polynomial interpolation or to numerically integrate more complex functions. The degree of a term in a polynomial is the sum of all of the exponents on the variables in that term, where a variable with no exponent is understood to have an exponent of 1. In knot theory the Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial are important knot invariants. en.wikipedia.org /wiki/Polynomial   (2422 words)

 Lagrange polynomial - Wikipedia, the free encyclopedia In numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolation polynomial for a given set of data points in the Lagrange form. The interpolation polynomial passes through all four control points, and each scaled basis polynomial passes through its respective control point and is 0 where x corresponds to the other three control points. The Lagrange basis polynomials are used in numerical integration to derive the Newton-Cotes formulas. en.wikipedia.org /wiki/Lagrange_polynomial   (434 words)

 PlanetMath: polynomial ring When a polynomial has all of its coefficients equal to 0, its degree is usually considered to be undefined, although some people adopt the convention that its degree is Similarly, a binomial is a polynomial with exactly two nonzero coefficients, and a trinomial is a polynomial with exactly three nonzero coefficients. This is version 5 of polynomial ring, born on 2001-10-23, modified 2005-07-24. planetmath.org /encyclopedia/Polynomial.html   (312 words)

 Polynomial Info - Encyclopedia WikiWhat.com   (Site not responding. Last check: 2007-10-22) Note that the polynomials of degree ≤ n are precisely those functions whose (n+1)st derivative is identically zero. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Weierstrass approximation theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial. In order to determine function values of polynomials for given values of the variable x, one does not apply the polynomial as a formula directly, but uses the much more efficient Horner scheme instead. www.wikiwhat.com /encyclopedia/p/po/polynomial.html   (1475 words)

 PlanetMath: minimal polynomial (endomorphism) The minimal polynomial is intimately related to the characteristic polynomial for Cross-references: algebraic multiplicity, between, difference, lemma, characteristic polynomial, finite dimensional, fundamental theorem of algebra, corollary, basis, factors, eigenvalues, roots, properties, minimal polynomial, polynomials, division algorithm, degree, minimal, vectors, dimension, monic polynomial, vector space, Endomorphism This is version 6 of minimal polynomial (endomorphism), born on 2002-11-21, modified 2004-04-30. planetmath.org /encyclopedia/MinimalPolynomialEndomorphism.html   (164 words)

 New View of Statistics: Polynomial Regression Polynomial regression is the answer for these data and for most curvilinear data that either show a maximum or a minimum in the curve, or that could show a max or min if you extrapolated the curve beyond your data. I deal with repeated-measures polynomials later, but the interpretation of the numbers describing the shape of the curve is the same, and I deal with that here. Polynomials are a special case of the more general non-linear models. www.sportsci.org /resource/stats/polynomial.html   (1000 words)

 Polynomials: Definitions / Evaluation Polynomial terms have variables to whole-number exponents; there are no square roots of exponents, no fractional powers, and no variables in the denominator. Polynomials are usually written this way, with the terms written in "decreasing" order; that is, with the highest exponent first, the next highest next, and so forth, until you get down to the plain old number. The first term in the polynomial, when it is written in decreasing order, is also the term with the biggest exponent, and is called the "leading term". www.purplemath.com /modules/polydefs.htm   (441 words)

 Hurwitz polynomial - Wikipedia, the free encyclopedia A Hurwitz polynomial is a polynomial whose coefficients are positive real numbers and whose zeros are located in the left half-plane of the complex plane, that is, the real part of every zero is negative. For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive. For all of a polynomial's roots to lie in the left half-plane, it is necessary and sufficient that the polynomial in question pass the Routh-Hurwitz stability criterion. en.wikipedia.org /wiki/Hurwitz_polynomial   (163 words)

 Functions A polynomial function of degree n is a function defined by an equation of the form: A complete analysis of graphs of polynomial functions of degree greater than 2 requires methods that you will learn in calculus. For now play with this applet which plots a graph of a polynomial function of degree at most 6. www.ma.utexas.edu /cgi-pub/kawasaki/plain/functions/2A.html   (245 words)

 Polynomial functions - Topics in precalculus The constant term of a polynomial is the term of degree 0; it is the term in which the variable does not appear. Now, to indicate a polynomial of the 50th degree, we cannot indicate the constants by resorting to different letters. A polynomial function of the first degree, such as y = 2x + 1, is called a linear function; while a polynomial function of the second degree, such as y = x² + 3x − 2, is called a quadratic. www.themathpage.com /aPreCalc/polynomial.htm   (1045 words)

 Polynomial Factorization   (Site not responding. Last check: 2007-10-22) A polynomial is of degree k if the largest power of the variable in any term is no greter than k. A polynomial with integer coefficients is "prime" if it cannot be expressed as the product of two lower-degree polynomials with integer coefficients. The polynomials must be printed in increasing order of degree, and when two or more degrees match they must be sorted by increasing value of coefficients from the greatest to the lowest degree. acm.uva.es /p/v4/463.html   (244 words)

 Polynomials   (Site not responding. Last check: 2007-10-22) Definition: A polynomial is an algebraic expression that is a sum of terms, where each term contains only variables with whole number exponents and integer coefficients. When we write a polynomial we follow the convention that says we write the terms in order of descending powers, from left to right. The degree of an individual term in a polynomial is the sum of powers of all the variables in that term. www.jamesbrennan.org /algebra/polynomials/polynomials.htm   (245 words)

 Polynomial Equations   (Site not responding. Last check: 2007-10-22) It is commonly thought that the roots of all polynomial equations up to the fourth degree with rational coefficients can be expressed in terms of those coefficients by means of the four arithmetic operations and the extraction of roots. These celebrated results have led to the notion that the roots of all polynomial equations of degree less than the fifth can be expressed in terms of the coefficients subjected to the operations of addition, subtraction, multiplication, division, and root-finding. For the purpose of this discussion only we shall refer to a polynomial that is irreducible over the quadratic numbers as an "irreducible polynomial." In more customary usage it means that it is irreducible over the field of its coefficients. www25.brinkster.com /ranmath/misund/poly01.htm   (608 words)

 Lab 6: Writing a Polynomial class using vectors Note that the actual degree of the polynomial may be smaller than (size of vector - 1), since some of the higher order terms may have zero coefficients. Two polynomials are equal if and only if (1) they have the same degree and (2) the coefficient of the terms of the same degree are equal. The polynomial is initialized to be a degree d polynomial. www.ugrad.cs.ubc.ca /~cs126/Labs/Polynomials/lab-polynomials.html   (841 words)

 The Prime Glossary: Matijasevic's polynomial   (Site not responding. Last check: 2007-10-22) Another approach might be to ask if there is a non-constant polynomial all of whose positive values (as the variables range in the set of non-negative integers) are all primes. Look at the special form of the second part: it is one minus a sum of squares, so the only way for it to be positive is for each of the squared terms to be zero (this is a trick of Putnam’s). Matijasevic, "Primes are enumerated by a polynomial in 10 variables (in Russian)," Zapiski Sem. primes.utm.edu /glossary/page.php?sort=MatijasevicPoly   (334 words)

 AllRefer.com - polynomial (Mathematics) - Encyclopedia polynomial, mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers. An example of a polynomial in one variable is 11x The degree of a polynomial in one variable is the highest power of the variable appearing with a nonzero coefficient; in the example given above, the degree is 4. reference.allrefer.com /encyclopedia/P/polynomi.html   (175 words)

 Solving Polynomial Inequalities As you can see, being familiar with polynomials and their shapes can make your life simpler for some of these problems. The factors multiply together to create the polynomial; the signs of the factors multiply together to give the sign of the polynomial. The easiest solution method for polynomial inequalities is using what you know about polynomial shapes, but the shape isn't always enough to give you the answer. www.purplemath.com /modules/ineqpoly.htm   (696 words)

 Polynomial Matrix Glossary A polynomial matrix P has full column rank (or full normal column rank) if it has full column rank everywhere in the complex plane except at a finite number of points. The roots or zeros of a polynomial matrix P are those points in the complex plane where P loses rank. The polynomials polynomials a, b and c are given while the polynomials x and y are unknown. www.polyx.com /glossary.htm   (1119 words)

 Roots or zeros of a polynomial - Topics in precalculus It is a polynomial set equal to 0. It is traditional to speak of a root of a polynomial. Therefore, the y-intercept of a polynomial is simply the constant term. www.themathpage.com /aPreCalc/roots-zeros-polynomial.htm   (378 words)

 Algebra (Math 1314) - Preliminaries - Polynomials   (Site not responding. Last check: 2007-10-22) In this section we will start looking at polynomials.  Polynomials will show up in pretty much every section of every chapter in the remainder of this material and so it is important that you understand them. The first one isn’t a polynomial because it has a negative exponent and all exponents in a polynomial must be positive. We can also talk about polynomials in three variables, or four variables or as many variables as we need.  The vast majority of the polynomials that we’ll see in this course are polynomials in one variable and so most of the examples in the remainder of this section will be polynomials in one variable. tutorial.math.lamar.edu /AllBrowsers/1314/Polynomials.asp   (1277 words)

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