Polynomial

In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate \(x\) is \(x^2 - 4x + 7\). An example with three indeterminates is \(x^3 + 2xyz^2 - yz + 1\).

Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and 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, which are central concepts in algebra and algebraic geometry.

The word polynomial joins two diverse roots: the Greek poly, meaning "many", and the Latin nomen, or "name". It was derived from the term binomial by replacing the Latin root bi- with the Greek poly-. That is, it means a sum of many terms (many monomials). The word polynomial was first used in the 17th century.

The \(x\) occurring in a polynomial is commonly called a variable or an indeterminate. When the polynomial is considered as an expression, \(x\) is a fixed symbol which does not have any value (its value is "indeterminate"). However, when one considers the function defined by the polynomial, then \(x\) represents the argument of the function, and is therefore called a "variable". Many authors use these two words interchangeably.

A polynomial in the indeterminate \(x\) is commonly denoted by an upper- or lower-case letter, like \(P\) or \(p\). However, a polynomial can be either denoted by a functional notation \(P(x)\) or \(p(x)\), the usage of which dates from a time when the distinction between a polynomial and the associated function was unclear. Moreover, the functional notation is often useful for specifying, in a single phrase, a polynomial and its indeterminate. For example, "let \(P(x)\) be a polynomial" is a shorthand for "let \(P\) be a polynomial in the indeterminate \(x\)". On the other hand, when it is not necessary to emphasize the name of the indeterminate, many formulas are much simpler and easier to read if the name(s) of the indeterminate(s) do not appear at each occurrence of the polynomial.

The ambiguity of having two notations for a single mathematical object may be formally resolved by considering the general meaning of the functional notation for polynomials. If \(a\) denotes a number, a variable, another polynomial, or, more generally, any expression, then \(P(a)\) denotes, by convention, the result of substituting \(a\) for \(x\) in \(P\). Thus, the polynomial \(P\) defines the function \[a\mapsto P(a),\] which is the polynomial function associated to \(P\). Frequently, when using this notation, one supposes that \(a\) is a number. However, one may use it over any domain where addition and multiplication are defined (that is, any ring). In particular, if \(a\) is a polynomial then \(P(a)\) is also a polynomial.

More specifically, when \(a\) is the indeterminate \(x\), then the image of \(x\) by this function is the polynomial \(P\) itself (substituting \(x\) for \(x\) does not change anything). In other words, \[P(x)=P,\] which formally justifies the existence of two notations for the same polynomial.

A polynomial is an expression that can be built from constants and symbols called variables or indeterminates by means of addition, multiplication and exponentiation to a non-negative integer power. The constants are generally numbers, but may be any mathematical objects that do not involve the indeterminates and that can be added and multiplied. Two polynomial expressions are considered as defining the same polynomial if they may be transformed, one into the other, by applying the usual properties of commutativity, associativity, and distributivity of addition and multiplication. For example \((x-1)(x-2)\) and \(x^2-3x+2\) are two polynomial expressions that represent the same polynomial; so, one has the equality \((x-1)(x-2)=x^2-3x+2\).

A polynomial in a single indeterminate x can always be written (or rewritten) in the form \[a_n x^n + a_{n-1}x^{n-1} + \dotsb + a_2 x^2 + a_1 x + a_0,\] where \(a_0, \ldots, a_n\) are constants that are called the coefficients of the polynomial, and \(x\) is the indeterminate. The word "indeterminate" means that \(x\) represents no particular value, although any value may be substituted for it. The mapping that associates the result of this substitution to the substituted value is a function, called a polynomial function; see § Polynomial functions.

This can be expressed more concisely by using summation notation: \[\sum_{k=0}^n a_k x^k\] That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a number – called the coefficient of the term – and a finite number of indeterminates, raised to non-negative integer powers.

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 term with a nonzero coefficient. Because \(x = x^1\), the degree of an indeterminate without a written exponent is one.

A term with no indeterminates and a polynomial with no indeterminates are called, respectively, a constant term and a constant polynomial. The degree of a constant term and of a nonzero constant polynomial is \(0\). The degree of the zero polynomial \(0\) (which has no terms at all) is generally treated as not defined (but see below).

For example: \[-5x^2y\] is a term. The coefficient is \(-5\), the indeterminates are \(x\) and \(y\), the degree of \(x\) is two, while the degree of \(y\) is one. 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: \[\underbrace{_\,3x^2}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{1}\end{smallmatrix}} \underbrace{-_\,5x}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{2}\end{smallmatrix}} \underbrace{+_\,4}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{3}\end{smallmatrix}}.\] It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.

Polynomials of small degree have been given specific names. 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, the term \(2x\) in \(x^2 + 2x + 1\) is a linear term in a quadratic polynomial.

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 −1 or \(-\infty\)). The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots. The graph of the zero polynomial, \(f(x) = 0\), is the \(x\)-axis.

In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree \(n\) if all of its non-zero terms have degree \(n\). The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. For example, \(x^3y^2 + 7x^2y^3 - 3x^5\) is homogeneous of degree \(5\). For more details, see homogeneous polynomials.

Condensed: the full section is in Wikipedia.

Polynomials can be added using the associative law of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the commutative law) and combining of like terms. For example, if \[P = 3x^2 - 2x + 5xy - 2\] and \[Q = -3x^2 + 3x + 4y^2 + 8\] then the sum \[P + Q = 3x^2 - 2x + 5xy - 2 - 3x^2 + 3x + 4y^2 + 8\] can be reordered and regrouped as \[P + Q = (3x^2 - 3x^2) + (- 2x + 3x) + 5xy + 4y^2 + (8 - 2)\] and then simplified to \[P + Q = x + 5xy + 4y^2 + 6.\] When polynomials are added together, the result is another polynomial.

Subtraction of polynomials is similar.

Polynomials can also be multiplied. To expand the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other. For example, if \[\begin{align} \color{Red} P \,&\color{Red}{= 2x + 3y + 5} \\ \color{Blue} Q \,&{\color{Blue}{= 2x + 5y + xy + 1}}, \end{align}\] then \[\begin{array}{rccrcrcrcr} {\color{Red}{P}} {\color{Blue}{Q}} & {{=}}&&({\color{Red}{2x}}\cdot{\color{Blue}{2x}}) &+&({\color{Red}{2x}}\cdot{\color{Blue}{5y}})&+&({\color{Red}{2x}}\cdot {\color{Blue}{xy}})&+&({\color{Red}{2x}}\cdot{\color{Blue}{1}}) \\&&+&({\color{Red}{3y}}\cdot{\color{Blue}{2x}})&+&({\color{Red}{3y}}\cdot{\color{Blue}{5y}})&+&({\color{Red}{3y}}\cdot {\color{Blue}{xy}})&+& ({\color{Red}{3y}}\cdot{\color{Blue}{1}}) \\&&+&({\color{Red}{5}}\cdot{\color{Blue}{2x}})&+&({\color{Red}{5}}\cdot{\color{Blue}{5y}})&+& ({\color{Red}{5}}\cdot {\color{Blue}{xy}})&+&({\color{Red}{5}}\cdot{\color{Blue}{1}}) \end{array}\] Carrying out the multiplication in each term produces \[\begin{array}{rccrcrcrcr} PQ & = && 4x^2 &+& 10xy &+& 2x^2y &+& 2x \\ &&+& 6xy &+& 15y^2 &+& 3xy^2 &+& 3y \\ &&+& 10x &+& 25y &+& 5xy &+& 5. \end{array}\] Combining similar terms yields \[\begin{array}{rcccrcrcrcr} PQ & = && 4x^2 &+&( 10xy + 6xy + 5xy ) &+& 2x^2y &+& ( 2x + 10x ) \\ && + & 15y^2 &+& 3xy^2 &+&( 3y + 25y )&+&5 \end{array}\] which can be simplified to \[PQ = 4x^2 + 21xy + 2x^2y + 12x + 15y^2 + 3xy^2 + 28y + 5.\] As in the example, the product of polynomials is always a polynomial.

Given a polynomial \(f\) of a single variable and another polynomial \(g\) of any number of variables, the composition \(f \circ g\) is obtained by substituting each copy of the variable of the first polynomial by the second polynomial. For example, if \(f(x) = x^2 + 2x\) and \(g(x) = 3x + 2\) then \[(f\circ g)(x) = f(g(x)) = (3x + 2)^2 + 2(3x + 2).\] A composition may be expanded to a sum of terms using the rules for multiplication and division of polynomials. The composition of two polynomials is another polynomial.

The division of one polynomial by another is not typically a polynomial. Instead, such ratios are a more general family of objects, called rational fractions, rational expressions, or rational functions, depending on context. This is analogous to the fact that the ratio of two integers is a rational number, not necessarily an integer. For example, the fraction \(1/(x^2+1)\) is not a polynomial, and it cannot be written as a finite sum of powers of the variable \(x\).

For polynomials in one variable, there is a notion of Euclidean division of polynomials, generalizing the Euclidean division of integers. This notion of the division \(a(x)/b(x)\) results in two polynomials, a quotient \(q(x)\) and a remainder \(r(x)\), such that \(a = bq + r\) and \(\deg(r) < \deg(b)\), where \(\deg(p)\) is the degree of \(p\). The quotient and remainder may be computed by any of several algorithms, including polynomial long division and synthetic division.

When the denominator \(b(x)\) is monic and linear, that is, \(b(x) = x-c\) for some constant \(c\), then the polynomial remainder theorem asserts that the remainder of the division of \(a(x)\) by \(b(x)\) is the evaluation \(a(c)\). In this case, the quotient may be computed by Ruffini's rule, a special case of synthetic division.

All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. 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. For example, the factored form of\[5x^3-5\] is \[5(x - 1)\left(x^2 + x + 1\right)\] over the integers and the reals, and \[5(x - 1)\left(x + \frac{1 + i\sqrt{3}}{2}\right)\left(x + \frac{1 - i\sqrt{3}}{2}\right)\] over the complex numbers.

The computation of the factored form, called factorization is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms are available in most computer algebra systems.

Calculating derivatives and integrals of polynomials is particularly simple, compared to other kinds of functions. The derivative of the polynomial \[P = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_2 x^2 + a_1 x + a_0 = \sum_{i=0}^n a_i x^i\] with respect to \(x\) is the polynomial \[n a_n x^{n - 1} + (n - 1)a_{n - 1} x^{n - 2} + \dots + 2 a_2 x + a_1 = \sum_{i=1}^n i a_i x^{i-1}.\] Similarly, the general antiderivative (or indefinite integral) of \(P\) is \[\frac{a_n x^{n + 1}}{n + 1} + \frac{a_{n - 1} x^{n}}{n} + \dots + \frac{a_2 x^3}{3} + \frac{a_1 x^2}{2} + a_0 x + c = c + \sum_{i = 0}^n \frac{a_i x^{i + 1}}{i + 1}\] where \(c\) is an arbitrary constant. For example, antiderivatives of \(x^2 + 1\) have the form \(\frac13x^3 + x + c\).

For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers modulo some prime number \(p\), or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient \(ka_k\) understood to mean the sum of \(k\) copies of \(a_k\). For example, over the integers modulo \(p\), the derivative of the polynomial \(x^p + x\) is the polynomial \(1\).

A polynomial function is a function defined by evaluating a polynomial. More precisely, a function \(f\) of one argument from a given domain is a polynomial function if there exists a polynomial \[a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0\] that evaluates to \(f(x)\) for all x in the domain of \(f\) (here, \(n\) is a non-negative integer and \(a_0, a_1, a_2, \ldots, a_n\) are constant coefficients). Generally, unless otherwise specified, polynomial functions have complex coefficients, arguments, and values. In particular, a polynomial, restricted to have real coefficients, defines a function from the complex numbers to the complex numbers. If the domain of this function is also restricted to the reals, the resulting function is a real function that maps reals to reals.

For example, the function \(f\), defined by \[f(x) = x^3 - x,\] is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in \[f(x,y)= 2x^3+4x^2y+xy^5+y^2-7.\] According to the definition of polynomial functions, there may be expressions that obviously are not polynomials but nevertheless define polynomial functions. An example is the expression \(\left(\sqrt{1-x^2}\right)^2,\) which takes the same values as the polynomial \(1-x^2\) on the interval \([-1,1]\), and thus both expressions define the same polynomial function on this interval.

Every polynomial function is continuous, smooth, and entire.

The evaluation of a polynomial is the computation of the corresponding polynomial function; that is, the evaluation consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and additions.

For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using Horner's method, which consists of rewriting the polynomial as \[(((((a_n x + a_{n-1})x + a_{n-2})x + \dotsb + a_3)x + a_2)x + a_1)x + a_0.\]

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 + a₂x, where a₂ ≠ 0 is a parabola.
• The graph of a degree 3 polynomial f(x) = a₀ + a₁x + a₂x + a₃x, where a₃ ≠ 0 is a cubic curve.
• The graph of any polynomial with degree 2 or greater f(x) = a₀ + a₁x + a₂x + ⋯ + a_nx, where a_n ≠ 0 and n ≥ 2 is a continuous non-linear curve.

A non-constant polynomial function tends to infinity when the variable increases indefinitely (in absolute value). If the degree is higher than one, the graph does not have any asymptote. It has two parabolic branches with vertical direction (one branch for positive x and one for negative x).

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

A polynomial equation, also called an algebraic equation, is an equation of the form \[a_n x^n + a_{n-1}x^{n-1} + \dotsb + a_2 x^2 + a_1 x + a_0 = 0.\] For example, \[3x^2 + 4x - 5 = 0\]is a polynomial equation.

When considering equations, the indeterminates (variables) of polynomials are also called unknowns, and the solutions are the possible values of the unknowns for which the equality is true (in general more than one solution may exist). A polynomial equation stands in contrast to a polynomial identity like \((x+y)(x-y) = x^2 - y^2\), where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality.

In elementary algebra, methods such as the quadratic formula are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the cubic and quartic equations. For higher degrees, the Abel-Ruffini theorem asserts that there can not exist a general formula in radicals. However, root-finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of any degree.

The number of 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.