Coefficient

Summary

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as a, b and c).[1] When the coefficients are themselves variables, they may also be called parameters.

For example, the polynomial has coefficients 2, −1, and 3, and the powers of the variable in the polynomial have coefficient parameters , , and .

The constant coefficient is the coefficient not attached to variables in an expression. For example, the constant coefficients of the expressions above are the number 3 and the parameter c, respectively. The coefficient attached to the highest degree of the variable in a polynomial is referred to as the leading coefficient. For example, in the expressions above, the leading coefficients are 2 and a, respectively.

Terminology and definitionEdit

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression. For example, in the polynomial

 
with 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. Following René Descartes, the variables are often denoted by x, y, ..., and the parameters by a, b, c, ..., but this is not always the case. For example, if y is considered a parameter in the above expression, then the coefficient of x would be −3y, and the constant coefficient (with respect to x) would be 1.5 + y.

When one writes

 
it is generally assumed that x is the only variable, and that a, b and c are parameters; thus the constant coefficient is c in this case.

Any polynomial in a single variable x can be written as

 
for some nonnegative 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 the leading coefficient of the polynomial. For example, the leading coefficient of the polynomial
 
is 4. This can be generalised to multivariate polynomials with respect to a monomial order, see Gröbner basis § Leading term, coefficient and monomial.

Linear algebraEdit

In linear algebra, a system of linear equations is frequently represented by its coefficient matrix. For example, the system of equations

 
the associated coefficient matrix is   Coefficient matrices are used in algorithms such as Gaussian elimination and Cramer's rule to find solutions to the system.

The leading entry (sometimes leading coefficient[citation needed]) of a row in a matrix is the first nonzero entry in that row. So, for example, in the matrix

 
the 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 as constants in elementary algebra, they can also be viewed as variables as the context broadens. For example, the coordinates   of a vector   in a vector space with basis   are the coefficients of the basis vectors in the expression

 

See alsoEdit

ReferencesEdit

  1. ^ Weisstein, Eric W. "Coefficient". mathworld.wolfram.com. Retrieved 2020-08-15.

Further readingEdit

  • Sabah Al-hadad and C.H. Scott (1979) College Algebra with Applications, page 42, Winthrop Publishers, Cambridge Massachusetts ISBN 0-87626-140-3 .
  • Gordon Fuller, Walter L Wilson, Henry C Miller, (1982) College Algebra, 5th edition, page 24, Brooks/Cole Publishing, Monterey California ISBN 0-534-01138-1 .