mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a constant term (sometimes referred to as a free term) is a term in an

algebraic expression In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers), variables, and the basic algebraic operations: addition (+), subtraction (-), multiplication (×), division (÷), whole number pow ...

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

quadratic polynomial In mathematics, a quadratic function of a single variable is a function of the form :f(x)=ax^2+bx+c,\quad a \ne 0, where is its variable, and , , and are coefficients. The expression , especially when treated as an object in itself rather tha ...

, :

x^2 + 2x + 3,\

The number 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 a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

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

coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...

x^0.

In particular, the constant term will always be the lowest degree 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 ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

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 is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

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 continuous function is a differentiable function whose derivative is equal to the original function . This can be stated ...

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

C

). For example, the antiderivative of

\cos x

\sin x

, since the derivative of

\sin x

is equal to

\cos x

based on the properties of trigonometric derivatives. However, the ''

integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...

'' of

\cos x

is equal to

\sin x

(the antiderivative), plus an arbitrary constant:

\int \cos x \, \mathrm dx = \sin x + C,

because for any constant

C

, the derivative of the right-hand side of the equation is equal to the left-hand side of the equation.

References

{{DEFAULTSORT:Constant yerm Polynomials Elementary algebra

Constant of integration

See also

References