Editing Constant term (section)

{{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.