In mathematics, a constant term is a term 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). ...

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 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 ex ...

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 ...

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 mathemati ...

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 con ...

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. ...

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