# Constant term

Constant term

In mathematics, a constant term is a term in an algebraic expression has a value that is constant or cannot change, because it does not contain any modifiable variables. For example, in the quadratic polynomial

$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, and has a constant term of c. If c = 0, then the constant term will not actually appear when the quadratic is written.

It is notable that a term that is constant, with a constant as a multiplicative coefficient added to it (although this expression could be more simply written as their product), still constitutes a constant term as a variable is still not present in the new term. Although the expression is modified, the term (and coefficient) itself classifies as constant. However, should this introduced coefficient contain a variable, while the original number has a constant meaning, this has no bearing if the new term stays constant as the introduced coefficient will always override the constant expression - for example, in (x + 1)(x − 2) when x is multiplied by 2, the result, 2x, is not constant; while 1 * -2 is -2 and still a constant.

Any polynomial written in standard form has a unique constant term, which can be considered a coefficient of x0. 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 x0y0, where the variables are become eliminated by exponentiated to 0 (any 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 extended to power series and other types of series, for example in this power series:

$a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots,$

a0 is the constant term. In general a constant term is one that does not involve any variables at all. However in expressions that involve terms with other types of factors than constants and powers of variables, the notion of constant term cannot be used in this sense, since that would lead to calling "4" the constant term of (x − 3)2 + 4, whereas substituting 0 for x in this polynomial makes it evaluate to 13.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Constant — or The Constant may refer to: Contents 1 In Mathematics 2 Other concepts 3 People 4 Organization 5 …   Wikipedia

• Constant (mathematics) — In mathematics, a constant is a non varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable (i.e. variable quantity), which is a symbol that stands for a value that may vary. In… …   Wikipedia

• constant of integration — noun An unspecified constant term added to a particular antiderivative to make it represent its whole family of antiderivatives …   Wiktionary

• Constant Tonegaru — Constant (Constantin) Tonegaru Born February 26, 1919(1919 02 26) Galaţi Died February 10, 1952(1952 02 10) (aged 32) Bucharest Occupation poet, journalist, activist, civil servant …   Wikipedia

• Constant de Kerchove de Denterghem — Born Constant de Kerchove de Denterghem 31 December 1790(1790 12 31) Ghent, Belgium Died 12 July 1865(1865 07 12) (aged 74) Wondelgem, Belgium …   Wikipedia

• Constant Troyon — (August 28, 1810 February 21, 1865), French painter, was born in Sèvres, near Paris, where his father was connected with the famous manufactory of porcelain. Troyon was an animal painter of the first rank, and was closely associated with the… …   Wikipedia

• Constant visual observation — Constant visual observation, often abbreviated to constant visual , is a term used in various Mental Health Services, Prisons and Special Schools to describe the status of a prisoner or patient who poses a threat to himself or a third party, and… …   Wikipedia

• Constant current — is a term most often used in electronics to describe a system that can vary the voltage across an electronic circuit to maintain a constant electric current. When a component is indicated to be driven by a constant current, the driving circuit in …   Wikipedia

• Constant bitrate — (CBR) is a term used in telecommunications, relating to the quality of service. Compare with variable bitrate. When referring to codecs, constant bit rate encoding means that the rate at which a codec s output data should be consumed is constant …   Wikipedia

• Constant bearing, decreasing range — (CBDR) means that some object, usually another ship viewed from the deck or bridge of one s own ship, is getting closer but maintaining the same true bearing. If this continues, the objects will collide. Contents 1 Lack of visual clues 2 Relative …   Wikipedia