Polynomials are among the few functions that can be understood from several viewpoints at once. Algebraically, a polynomial is a finite linear combination of powers of a variable. Analytically, it is a smooth function whose local behaviour is controlled by finitely many coefficients. Geometrically, the graph of a real polynomial is a curve with a rigid large-scale shape. This mixture of algebraic finiteness and analytic flexibility is why polynomials appear throughout [Function](/page/Function), [Derivative](/page/Derivative), [Limit](/page/Limit), [Continuity](/page/Continuity), [approximation theory](/page/Approximation%20Theory), and differential equations.

The finiteness condition is the central point. Infinite series such as [power series](/page/Power%20Series) can model a much larger class of functions, but they require convergence questions before they define functions. A polynomial does not: after choosing a coefficient ring or field and a variable, only finitely many arithmetic operations are involved. That makes polynomials the first robust class of functions in which algebraic operations, roots, derivatives, integrals, and asymptotic behaviour can be studied together.

## Definition

Before treating polynomials as functions, it is useful to separate the symbolic object from the function it may define. The expression $x^2-1$ is not merely a rule on [real numbers](/page/Real%20Numbers); it can be manipulated over $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, or a more general [Ring](/page/Ring). This distinction matters because different coefficient systems can change factorisation, roots, and even whether two polynomial expressions define the same function. The formal definition records exactly the finite coefficient data that survives all these changes of setting.

[definition: Polynomial] Let $R$ be a commutative ring with identity. A polynomial in one variable $x$ with coefficients in $R$ is a formal finite sum \begin{align*} p = \sum_{i=0}^{n} a_i x^i, \end{align*} where $n \in \mathbb{N} \cup \{0\}$, each $a_i \in R$, and all but finitely many coefficients are zero. The set of all such polynomials is denoted $R[x]$. [/definition]

[example: A Quadratic as a Formal Polynomial] In $\mathbb{R}[x]$, the expression \begin{align*} p=x^2-3x+2 \end{align*} is the finite coefficient list $a_0=2$, $a_1=-3$, $a_2=1$, and $a_i=0$ for all $i\ge 3$. Its degree is $2$, and its leading coefficient is $1$, so it is monic. When it is evaluated as a real function, the input $t=3$ gives \begin{align*} \tilde p(3)=3^2-3\cdot 3+2=2. \end{align*} [/example]

The phrase "formal finite sum" means that the coefficients are part of the data. Addition and multiplication are defined by collecting coefficients, rather than by evaluating at a point. The notation $p(x)$ is often used informally for such an element of $R[x]$, but evaluation at a specific input is a separate operation introduced below. Multivariable polynomials are the standard extension obtained by allowing finitely many monomials in variables $x_1, \dots, x_m$; this page focuses on the one-variable case because degree, roots, and polynomial functions already show the essential issues there. To compare polynomials by complexity, to run induction on them, or to state how roots constrain them, we need to know the largest power whose coefficient is actually present.

[definition: Degree of a Polynomial] Let $R$ be a commutative ring with identity, and let \begin{align*} p(x) = \sum_{i=0}^{n} a_i x^i \in R[x]. \end{align*} If $p \ne 0$, the degree of $p$ is \begin{align*} \deg p = \max\{i \in \{0, \dots, n\} : a_i \ne 0\}. \end{align*} [/definition]

To predict division, factorisation, and growth at infinity, a polynomial needs a distinguished top coefficient as well as a highest power. That coefficient is the part of the polynomial that survives after all lower powers become negligible or after a [division algorithm](/theorems/725) normalises the top term.

[definition: Leading Coefficient] Let $R$ be a commutative ring with identity, and let $p \in R[x]$ be nonzero. If $\deg p = d$ and \begin{align*} p(x) = \sum_{i=0}^{d} a_i x^i \end{align*} with $a_d \ne 0$, then $a_d$ is the leading coefficient of $p$. [/definition]

To compare polynomials up to multiplication by an invertible scalar, it is useful to choose a standard representative whenever possible. The simplest normalisation is to require the leading coefficient to be the multiplicative identity.

[definition: Monic Polynomial] Let $R$ be a commutative ring with identity. A nonzero polynomial $p \in R[x]$ is monic if its leading coefficient is $1_R$. [/definition]

In analysis, a polynomial is often used as a function on $\mathbb{R}$ or $\mathbb{C}$. That requires evaluation: a formal expression becomes a rule that sends an input to the corresponding finite arithmetic combination of powers. The next definition names this functional interpretation while keeping it separate from the formal polynomial.

[definition: Polynomial Function] Let $k$ be a field, and let $p \in k[x]$ be given by \begin{align*} p(x) = \sum_{i=0}^{n} a_i x^i. \end{align*} The polynomial function associated to $p$ is the map $\tilde{p}: k \to k$ defined by \begin{align*} \tilde{p}(t)=\sum_{i=0}^{n} a_i t^i. \end{align*} [/definition]

Over infinite fields such as $\mathbb{R}$ and $\mathbb{C}$, distinct formal polynomials define distinct polynomial functions. Over finite fields this may fail, so the symbolic and functional viewpoints should not be silently identified in every setting.

## Equivalent Characterisations

For real or complex analysis, the defining finite sum is often the most convenient form. Another way to recognise polynomials is through their derivatives: after enough differentiations, a polynomial becomes zero. This reformulation connects polynomials to [Taylor's theorem](/theorems/827) and to local approximation by finite jets.

[quotetheorem:8293]

This result explains why polynomials are the model finite-order smooth functions. A general smooth function can have derivatives of every order that carry new information, while a polynomial stores all of its information in finitely many derivatives at a single point. To use this as a test for whether a smooth function is polynomial, we need a converse statement on domains where local information cannot split into unrelated pieces.

[quotetheorem:8294]