English

Pronunciation

• /pɑ.lə.noʊ.mi.l/

Noun

polynomials

1. Plural of polynomial

In mathematics, a polynomial is an expression constructed from one or more variables and constants, using the operations of addition, subtraction, multiplication, and constant positive whole number exponents. For example, x^2 - 4x + 7\, is a polynomial, but x^2 - 4/x + 7x^\, is not because it involves division by a variable and has an exponent that is not a positive whole number.

Polynomials are one of the most important concepts in algebra and throughout mathematics and science. They are used to form polynomial equations, which encode a wide range of problems, from elementary story 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 are used in calculus and numerical analysis to approximate other functions. Polynomials are used to construct polynomial rings, one of the most powerful concepts in algebra and algebraic geometry.

Overview

A polynomial is either zero, or can be written as the sum of one or more non-zero terms. The number of terms is finite. These terms consist of a constant (called the coefficient of the term) multiplied by zero or more variables (which are usually represented by letters). Each variable may have an exponent which is a non-negative integer. The exponent on a variable in a term is equal to the degree of that variable in that term. Since x=x^1, the degree of a variable without a written exponent is one. A term with no variables is called a constant term, or just a constant. The degree of a constant term is 0. The coefficient of a term may be any number, including fractions, irrational numbers, negative numbers, and complex numbers.

For example,

-5x^2y\,

is a term. The coefficient is –5, the variables are x and y, the degree of x is two, and the degree of y is one.

The degree of the entire term is the sum of the degrees of each variable in it. In the example above, the degree is 2 + 1 = 3.

A polynomial is a sum of terms. For example, the following is a polynomial:

3x^2 - 5x + 4\,.

It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero. Here "" stands for "", so the coefficient of the middle term is −5.

When a polynomial in one variable is arranged in the traditional order, the terms of higher degree come before the terms of lower degree. In the first term above, the coefficient is 3, the variable is x, and the exponent is 2. In the second term, the coefficient is –5. The third term is a constant. The degree of a non-zero polynomial is the largest degree of any one term. In the example, the polynomial has degree two.

Alternative forms

An expression that can be converted to polynomial form through a sequence of applications of the commutative, associative, and distributive laws is usually considered to be a polynomial. For instance

(x+1)^3

is a polynomial because it can be worked out to x^3+3x^2+3x+1. Similarly

\frac

is considered a valid term in a polynomial, even though it involves a division, because it is equivalent to \tfracx^3 and \tfrac is just a constant. The coefficient of this term is therefore \tfrac. For similar reasons, if complex coefficients are allowed, one may have a single term like (2+3i)x^3; even though it looks like it should be worked out to two terms, the complex number 2+3i is in fact just a single coefficient in this case that happens to require a "+" to be written down.

Division by an expression containing a variable is not generally allowed in polynomials. For example,

\,

is not a polynomial because it includes division by a variable. Similarly,

( 5 + y ) ^ x ,\,

is not a polynomial, because it has a variable exponent.

Since subtraction can be treated as addition of the additive opposite, and since exponentiation to a constant positive whole number power can be treated as repeated multiplication, polynomials can be constructed from constants and variables with just the two operations addition and multiplication.

Polynomial functions

A polynomial function is a function defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies

ƒ(x) = anxn + an−1xn−1 + ... + a2x2 + a1x + a0

for all arguments x, where n is a nonnegative integer and a0, a1,a2, ..., an are constant coefficients.

For example, the function f, taking real numbers to real numbers, defined by

f(x) = x^3 - x

is a polynomial function of one argument. Polynomial functions of multiple arguments can also be defined, using polynomials in multiple variables, as in

f(x,y)= 2x^3+4x^2y+xy^5+y^2-7.

Polynomial functions are an important class of smooth functions.

Polynomial equations

A polynomial equation is an equation in which a polynomial is set equal to another polynomial.

3x^2 + 4x -5 = 0 \,

is a polynomial equation. In case of a polynomial equation the variable is considered an unknown, and one seeks to find the possible values for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation is to be contrasted with a polynomial identity like (x + y)(x – y) = x2–y2, where both members represent the same polynomial in different forms, and as a consequence any evaluation of both members will give a valid equality.

Elementary properties of polynomials

1. A sum of polynomials is a polynomial
2. A product of polynomials is a polynomial
3. The derivative of a polynomial function is a polynomial function
4. Any primitive or antiderivative of a polynomial function is a polynomial function

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. All polynomials with real or complex coefficients also have a factored form in which the polynomial is written as a product of linear polynomials. For example, the polynomial

x^2 - 2x - 3 \,

is the expanded form of the polynomial

(x - 3)(x + 1)\,,

which is written in factored form. Note that the constants in the linear polynomials (like -3 and +1 in the above example) may be complex numbers in certain cases, even if all coefficients of the expanded form are real numbers. This is because the field of real numbers is not algebraically closed; however, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

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

a_n x^n + a_x^ + \cdots + a_2 x^2 + a_1 x + a_0.

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

((\ldots(a_n x + a_)x + ... + a_2)x + a_1)x + a_0\,.

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.

More advanced examples of polynomials

In 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.

History

Determining 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 3x + 2y + z = 29.

Notation

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. René Descartes, in La geometrie, 1637, introduced the concept of the graph of a polynomial equation. He 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 variable, where the a 's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.

Solving polynomial equations

Every 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.

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 called a system of linear equations, for which a range of different solution methods exist, including the classical 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.

Graphs

A polynomial function in one real variable can be represented by a graph.

• The graph of the zero polynomial

f(x) = 0

is the x-axis.

• The graph of a degree 0 polynomial

f(x) = a0 , where a0 ≠ 0,

is a horizontal line with y-intercept a0

• The graph of a degree 1 polynomial (or linear function)

f(x) = a0 + a1x , where a1 ≠ 0,

is an oblique line with y-intercept a0 and slope a1.

• The graph of a degree 2 polynomial

f(x) = a0 + a1x + a2x2, where a2 ≠ 0

is a parabola.

• The graph of any polynomial with degree 2 or greater

f(x) = a0 + a1x + a2x2 + . . . + anxn , where an ≠ 0 and n ≥ 2

is a continuous non-linear curve.

Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.

The illustrations below show graphs of polynomials. | x + 1 |- | 2 | quadratic | x^2 + 1 |- | 3 | cubic | x^3 + 1 |- | 4 | quartic or biquadratic | x^4 + 1 |- | 5 | quintic | x^5 + 1 |- | 6 | sextic or hexic | x^6 + 1 |- | 7 | septic or heptic | x^7 + 1 |- | 8 | octic | x^8 + 1 |- | 9 | nonic | x^9 + 1 |- | 10 | decic | x^ + 1 |- |}

The names for degrees higher than 3 are less common. The names for the degrees may be applied to the polynomial or to its terms. For example, a constant may refer to a zero degree polynomial or to a zero degree term.

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 to be negative (either –1 or –∞)http://mathworld.wolfram.com/ZeroPolynomial.html. The latter convention is important when defining Euclidean division of polynomials.

The word monomial can be ambiguous, as it is also often used to denote just a power of the variable, or in the multivariate case product of such powers, without any coefficient. Two or more terms which involve the same monomial in the latter sense, in other words which differ only in the value of their coefficients, are called similar terms; they can be combined into a single term by adding their coefficients; if the resulting term has coefficient zero, it may be removed altogether. The above classification according to the number of terms assumes that similar terms have been combined first.

Extensions of the concept of a polynomial

One also speaks of polynomials in several variables, obtained by taking the ring of polynomials of a ring of polynomials: R[X,Y] = (R[X])[Y] = (R[Y])[X]. These are of fundamental importance in algebraic geometry which studies the simultaneous zero sets of several such multivariate polynomials.

Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element.

Other related objects studied in abstract algebra are formal power series, which are like polynomials but may have infinite degree, and the rational functions, which are ratios of polynomials.

See also

Please see List of polynomial topics

References

• R. Birkeland. Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen. Mathematische Zeitschrift vol. 26, (1927) pp. 565-578. Shows that the roots of any polynomial may be written in terms of multivariate hypergeometric functions. Paper is available here.
• F. von Lindemann. Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen. Nachrichten von der Königl. Gesellschaft der Wissenschaften, vol. 7, 1884. Polynomial solutions in terms of theta functions. Paper available here.
• F. von Lindemann. Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen II. Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, 1892 edition. Paper available here.
• K. Mayr. Über die Auflösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen. Monatshefte für Mathematik und Physik vol. 45, (1937) pp. 280-313.
• H. Umemura. Solution of algebraic equations in terms of theta constants. In D. Mumford, Tata Lectures on Theta II, Progress in Mathematics 43, Birkhäuser, Boston, 1984.

External links

• Euler's work on Imaginary Roots of Polynomials at Convergence
• Free online polynomial calculator