Polynomial - Wikipedia, the free encyclopedia. The graph of a polynomial function of degree 3. In mathematics, a polynomial is an expression consisting of variables and coefficients which only employs the operations of addition, subtraction, multiplication, and non- negative integerexponents. An example of a polynomial of a single variable x is x. An example in three variables is x. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry. Etymology. The word polynomial joins two diverse roots: the Greek poly, meaning . It was derived from the term binomial by replacing the Latin root bi- with the Greek poly- . The word polynomial was first used in the 1. Two such expressions that may be transformed, one to the other, by applying the usual properties of commutativity, associativity and distributivity of addition and multiplication are considered as defining the same polynomial. A polynomial in a single indeterminate x can always be written (or rewritten) in the formanxn+an. The mapping that associates the result of this substitution to the substituted value is a function, called a polynomial function. This can be expressed more concisely by using summation notation. Each term consists of the product of a number. The exponent on an indeterminate in a term is called the degree of that indeterminate in that term; the degree of the term is the sum of the degrees of the indeterminates in that term, and the degree of a polynomial is the largest degree of any one term with nonzero coefficient. Because x = x. 1, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called respectively a constant term and a constant polynomial. The degree of the zero polynomial (which has no term) is generally treated as not defined (but see below). However, efficient polynomial factorization algorithms are available in most computer algebra systems. A formal quotient of polynomials, that is. Solving Polynomial Equations: Foundations, Algorithms, and Applications. 10-07-2016 1/4 Polynomial Algorithms In Computer Algebra.
The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is 2 + 1 = 3. Forming a sum of several terms produces a polynomial. For example, the following is a polynomial: 3x. A polynomial of degree zero is a constant polynomial or simply a constant. Polynomials of degree one, two or three are respectively linear polynomials,quadratic polynomials and cubic polynomials. For higher degrees the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, in x. The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero. Rather the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either . The zero polynomial is also unique in that it is the only polynomial having an infinite number of roots. In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all its non- zero terms have degree n. The zero polynomial is homogeneous, and, as homogeneous polynomial, its degree is undefined. For more details, see homogeneous polynomial. The commutative law of addition can be used to rearrange terms into any preferred order. In polynomials with one indeterminate, the terms are usually ordered according to degree, either in . The polynomial in the example above is written in descending powers of x. The first term has coefficient 3, indeterminate x, and exponent 2. In the second term, the coefficient is . The third term is a constant. Because the degree of a non- zero polynomial is the largest degree of any one term, this polynomial has degree two. It may happen that this makes the coefficient 0. A polynomial with two indeterminates is called a bivariate polynomial. These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance when working with univariate polynomials one does not exclude constant polynomials (which may result, for instance, from the subtraction of non- constant polynomials), although strictly speaking constant polynomials do not contain any indeterminates at all. It is possible to further classify multivariate polynomials as bivariate, trivariate, and so on, according to the maximum number of indeterminates allowed. Again, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on. It is common, also, to say simply . For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using Horner's method: (((. This is more efficient than the usual algorithm of division when the quotient is not needed. A sum of polynomials is a polynomial. If the set of the coefficients does not contain the integers (for example if the coefficients are integers modulo some prime numberp), then kak should be interpreted as the sum of ak with itself, k times. For example, over the integers modulo p, the derivative of the polynomial xp + 1 is the polynomial 0. For instance, the antiderivatives of x. As for the integers, two kinds of divisions are considered for the polynomials. The Euclidean division of polynomials that generalizes the Euclidean division of the integers. It results in two polynomials, a quotient and a remainder that are characterized by the following property of the polynomials: given two polynomials a and b such that b . By hand as well as with a computer, this division can be computed by the polynomial long division algorithm. This factored form is unique up to the order of the factors and their multiplication by an invertible constant. In the case of the field of complex numbers, the irreducible factors are linear. Over the real numbers, they have the degree either one or two. Over the integers and the rational numbers the irreducible factors may have any degree. However, efficient polynomial factorizationalgorithms are available in most computer algebra systems. A formal quotient of polynomials, that is, an algebraic fraction wherein the numerator and denominator are polynomials, is called a . Division of a polynomial by a number, however, yields another polynomial. For example, x. 3/1. When this expression is used as a term, its coefficient is therefore 1/1. For similar reasons, if complex coefficients are allowed, one may have a single term like (2 + 3i) x. The expression 1/(x. The expression (5 + y)x is not a polynomial, because it contains an indeterminate used as exponent. Because subtraction can be replaced by addition of the opposite quantity, and because positive integer exponents can be replaced by repeated multiplication, all polynomials can be constructed from constants and indeterminates using only addition and multiplication. Polynomial functions. A polynomial function is a function that can be defined by evaluating a polynomial. A function f of one argument is thus a polynomial function if it satisfies. Polynomial functions of multiple variables are similarly defined, using polynomials in multiple indeterminates, as inf(x,y)=2x. They are all continuous, smooth, entire, computable, etc. Graphs. Polynomial of degree 2: f(x) = x. A polynomial equation stands in contrast to a polynomial identity like (x + y)(x . There are also formulas for the cubic and quartic equations. For higher degrees, the Abel. However, root- finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of any degree. The number of real solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when the complex solutions are counted with their multiplicity. This fact is called the fundamental theorem of algebra. Solving equations. Every polynomial P in x corresponds to a function, f(x) = P (where the occurrences of x in P are interpreted as the argument of f), called the polynomial function of P; the equation in x setting f(x) = 0 is the polynomial equation corresponding to P. The solutions of this equation are called the roots of the polynomial; they are the zeroes of the function f (corresponding to the points where the graph of f meets the x- axis). A number a is a root of P if and only if the polynomial x . It may happen that x . If P is a nonzero polynomial, there is a highest power m such that (x . When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined (or infinite) multiplicity. With this exception made, the number of roots of P, even counted with their respective multiplicities, cannot exceed the degree of P. If, however, the set of allowed candidates is expanded to the complex numbers, every non- constant polynomial has at least one root; this is the fundamental theorem of algebra. By successively dividing out factors x . Formulas for expressing the roots of polynomials of degree 2 in terms of square roots have been known since ancient times (see quadratic equation), and for polynomials of degree 3 or 4 similar formulas (using cube roots in addition to square roots) were found in the 1. Niccol. But formulas for degree 5 eluded researchers for several centuries.
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