coefficients

Coefficients are numerical factors that multiply variables or terms in algebraic expressions. They indicate how much of a given variable is present in each term, and can be integers, fractions, or more general numbers. In a polynomial, p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, the numbers a_n, a_{n-1}, ..., a_0 are the coefficients of the corresponding powers of x. The leading coefficient is a_n, and the constant term is a_0. The degree of the polynomial is the highest power with a nonzero coefficient.

In systems of linear equations, coefficients accompany the variables, forming the coefficient matrix. These coefficients determine how the variables contribute to each equation and influence the solution set.

In statistics, coefficients measure relationships between variables. A regression coefficient quantifies the expected change in the dependent variable per unit change in a predictor, while the correlation coefficient (often denoted r) measures the strength and direction of a linear relationship. Standardized coefficients are unitless and allow comparison across variables.

In physics and engineering, coefficients describe proportional relationships between physical quantities, such as the drag coefficient, friction coefficient, or heat transfer coefficient. In chemistry, stoichiometric coefficients indicate the relative amounts of reactants and products in a balanced chemical equation.

Across disciplines, coefficients serve as multipliers or proportionality factors that translate a variable into a measurable contribution within a formula or model.