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Overview
For example,
is a monomial. The coefficient is negative five, the variables x and y, the degree of x is two and the degree of y is one. The degree of a monomial is the sum of the degrees of the variables in that monomial. In the example above, the degree is 2 + 1 = 3. A constant is said to have degree zero.
consists of three monomials (called terms), the first of degree two, the second of degree one, and the third of degree zero. It is arranged in the traditional order, where the terms of higher degree come before the terms of lower degree. In the first term, the coefficient is three, the variable is x, and the exponent is two. In the second term, the coefficient is negative five. The degree of a polynomial is the largest degree of any one term. In the example, the polynomial has degree two. A polynomial of degree zero is called a constant, of degree one is called linear, of degree two is called quadratic, of degree three is called cubic. Less commonly used, degree four is called quartic and degree five quintic. A polynomial with one term is called a monomial, two terms binomial, three terms trinomial. An expression that can be changed to a polynomial through a sequence of applications of the commutative, associative, and distributive laws is usually considered to be a polynomial. For example
is considered a polynomial, because it is equivalent to <math>\tfrac{1}{12}x^3</math>. The coefficient is <math>\tfrac{1}{12}</math>. But,
is not a polynomial because it includes division by a variable, and neither is
because it has a variable exponent. Since subtraction can be treated as addition of the opposite and since a constant positive whole number exponent can be treated as repeated multiplication, polynomials can be constructed with just the operations of addition and multiplication. A polynomial function is a function defined by evaluating a polynomial. For example, the function f defined by
is a polynomial function. Polynomial functions are an important class of smooth functions. The adjective smooth comes from calculus. It means that it is always possible to take the derivative of a polynomial, repeatedly, as often as is desired. The word smooth is descripitve of the appearance of the graph of a polynomial function. A polynomial equation is an equation in which a polynomial is set equal to another polynomial.
is a polynomial equation. Elementary properties of polynomials
Polynomials serve to approximate other functions, such as sine, cosine, and exponential. All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses, and a factored form, where the polynomial is written as a product of polynomials. For example, the polynomial
is the expanded form and is equal to the polynomial
which is written in factored form. In school algebra, students learn to move easily from one form to the other (see: factoring). Every polynomial in one variable is equivalent to a polynomial with the form
This form is sometimes taken as the definition of a polynomial in one variable. Evaluation of a polynomial consists of assigning a number to each variable and carrying out the indicated multiplications and additions. Evaluation is sometimes performed more efficiently using the Horner scheme
In elementary algebra, methods are given for solving all first degree and second degree polynomial equations in one variable. In the case of polynomial equations, the variable is often called an unknown. The number of solutions may not exceed the degree, and will equal the degree when multiplicity of solutions and complex number solutions are counted. This fact is called the fundamental theorem of algebra. A system of polynomial equations is a set of equations in which a given variable must take on the same value everywhere it appears in any of the equations. Systems of equations are usually grouped with a single open brace on the left. In elementary algebra, methods are given for solving a system of linear equations in several unknowns. To get a unique solution, the number of equations should equal the number of unknowns. If there are more unknowns than equations, the system is called underdetermined. If there are more equations than unknowns, the system is called overdetermined. This important subject is studied extensively in the area of mathematics known as linear algebra. Overdetermined systems are common in practical applications. For example, one U.S. mapping survey used computers to solve 2.5 million equations in 400,000 unknowns. [2] More advanced examples of polynomialsIn linear algebra, the characteristic polynomial of a square matrix encodes several important properties of the matrix. In graph theory the chromatic polynomial of a graph encodes the different ways to vertex color the graph using x colors. In abstract algebra, one may define polynomials with coefficients in any ring. In knot theory the Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial are important knot invariants. HistoryDetermining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations are written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, circa 200 BCE, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou." We would write <math>3x + 2y + z = 29</math> The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, - for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithemetica integra, 1544. Rene Descartes, in La geometrie, 1637, introduced the concept of the graph of a polynomial equation. He also popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen above, in the general formula for a polynomial in one veriable, where the a 's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well. [3] Solving polynomial equationsEvery polynomial corresponds to a polynomial function, where f(x) is set equal to the polynomial, and to a polynomial equation, where the polynomial is set equal to zero. The solutions to the equation are called the roots of the polynomial and they are the zeroes of the function and the x-intercepts of its graph. If x = a is a root of a polynomial, then (x - a) is a factor of that polynomial. Some polynomials, such as f(x) = x2 + 1, do not have any roots among the real numbers. If, however, the set of allowed candidates is expanded to the complex numbers, every (non-constant) polynomial has at least one distinct root; this follows from the fundamental theorem of algebra. There is a difference between approximating roots and finding exact roots. Formulas for the roots of polynomials up to a degree of 2 have been known since ancient times (see quadratic equation) and up to a degree of 4 since the 16th century (see Gerolamo Cardano, Niccolo Fontana Tartaglia). But formulas for degree 5 eluded researchers. In 1824, Niels Henrik Abel proved the striking result that there can be no general formula (involving only the arithmetical operations and radicals) for the roots of a polynomial of degree 5 or greater in terms of its coefficients (see Abel-Ruffini theorem). This result marked the start of Galois theory which engages in a detailed study of relationships among roots of polynomials. Approximate solutions to any polynomial equation can be found either by Newton's method or by one of the many more modern methods of approximating solutions. For a polynomial in Chebyshev form the Clenshaw algorithm can be used. As a practical matter, an approximate solution that is accurate to a desired precission may be as useful as an exact solution. The difference engine of Charles Babbage was designed to create large tables of approximate values of logarithms and trigonometric functions automatically, by evaluating approximating polynomials at many points using Newton's method. Numerically solving a polynomial equation in one unknown is easily done on computer by the Durand-Kerner method or by some other root-finding algorithm. The reduction of equations in several unknowns to equations each in one unknown is discussed in the article on the Buchberger's algorithm. The special case where all the polynomials are of degree one is the subject of the article gaussian elimination. It has been shown by Richard Birkeland and Karl Meyr that the roots of any polynomial may be expressed in terms of multivariate hypergeometric functions. Ferdinand von Lindemann and Hiroshi Umemura showed that the roots may also be expressed in terms of Siegel modular functions, generalizations of the theta functions that appear in the theory of elliptic functions. These characterizations of the roots of arbitrary polynomials are generalizations of the methods previously discovered to solve the quintic equation. GraphsA polynomial function in one real variable can be represented by a graph.
is the x-axis.
is a horizontal line with y-intercept a0
is an oblique line with y-intercept a0 and slope a1.
is a parabola.
is a continuous non-linear curve. Polynomial graphs are analized in calculus using intercepts, slopes, concavity, and end behavior.
Polynomials and calculusOne important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomials. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Stone-Weierstrass 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. Polynomials are also frequently used to interpolate functions. Quotients of polynomials are called rational expressions, and functions that evaluate rational expressions are called rational functions. Piecewise rationals are the only functions that can be evaluated directly on a computer, since typically only the operations of addition, multiplication, division and comparison are implemented in hardware. All the other functions that computers need to evaluate, such as trigonometric functions, logarithms and exponential functions, must then be approximated in software by suitable piecewise rational functions. Calculating derivatives and integrals is particularly easy. For the polynomial
the derivative with respect to x is
and the indefinite integral is
Abstract algebra
In abstract algebra, one must take care to distinguish between polynomials and polynomial functions. A polynomial f is defined to be a formal expression of the form
where the coefficients a0, ..., an are elements of some ring R and X is considered to be a formal symbol. Two polynomials are considered to be equal if and only if the sequences of their coefficients are equal. Polynomials with coefficients in R can be added by simply adding corresponding coefficients and multiplied using the distributive law and the rules
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