Inmathematics, acoefficient is amultiplicative factor involved in someterm of apolynomial, aseries, or any other type ofexpression. It may be anumber without units, in which case it is known as anumerical factor.[1] It may also be aconstant withunits of measurement, in which it is known as aconstant multiplier.[1] In general, coefficients may be anyexpression (includingvariables such asa,b andc).[2][1] When the combination of variables and constants is not necessarily involved in aproduct, it may be called aparameter.[1] For example, the polynomial has coefficients 2, −1, and 3, and the powers of the variable in the polynomial have coefficient parameters,, and.
Aconstant coefficient, also known asconstant term or simplyconstant, is a quantity either implicitly attached to thezeroth power of a variable or not attached to other variables in an expression; for example, the constant coefficients of the expressions above are the number 3 and the parameterc, involved in 3=c⋅x0. The coefficient attached to the highest degree of the variable in a polynomial of one variable is referred to as theleading coefficient; for example, in the example expressions above, the leading coefficients are 2 anda, respectively.
In the context ofdifferential equations, these equations can often be written in terms of polynomials in one or more unknown functions and their derivatives. In such cases, the coefficients of the differential equation are the coefficients of this polynomial, and these may be non-constant functions. A coefficient is aconstant coefficient when it is aconstant function. For avoiding confusion, in this context a coefficient that is not attached to unknown functions or their derivatives is generally called aconstant term rather than a constant coefficient. In particular, in alinear differential equation with constant coefficient, the constant coefficient term is generally not assumed to be a constant function.
In mathematics, acoefficient is a multiplicative factor in some term of a polynomial, a series, or any expression. For example, in the polynomialwith variables and, the first two terms have the coefficients 7 and −3. The third term 1.5 is the constant coefficient. In the final term, the coefficient is 1 and is not explicitly written.
In many scenarios, coefficients are numbers (as is the case for each term of the previous example), although they could be parameters of the problem—or any expression in these parameters. In such a case, one must clearly distinguish between symbols representing variables and symbols representing parameters. FollowingRené Descartes, the variables are often denoted byx,y, ..., and the parameters bya,b,c, ..., but this is not always the case. For example, ify is considered a parameter in the above expression, then the coefficient ofx would be−3y, and the constant coefficient (with respect tox) would be1.5 +y.
When one writesit is generally assumed thatx is the only variable, and thata,b andc are parameters; thus the constant coefficient isc in this case.
Anypolynomial in a single variablex can be written asfor somenonnegative integer, where are the coefficients. This includes the possibility that some terms have coefficient 0; for example, in, the coefficient of is 0, and the term does not appear explicitly. For the largest such that (if any), is called theleading coefficient of the polynomial. For example, the leading coefficient of the polynomialis 4. This can be generalised to multivariate polynomials with respect to amonomial order, seeGröbner basis § Leading term, coefficient and monomial.
Theleading entry (sometimesleading coefficient[citation needed]) of a row in a matrix is the first nonzero entry in that row. So, for example, in the matrixthe leading coefficient of the first row is 1; that of the second row is 2; that of the third row is 4, while the last row does not have a leading coefficient.
Though coefficients are frequently viewed asconstants in elementary algebra, they can also be viewed as variables as the context broadens. For example, thecoordinates of avector in avector space withbasis are the coefficients of the basis vectors in the expression
Sabah Al-hadad and C.H. Scott (1979)College Algebra with Applications, page 42, Winthrop Publishers, Cambridge MassachusettsISBN0-87626-140-3 .
Gordon Fuller, Walter L Wilson, Henry C Miller, (1982)College Algebra, 5th edition, page 24, Brooks/Cole Publishing, Monterey CaliforniaISBN0-534-01138-1 .
