Constant term

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In mathematics, a constant term is a term in an algebraic expression that does not contain any variables, and therefore has constant value. For example, in the quadratic polynomial

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

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.

Any polynomial written in standard form has a unique constant term, which can be considered the coefficient of x0. In particular, the constant term is always the lowest degree term of the polynomial. This also applies to multivariate polynomials. For example, the polynomial

has a constant term of −4, which can be considered the coefficient of x0y0. For any polynomial, the constant term can be obtained by substituting in 0 for all of the variables.

The concept can be extended to power series and other types of series: in the power series

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.

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