Editing Constant term

{{Short description|Term in an algebraic expression which does not contain any variables}}

In [[mathematics]], a '''constant term''' (sometimes referred to as a '''free term''') is a [[Term (logic)|term]] in an [[algebraic expression]] that does not contain any [[Variable (mathematics)|variables]] and therefore is [[constant function|constant]]. For example, in the [[quadratic polynomial]],
:<math>x^2 + 2x + 3,\ </math>
The number 3 is a constant term.<ref>{{cite book |author=Fred Safier |year=2012 |title=Schaum's Outline of Precalculus |url=https://books.google.com/books?id=OxiJgkX4vtMC&dq=%22constant+term%22&pg=PA7 |edition=3rd |publisher=McGraw-Hill Education |page=7 |isbn=978-0-07-179560-9 }}</ref>

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
:<math>ax^2+bx+c,\ </math>
where <math>x</math> is the variable, as having a constant term of <math>c.</math>  If the constant term is 0, then it will conventionally be omitted when the quadratic is written out.

Any [[polynomial]] written in standard form has a unique constant term, which can be considered a [[coefficient]] of <math>x^0.</math>  In particular, the constant term will always be the lowest [[Degree of a polynomial|degree]] term of the polynomial.  This also applies to multivariate polynomials. For example, the polynomial
:<math>x^2+2xy+y^2-2x+2y-4\ </math>
has a constant term of &minus;4, which can be considered to be the coefficient of <math>x^0y^0,</math> 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]] and other types of series, for example in this power series:
:<math>a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots,</math>
<math>a_0</math> is the constant term.

==Constant of integration==
{{main|Constant of integration}}

The [[derivative]] 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]] is only determined up to an unknown constant term, which is called "the constant of integration" and added in symbolic form (usually denoted as <math>C</math>).<ref>{{cite book |author=Arthur Sherburne Hardy |year=1892 |title=Elements of the Differential and Integral Calculus |url=https://books.google.com/books?id=2QgUAAAAYAAJ&dq=%22constant+term%22+%22constant+of+integration%22&pg=PA168 |page=168 |publisher=Ginn & Company}}</ref>

For example, the antiderivative of <math>\cos x</math> is <math>\sin x</math>, since the derivative of <math>\sin x</math> is equal to <math>\cos x</math> based on the [[Differentiation_rules#Derivatives_of_trigonometric_functions|properties of trigonometric derivatives]]. 

However, the ''[[integral]]'' of <math>\cos x</math> is equal to <math>\sin x</math> (the antiderivative), plus an arbitrary constant:

<math display="block"> \int \cos x \, \mathrm dx = \sin x + C,</math>

because for any constant <math>C</math>, the derivative of the right-hand side of the equation is equal to the left-hand side of the equation.

==See also==
* [[Constant (mathematics)]]

==References==
{{reflist}}

{{DEFAULTSORT:Constant yerm}}
[[Category:Polynomials]]
[[Category:Elementary algebra]]