Constant term
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In mathematics, the constant term of a polynomial is the term of degree 0. For example, in the polynomial
- X3 + 2X + 3
over the variable X, the constant term is 3. Here, the constant term is given by a numeral, but it may also be specified by a letter that is a parameter rather than a variable, as in the polynomial
- ax2 + bx + c,
in the variable x, where a, b, and c are parameters so that c is the constant term. Since the constant term contains no nonzero powers of any variables, the constant term can also be descibed as the coefficient of the term of degree 0, which can be made explicit by expressing the term of degree 0 in the form cx0.
In a polynomial like x4 + 2x2, in which no term of degree 0 is apparent, it is taken to be 0x0, or simply 0.
While these examples are for the simple case of a univariate polynomial (a polynomial in one variable), the definition equally applies to a multivariate polynomial. For example, the polynomial
- X2 + 2XY + Y2 − 2X + 2Y − 4
in the variables X and Y has constant term −4. In general, the constant term of a polynomial in a set of variables can be obtained by substituting 0 for all variables in that set.
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.
