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Polynomial (mathematics): definition, structure, uses and properties

A polynomial is an algebraic expression built from variables and coefficients using only addition, subtraction, multiplication and whole-number exponents. Overview, structure, history and applications.

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Overview

A polynomial is an algebraic expression formed by summing one or more terms each of which is a constant multiplied by a variable raised to a whole-number exponent. Terms of a polynomial are often called monomials. Polynomials appear throughout mathematics and its applications: as formulas to describe shapes and motion, as objects of study in algebra, and as building blocks for approximation in numerical analysis. For a basic treatment see polynomial resources and introductions at elementary and collegiate levels.

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Structure and basic terminology

Key parts of a polynomial include coefficients, variables (or indeterminates), terms, and the degree. A general single-variable polynomial in x has the form a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where the a_i are coefficients and n is a nonnegative integer called the degree when a_n ≠ 0. Terms are combined by addition; multiplication may produce higher-degree terms but only nonnegative integer exponents are permitted. If division by a variable, fractional exponents, or operations like roots appear, the expression is no longer a polynomial. For definitions and examples, consult introductory algebra.

Operations and algebraic properties

Polynomials are closed under addition, subtraction and multiplication: combining two polynomials yields another polynomial. Division of polynomials may produce a quotient and remainder; the quotient is a polynomial only when the divisor fits evenly into the dividend. Important algebraic ideas include factorization (expressing a polynomial as a product of lower-degree polynomials), the remainder theorem, and the relation between coefficients and roots. Over the complex numbers every nonconstant single-variable polynomial factors completely into linear factors, a statement closely related to the Fundamental Theorem of Algebra; for further reading see factorization and related theory.

History and development

The study of polynomial expressions stretches back to ancient civilizations that solved linear and quadratic problems. Systematic symbolic notation and general methods for manipulating polynomials developed in medieval and early modern algebra. Over time polynomials became central to the development of equations, polynomial rings, and modern algebraic structures. For historical context and classic sources, follow historical accounts or textbooks that track the subject's evolution.

Uses, examples and importance

Polynomials model many practical relationships because they are easy to compute and differentiate. In physics and engineering low-degree polynomials approximate motion, fit measurement data (polynomial regression), and appear in control systems. In numerical analysis polynomials are used for interpolation and polynomial approximation of more complicated functions; see approximation methods. In algebraic geometry, varieties defined by polynomial equations connect algebra and geometry; introductory material is available at algebraic geometry basics. Concrete examples include quadratic polynomials (used for projectile motion), cubic polynomials (used in curve fitting), and higher-degree polynomials used in signal processing and coding theory.

Distinctions and notable facts

  • Polynomial vs rational expression: a rational expression permits division by polynomials, producing quotients that need not be polynomials. See rational expressions.
  • Power series and analytic functions may resemble polynomials locally but include infinitely many terms; these are distinct concepts discussed in analysis texts at power series.
  • Roots (zeros) of polynomials are values of the variable that make the polynomial equal to zero. The number and nature of roots depend on the field in which one works; complex roots and multiplicities are treated in standard algebra sources such as root theory.
  • Polynomials with coefficients in a ring form an algebraic structure called a polynomial ring; this is a central object in abstract algebra and commutative algebra; see polynomial rings.

Polynomials combine conceptual simplicity with practical power: they are computationally convenient, serve as approximations to more complicated functions, and lead to rich theoretical frameworks across algebra, analysis, and geometry. Whether one studies their arithmetic, factorization, or application to modeling, polynomials remain fundamental tools in mathematics.

Etymology

The word polynomial means something like "multi-named". It comes from the Greek πολύ polý "much" and όνομα onoma "name". This name goes back to Euclid's Elements. In Book X he calls a two-membered sum a+b δύο ὀνομάτων (ek dýo onomátōn): "consisting (of) two names". The term polynomial goes back to Viëta: in his Isagoge (1591) he uses the expression polynomia magnitudo for a multimember quantity.

Polynomials in abstract algebra

→ Main article: Polynomial ring

Definition

In abstract algebra one defines a polynomial as an element of a polynomial ring R[X]. This in turn is the extension of the coefficient ring Rby an indefinite, (algebraically) free element X. Thus contains R[X]the powers X^{n}, n\in \mathbb {N} and their linear combinations {\displaystyle \textstyle a_{0}+\sum _{k=1}^{n}a_{k}X^{k}}with a_k\in R. These are also already all elements, i.e., each polynomial is uniquely defined by the sequence

(a_0,a_1,\dots,a_n,0,0,\dots)\in R\times R\times R\times\dots

of its coefficients.

Construction

Conversely, a model of the polynomial ring can be R\times R\times R\times\dotsconstructed R[X]by the set of finite sequences in For this purpose, on R[X]an addition " +" is defined as a limb-wise sum of the sequences and a multiplication " \cdot " is defined by convolution of the sequences. Thus, if a=(a_n)_{n\in \N_0}and b=(b_n)_{n\in \N_0}, then

a+b:=(a_n+b_n)_{n\in \N_0}

and

a\cdot b:=\left(\sum_{i=0}^{n} a_ib_{n-i}\right)_{n\in \N_0}=\left(\sum_{i+j=n} a_ib_j\right)_{n\in \N_0},

Now R[X]with these links is itself a commutative ring, the polynomial ring (in an indefinite) over R.

Identifying the indefinite as a sequence , so that X^2=X\cdot X=(0,0,1,0,0,\dotsc), X^3=X^2\cdot X=(0,0,0,1,0,0,\dotsc)etc., so any sequence can (a_0,a_1,a_2,\dotsc)\in R[X]again be represented in the intuitive sense as a polynomial as

(a_0,a_1,a_2,\dotsc) = a_0 + a_1\cdot X + a_2\cdot X^2 + \dotsb = a_0+\sum_{n\in\N_{>0}} a_n\cdot X^n.

Connection with the analytical definition

Now consider that, according to the premise, there exists a natural number n\in \N_0exists such that a_{i}=0for all i>n, then according to the above considerations, any polynomial f\in R[X]over a commutative unitary ring can be written uniquely as f=a_0 + a_1\cdot X + \dotsb + a_n\cdot X^n. fHowever, is not a function as in analysis or elementary algebra, but an infinite sequence (an element of the ring R[X]) and Xis not an "unknown", but the sequence (0,1,0,0,\dotsc). However, one can use fas a "pattern" to form a polynomial function (i.e., a polynomial in the ordinary analytic sense) afterwards. For this one uses the so-called substitution homorphism.

Note, however, that different polynomials can induce the same polynomial function. For example, if Rthe residue class ring \mathbb Z/3\mathbb Z=\{\bar0,\bar1,\bar2\}, then the polynomials induce {\displaystyle f,g \in (\mathbb Z/3\mathbb Z)[X]

f=X(X-\bar1)(X-\bar2)=X^3-\bar3X^2+\bar2X=X^3-X

and

the zero polynomial g=0

both the zero mapping 0\in \operatorname{Abb}\left(\mathbb Z/3\mathbb Z,\mathbb Z/3\mathbb Z\right), that is: f(x)=g(x)=\bar 0=0(x)for all x\in\mathbb Z/3\mathbb Z.

However, for polynomials over the real or integers, or in general any infinite integrity ring, a polynomial is determined by the induced polynomial function.

The set of polynomial functions with values in also Rforms a ring (subring of the function ring), but this is rarely considered. There is a natural ring homomorphism from R[X]into the ring of polynomial functions whose kernel is the set of polynomials inducing the zero function.

Generalizations

Polynomials in several indefinites

general, any sum of monomials of the form a_{i_1,\dotsc,i_n}X_1^{i_1}\dotsm X_n^{i_n}is understood to be a polynomial (in several indefinites):

P(X_1, \dotsc, X_n) = \sum_{i_1,\dotsc,i_n}a_{i_1,\dotsc,i_n}X_1^{i_1}\dotsm X_n^{i_n}

Read: "Large-p from large-x-1 to large-x-n (is) equal to the sum over all i-1 to i-n from a-i-1-to-l-n times large-x-1 to the power of i-1 to large-x-n to the power of i-n."

By a monomial ordering it is possible to arrange the monomials in such a polynomial and thus generalize terms like leading coefficient.

The quantity i_1+\dotsb+i_nis called the total degree of a monomial X_1^{i_1}\dotsm X_n^{i_n}. If all (nonvanishing) monomials in a polynomial have the same total degree, it is called homogeneous. The maximum total degree of all non-vanishing monomials is the degree of the polynomial.

The maximum number of possible monomials of a given degree is

\binom{n+k-1}{k},

Read: "n+k-1 over k" or "k from n+k-1".

where nis the number of occurring indeterminates and kthe degree. Illustratively, here we consider a problem of combinations with repetition (reclining).

Summing the number of possible monomials of degree {\displaystyle 0}to we kget the number of possible monomials in a polynomial of a certain degree:

\binom{n+k}{k}

Read: "n+k over k" or "k from n+k".

If all indeterminates are in a certain way "equal", the polynomial is called symmetric. What is meant is: if the polynomial does not change when the indeterminates are interchanged.

Also, the polynomials in the nindefinites X_1, \dotsc, X_nover the ring Rform a polynomial ring written as R[X_1, \dotsc, X_n].

Formal power series

Going to infinite series of the form

f = \sum_{i=0}^\infty a_i X^i

Read: "f (is) equal to the sum of i equals zero to infinity of a-i (times) (upper case) x to the power of i."

over, one obtains formal power series.

Laurent polynomials and Laurent series

If one also allows negative exponents in a polynomial, one obtains a Laurent polynomial. Corresponding to the formal power series, formal Laurent series can also be considered. These are objects of the form

{\displaystyle f=\sum _{i=-N}^{\infty }a_{i}X^{i}.}
 Read: "f (is) equal to the sum of i equals minus (upper case) n to infinity of a-i (times) (upper case) x to the power of i."

Posynomial functions

If one allows several variables and arbitrary real powers, one obtains the notion of a posynomial function.

Questions and answers

Q: What is a polynomial?

A: A polynomial is a kind of mathematical expression that is a sum of several mathematical terms called monomials, which are numbers, variables, or products of numbers and several variables.

Q: How do mathematicians, scientists and engineers use polynomials?

A: Mathematicians, scientists and engineers all use polynomials to solve problems.

Q: What operations can be used in an algebraic expression to make it a polynomial?

A: In order for an algebraic expression to be considered a polynomial, the only arithmetic operations that can be used are addition, subtraction, multiplication and whole number exponentiation. If harder operations such as division or square roots are used then the algebraic expression is not considered a polynomial.

Q: What type of equations can be formed using polynomials?

A: Polynomials are often used to form both polynomial equations (such as 7x^4(-3)x^3+19x^2(-8)x+197=0) and also polynomial functions (such as f(x)=7x^4(-3)x^3+19x^2(-8)x+197).

Q: What subject does one need to understand in order to work with polynomials?

A: In order to work with polynomials one needs to understand algebra which is a gateway course into all technical subjects.

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