Coefficient

For other uses, see Coefficient (disambiguation).

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression (including variables such as a, b and c). When variables appear in the coefficients, they are often called parameters, and must be clearly distinguished from those representing other variables in an expression.

For example,

$2x^{2}-x+3$, has the real coefficients 2, -1, and 3 respectively, and

$ax^{2}+bx+c$, has coefficient parameters a, b, and c respectively assuming x is the variable of the equation.

The constant coefficient is the coefficient not attached to variables in an expression. For example, the constant coefficients of the expressions above are the real coefficient 3 and the parameter represented by c.

Similarly, the coefficient attached to the highest multiplicity 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 the parameter represented by a.

The binomial coefficients occur in the expanded form of $(x+y)^{n}$, and are tabulated in Pascal's triangle.

Terminology and Definition

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression;

For example, in

$7x^{2}-3xy+1.5+y,$

the first two terms have the coefficients 7 and −3, respectively. The third term 1.5 is a constant coefficient. The final term does not have any explicitly-written coefficient factor that would not change the term; the coefficient is thus taken to be 1 (since variables without number have a coefficient of 1).

In many scenarios, coefficients are numbers (as is the case for each term of the above 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 (always with respect to x) would be 1.5 + y.

When one writes

$ax^{2}+bx+c,$

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.

Similarly, any polynomial in one variable x can be written as

$a_{k}x^{k}+\dotsb +a_{1}x^{1}+a_{0}$

for some positive integer $k$, where $a_{k},\dotsc ,a_{1},a_{0}$ are coefficients; to allow this kind of expression in all cases, one must allow introducing terms with 0 as coefficient. For the largest $i$ with $a_{i}\neq 0$ (if any), $a_{i}$ is called the leading coefficient of the polynomial. For example, the leading coefficient of the polynomial

$\,4x^{5}+x^{3}+2x^{2}$

is 4.

Some specific coefficients that occur frequently in mathematics have dedicated names. For example, the binomial coefficients occur in the expanded form of $(x+y)^{n}$, and are tabulated in Pascal's triangle.

Linear algebra

In linear algebra, a system of linear equations is associated with a coefficient matrix, which is used in Cramer's rule to find a solution to the system.

The leading entry (sometimes leading coefficient) of a row in a matrix is the first nonzero entry in that row. So, for example, given the matrix described as follows:

${\begin{pmatrix}1&2&0&6\\0&2&9&4\\0&0&0&4\\0&0&0&0\end{pmatrix}},$

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 $(x_{1},x_{2},\dotsc ,x_{n})$ of a vector $v$ in a vector space with basis $\lbrace e_{1},e_{2},\dotsc ,e_{n}\rbrace$, are the coefficients of the basis vectors in the expression

$v=x_{1}e_{1}+x_{2}e_{2}+\dotsb +x_{n}e_{n}.$

See also

• Correlation coefficient
• Degree of a polynomial
• Monic polynomial

Further reading

• 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 .