In mathematics, a polynomial is an expression in which a finite number of constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). Thus,

$2\; x^2\; y\; z^3\; -\; 3\; y^2\; +\; 5\; y\; z\; -\; 2\; \backslash ,$

is a polynomial, but

$\{1\; \backslash over\; x^2\; +\; 1\}\; \backslash ,$

is not a polynomial.

A polynomial function is a function defined by evaluating a polynomial. For example, the function f defined by f(x) = x³−x is a polynomial function. Polynomial functions are an important class of smooth functions; smooth meaning that they are infinitely differentiable, i.e., they have derivatives of all finite orders.

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.

Elementary properties of polynomials

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All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have a factored form, in which parentheses appear.) In expanded form, a term of a polynomial is a part of the polynomial that is the product of a number (called the coefficient) and zero or more variables, where a variable that appears more than once is expressed as a power of that variable. Every polynomial in expanded form is a sum of terms, where subtraction is carried out by addition of terms with negative coefficients.

Polynomials are classified by their degree and number of variables. 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 particular, a term with no variables has degree zero. The degree of the polynomial is the largest degree of any one term, where we ignore terms with coefficient zero. If all the coefficients are zero, the degree is not zero, but is undefined or defined to have some other value for use in particular contexts.

Example:

$3\; x\; (\; x\; -\; y\; )\; +\; z\; \backslash ,$

is equivalent to the expanded form

$3\; x^2\; -\; 3\; x\; y\; +\; z\; \backslash ,$

In this expanded form, the second term is −3xy and its degree is 2. The polynomial is a second degree polynomial in three variables. If x = 10, y = 5, z = 100 then the evaluation of the polynomial is 250.

Every polynomial in one variable is equivalent to a polynomial with the form

a_nxⁿ + a_n−1xⁿ⁻¹ + ... + a₂x² + a₁x + a₀.

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

((...(a_nx + a_n−1)x + ... + a₂)x + a₁)x + a₀.

A polynomial equation is an equation in the form of a polynomial equal to zero or equal to another polynomial. The latter form is usually converted to the former by subtracting the second polynomial from both sides of the equation. When written as a polynomial equal to zero, the degree of the equation is the degree of the polynomial. The variables are often referred to as "unknowns", in the sense that the equation is a problem to be solved by finding numbers to assign to the variables to make the equation true when the polynomial is evaluated.

In elementary algebra, methods are given for solving all first degree and second degree polynomial equations in one unknown. The number of solutions may equal the degree, but it is necessary to consider multiplicity of solutions and complex numbers for this to be universally true, as described in the fundamental theorem of algebra.

A system of polynomial equations is a set of equations that must be evaluated with the same assignment of numbers to variables in every equation. Systems of equations are usually grouped with a single open brace on the left. In elementary algebra, methods are given for solving systems of linear equations in several unknowns. To get a unique solution, the number of equations usually must equal the number of unknowns. Systems of linear equations in several unknowns are treated in linear algebra.

More advanced examples of polynomials

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Also 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

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Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. 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 a root: this is the statement of the fundamental theorem of algebra.

There is a difference between approximating roots and finding concrete closed formulas for them. 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 for a long time. 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 relations among roots of polynomials.

The difference engine of Charles Babbage was designed to create large tables of values of logarithms and trigonometric functions automatically, by evaluating approximating polynomials at many points using Newton's difference method.

Polynomial functions

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For given numerical constants a₀, …, a_n, n > 0, in some field (often R or C), with a_n non-zero, then a polynomial (function) of degree n is a function of the form

$f(x)\; =\; a\_0\; +\; a\_1\; x\; +\; \backslash cdots\; +\; a\_\{n\; -\; 1\}\; x^\{n\; -\; 1\}\; +\; a\_n\; x^n.$

More concisely, a polynomial function can be written in sigma notation as

$f(x)\; =\; \backslash sum\_\{i\; =\; 0\}^\{n\}\; a\_\{i\}\; x^\{i\}.$

The constants a₀, …, a_n are called the coefficients of the polynomial. a₀ is called the constant coefficient and a_n is called the leading coefficient. When the leading coefficient is 1, the polynomial is said to be monic or normed.

Each summand a_i xⁱ of the polynomial is called a term. A polynomial with one, two or three terms is called monomial, binomial or trinomial respectively.

Polynomial functions of

• degree 0 are called constant functions (excluding the zero polynomial, which has indeterminate degree),
• degree 1 are called linear functions,
• degree 2 are called quadratic functions,
• degree 3 are called cubic functions,
• degree 4 are called quartic functions and
• degree 5 are called quintic functions.

Graphs

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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 or higher polynomial

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

is a continuous non-linear curve.

The best way to analyze the graph of a degree 2 or higher polynomial function is by its end behavior, the number of x-intercepts and the number of turning points.

End behavior

There are four end behaviors which are direct results of whether a_n, the leading coefficient, is positive or negative and whether n, the degree of the polynomial, is even or odd.

• If a_n is positive and n is even, the right end of the polynomial is in quadrant I while the left end is in quadrant II.
• If a_n is negative and n is even, the right end is in quadrant IV while the left end is in quadrant III.
• If a_n is positive and n is odd, the right end is in quadrant I while the left end is in quadrant III.
• If a_n is negative and n is odd, the right end is in quadrant IV while the left end is in quadrant II.

Number of x-intercepts

From the fundamental theorem of algebra, a polynomial of degree n has exactly n complex roots, which may or may not be real. Therefore, the number of x-intercepts cannot exceed $n$. It also follows from the Fundamental Theorem of Algebra that the complex roots of a polynomial with real coefficients must exist in complex-conjugate pairs. This in particular implies that an even-degree polynomial with real coefficients may have no x-intercepts (because all its roots may be complex); an odd-degree polynomial with real coefficients, on the other hand, must have at least one x-intercept, since any pairing of roots into complex-conjugate pairs will necessarily leave at least one root unpaired for odd n. This (these) "unpaired" root(s) must therefore be real. For example, a degree-4 polynomial function with real coefficients can have 0, 2 or 4 x-intercepts, whereas a similar degree-5 polynomial function can have 1, 3 or 5 x-intercepts. This assumes that x-intercepts are counted with their appropriate multiplicity - for example, the x-intercept of x² at the origin is counted twice because 0 is a double root of x².

Number of turning points

The number of turning points of an even-degree polynomial is any odd number less than the degree, while the number of turning points of an odd-degree polynomial is any even number less than the degree. For example, a degree 4 polynomial function can have 1 or 3 turning points whereas a degree 5 polynomial function can have 0, 2, or 4 turning points.

The following are some examples of polynomials of low degree.

Examples

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The function

$f(x)=\; -7x^3\; +\; \backslash begin\{matrix\}\backslash frac\{2\}\{3\}\backslash end\{matrix\}\; x^2\; -\; 5x\; +\; 3$

is an example of a cubic function with leading coefficient −7 and constant coefficient 3.

Roots

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A root or zero of a polynomial p is a number ζ so that p(ζ) = 0. The fundamental theorem of algebra states that a polynomial of degree n over the complex numbers has exactly n complex roots (not necessarily distinct ones). Therefore a polynomial can be factored as

$p(x)\; =\; a\_n(x-\backslash zeta\_1)\backslash cdots(x-\backslash zeta\_\{n\})$

where each $\backslash zeta\_i$ is a root of the polynomial p. For example, the polynomial x³−x can be factored as x(x−1)(x+1). An equation of the form p(x)=0 is called a polynomial equation in the unknown x. The solutions to the equation are the roots of the polynomial. The real roots of a polynomial (without non-real coefficients) are the x-intercepts of the graph of p(x) viewed as a polynomial function.

Solving a polynomial equation in one unknown is easily done on computer by the Durand-Kerner method. The reduction of equations in several unknowns to equations in one unknown each is discussed in the article on the Buchberger's algorithm. The special case where all the polynomials are of degree one is the gaussian elimination.

Equations of degree two can without computer be reduced to square roots. See quadratic equations.

Polynomials and calculus

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One 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.

Evaluation of polynomials

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To evaluate a polynomial in monomial form one can use the Horner scheme. For a polynomial in Chebyshev form the Clenshaw algorithm can be used. If several equidistant x_n have to be calculated one would use Newton's difference method.

Abstract algebra

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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

$f\; =\; a\_n\; X^n\; +\; a\_\{n\; -\; 1\}\; X^\{n\; -\; 1\}\; +\; \backslash cdots\; +\; a\_1\; X\; +\; a\_0$

where the coefficients a₀, ..., a_n 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

$$

X \; a = a \; X