mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a constant term is a

term Term may refer to: * Terminology, or term, a noun or compound word used in a specific context, in particular: **Technical term, part of the specialized vocabulary of a particular field, specifically: ***Scientific terminology, terms used by scient ...

in an

algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For ex ...

that does not contain any variables and therefore is constant. For example, in the

quadratic polynomial In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomia ...

x^2 + 2x + 3,\

the 3 is a constant term. After

like terms In mathematics, like terms are summands in a sum that differ only by a numerical factor. Like terms can be regrouped by adding their coefficients. Typically, in a polynomial expression, like terms are those that contain the same variables to t ...

are combined, an algebraic expression will have at most one constant term. Thus, it is common to speak of the quadratic polynomial :

ax^2+bx+c,\

where

x

is the variable, as having a constant term of

c.

If the constant term is 0, then it will conventionally be omitted when the quadratic is written out. Any

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

written in standard form has a unique constant term, which can be considered a

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...

x^0.

In particular, the constant term will always be the lowest

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

term of the polynomial. This also applies to multivariate polynomials. For example, the polynomial :

x^2+2xy+y^2-2x+2y-4\

has a constant term of −4, which can be considered to be the coefficient of

x^0y^0,

where the variables are eliminated by being exponentiated to 0 (any non-zero number exponentiated to 0 becomes 1). For any polynomial, the constant term can be obtained by substituting in 0 instead of each variable; thus, eliminating each variable. The concept of exponentiation to 0 can be applied to

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...

and other types of series, for example in this power series: :

a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots,

a_0

is the constant term.

Constant of integration

The

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

of a constant term is 0, so when a term containing a constant term is differentiated, the constant term vanishes, regardless of its value. Therefore the

antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...

is only determined up to an unknown constant term, which is called "the constant of integration" and added in symbolic form.

References

{{DEFAULTSORT:Constant yerm Polynomials Elementary algebra

Constant of integration

See also

References