Inmathematics, aconstant term (sometimes referred to as afree term) is aterm in analgebraic expression that does not contain anyvariables and therefore isconstant. For example, in thequadratic polynomial,

${\displaystyle x^2+2x+3,}$

The number 3 is a constant term.^[1]

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

${\displaystyle a{x}^2+bx+c,}$

where${\displaystyle x}$ is the variable, as having a constant term of${\displaystyle c.}$ If the constant term is 0, then it will conventionally be omitted when the quadratic is written out.

Anypolynomial written in standard form has a unique constant term, which can be considered acoefficient of${\displaystyle x^0.}$ In particular, the constant term will always be the lowestdegree term of the polynomial. This also applies to multivariate polynomials. For example, the polynomial

${\displaystyle x^2+2xy+y^2-2x+2y-4}$

has a constant term of −4, which can be considered to be the coefficient of${\displaystyle x^0{y}^0,}$ 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 topower series and other types of series, for example in this power series:

${\displaystyle a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\cdots ,}$

${\displaystyle a_0}$ is the constant term.

Thederivative 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 theantiderivative is only determined up to an unknown constant term, which is called "the constant of integration" and added in symbolic form (usually denoted as${\displaystyle C}$).^[2]

For example, the antiderivative of${\displaystyle \cos x}$ is${\displaystyle \sin x}$, since the derivative of${\displaystyle \sin x}$ is equal to${\displaystyle \cos x}$ based on theproperties of trigonometric derivatives.

However, theintegral of${\displaystyle \cos x}$ is equal to${\displaystyle \sin x}$ (the antiderivative), plus an arbitrary constant:

$${\displaystyle \int \cos x\mathrm{d}x=\sin x+C,}$$

because for any constant${\displaystyle C}$, the derivative of the right-hand side of the equation is equal to the left-hand side of the equation.

• Constant (mathematics)

• Polynomials
• Elementary algebra

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