Coefficient

A coefficient is a numerical or symbolic factor that multiplies a variable or term in an algebraic expression. In the term 5x^2, the coefficient is 5, and in -3x it is -3. Coefficients can be integers, fractions, or real numbers, and they determine the size or scale of the associated variable.

In polynomials, coefficients are the numbers that multiply the powers of the variable. For example, in 4x^3 − x^2 + 7x + 6, the coefficients are 4, -1, 7, and 6. The leading coefficient is the coefficient of the term with the highest degree (4 in this example). The constant term is the coefficient of x^0, which is 6.

The term coefficient is also used in many other fields. In statistics, the correlation coefficient measures the strength of a linear relationship between two variables. In physics and engineering, coefficients such as the drag coefficient, friction coefficient, and thermal expansion coefficient describe material or system properties. In chemistry, rate coefficients appear in expressions that describe how quickly reactions proceed, often dependent on temperature and other conditions.

Etymology traces the word to Latin coefficientem, from com- “together” and facere “to make,” referring to something that accompanies or multiplies another quantity. In mathematics, coefficients help express linear relationships, scale variables, and convey essential information about the structure of expressions and models.