Polynomial

A polynomial is an expression consisting of a finite sum of terms, each formed by multiplying a coefficient by variables raised to nonnegative integer powers. Polynomials are fundamental objects in algebra and define an important class of functions. They can be studied both as formal expressions and as functions obtained by substituting values for their variables. (openstax.org)

Definition and terminology

A polynomial in one variable has the form

where the coefficients $a_i$ belong to a specified number system or algebraic structure. Common choices include the integers, rational numbers, real numbers, and complex numbers. The symbol $x$ is an indeterminate: its powers are manipulated formally without requiring an assigned numerical value. (jmilne.org)

For a nonzero polynomial, the degree is the largest exponent with a nonzero coefficient. That coefficient is the leading coefficient, and $a_0$ is the constant term. A polynomial with leading coefficient $1$ is monic. Nonzero constants have degree zero; the zero polynomial usually has no degree, although some conventions assign it degree $-\infty$. Degree-one, degree-two, and degree-three polynomials are called linear, quadratic, and cubic, respectively. (openstax.org)

For example, $3x^4-2x+7$ has degree four. By contrast, $x^{-1}$, $x^{1/2}$, and $1/(x+1)$ are not polynomials in $x$, because polynomial exponents must be nonnegative integers. Coefficients themselves may be negative or fractional. An infinite sum of powers is a power series, rather than a polynomial, unless only finitely many coefficients are nonzero. (openstax.org)

Arithmetic and division

Polynomials are added by combining coefficients of matching powers. Multiplication uses the distributive law and $x^ix^j=x^{i+j}$. Sums, differences, and products remain polynomials. Over a field, nonzero polynomials satisfy

Cancellation can lower the degree of a sum; for example, $(x^2+x)+(-x^2+1)=x+1$. (jmilne.org)

The division algorithm states that, for polynomials $P$ and nonzero $D$ over a field, unique polynomials $Q$ and $R$ satisfy

where $R=0$ or $\deg R<\deg D$. Thus division does not generally produce a polynomial quotient without a remainder. Repeated division also extends the Euclidean algorithm to computing greatest common divisors of polynomials. (jmilne.org)

Roots and factorization

A root, or zero, of $P$ is a value $r$ satisfying the equation $P(r)=0$. The factor theorem states that $r$ is a root exactly when $x-r$ divides $P(x)$. The remainder on division by $x-r$ is $P(r)$. A root has multiplicity $m$ when $(x-r)^m$ divides the polynomial but $(x-r)^{m+1}$ does not. (openstax.org)

The fundamental theorem of algebra implies that every degree-$n$ complex polynomial, with $n>0$, factors as

with roots counted according to multiplicity. Consequently, a nonzero degree-$n$ polynomial has at most $n$ distinct roots. For real coefficients, nonreal complex roots occur in conjugate pairs. (openstax.org)

Factorization depends on the coefficient system. For example, $x^2+1$ has no real root but equals $(x-i)(x+i)$ over the complex numbers. A nonconstant polynomial that cannot be expressed as a product of two lower-degree polynomials over its coefficient field is called irreducible. (openstax.org)

Polynomial functions and calculus

A real polynomial defines a function on the entire real line. Its graph is continuous and smooth, without breaks or corners. For a nonconstant polynomial, the leading term determines its behavior as $x$ tends toward positive or negative infinity; the degree’s parity and the leading coefficient’s sign determine the directions of its two ends. (openstax.org)

In calculus, derivatives and integrals are computed term by term:

Both operations produce polynomials, with the antiderivative determined up to an additive constant. (openstax.org)

Several variables and algebraic structure

A polynomial may involve several indeterminates, as in $2x^2y-3xy+5$. Its total degree is the greatest sum of exponents in any nonzero term: this example has total degree three. Systems of polynomial equations define geometric objects studied in algebraic geometry. (jmilne.org)

In abstract algebra, polynomials with coefficients in a commutative ring $R$ form the polynomial ring $R[x]$. Over a field, they also form a vector space, with basis $1,x,x^2,\ldots$. Formal expressions must be distinguished from their evaluated functions: over the two-element finite field, the nonzero polynomial $x^2-x$ evaluates to zero at both elements. This follows directly by substituting $0$ and $1$. (jmilne.org)

Interpolation, approximation, and computation

Polynomial interpolation constructs a polynomial passing through specified data points. Given $n+1$ distinct real input values and corresponding outputs, exactly one polynomial of degree at most $n$ interpolates them. Lagrange and Newton forms provide alternative representations of this polynomial. (dlmf.nist.gov)

In approximation theory, the Weierstrass approximation theorem states that every continuous real-valued function on a closed, bounded interval can be approximated uniformly, to arbitrary accuracy, by polynomials. This does not mean that every continuous function equals a convergent power series. (jirilebl.github.io)

For numerical evaluation, Horner’s method rewrites a polynomial in nested form:

For degree $n$, this evaluates the polynomial using $n$ multiplications and $n$ additions, without computing each power separately. (dlmf.nist.gov)