Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of an expression (or of a series); it is usually a number, but in any case does not involve any variables of the expression. For instance in :$7x^2-3xy+1.5+y$ the first three terms respectively have coefficients 7, −3, and 1.5 (in the third term there are no variables, so the coefficient is the term itself; it is called the constant term or constant coefficient of this expression). The final term does not have any explicitly written coefficient, but is usually considered to have coefficient 1, since multiplying by that factor would not change the term. Often coefficients are numbers as in this example, although they could be parameters of the problem, as a, b, and c in :$ax^2+bx+c$ when it is understood that these are not considered as variables.

Thus a polynomial in one variable x can be written as :$a\_k\; x^k\; +\; \backslash cdots\; +\; a\_1\; x^1\; +\; a\_0,$ for some integer k, where a_k, ... a₁, a₀ 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 (if any), a_i is called the leading coefficient of the polynomial. So for example the leading coefficient of the polynomial

:$\backslash ,\; 4x^5\; +\; x^3\; +\; 2x^2$

is 4.

Specific coefficients arise in mathematical identities, such as the binomial theorem which involves binomial coefficients; these particular coefficients are tabulated in Pascal's triangle.

Linear algebra

:

The leading coefficient of the first row is 1; 2 is the leading coefficient of the second row; 4 is the leading coefficient of the third row, and the last row does not have a leading coefficient.

Though coefficients are frequently viewed as constants in elementary algebra, they can be variables more generally. For example, the coordinates $(x\_1,\; x\_2,\; ...,\; x\_n)$ of a vector v in a vector space with basis $\backslash lbrace\; e\_1,\; e\_2,\; ...,\; e\_n\; \backslash rbrace$, are the coefficients of the basis vectors in the expression :$v\; =\; x\_1\; e\_1\; +\; x\_2\; e\_2\; +\; ...\; +\; x\_n\; e\_n\; .$

Examples of physical coefficients

Chemistry

See also

References

Category:Polynomials Category:Mathematical terminology Category:Algebra