Coefficient Definition, Types & Examples Video & Lesson Transcript

The constant coefficient (or constant term) 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.

If a variable is written without a coefficient, then it is assumed to have a coefficient of 1; frequently, when the coefficient is 1, it is not written. There is no coefficient presented with the variable, so the term x has a coefficient of 1. A coefficient is any number that is being multiplied to a variable. The variable is the unknown symbol while the number being multiplied to it is called the coefficient. These examples are programmatically compiled from various online sources to illustrate current usage of the word ‘coefficient.’ Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors.

Other Word Forms of Coefficient

In this lesson, the definition of coefficient was examined in depth. In short, a coefficient is any numerical value or number, that is being multiplied by a variable or variables. For example, it can be real, imaginary, positive, negative, integer, fractional, whole, decimal, rational, or irrational. Many numbers can appear in an equation but any groupings of numbers, variables, or both are considered a term. Any number by itself is called a constant because it remains unchanging and is not being multiplied by a variable. The variable was defined as the symbol that stands for an unknown number.

In the context of differential equations, an equation can often be written as equating to zero a polynomial in the unknown functions and their derivatives. In this case, the coefficients of the differential equation are the coefficients of this polynomial, and are generally non-constant functions. A coefficient is a constant coefficient when it is a constant function. For avoiding confusion, the coefficient that is not attached to unknown functions and their derivative is generally called the constant term rather the constant coefficient. In particular, in a linear differential equation with constant coefficient, the constant term is generally not supposed to be a constant function.

More from Merriam-Webster on coefficient

For example, in the expressions above, the leading coefficients are 2 and a, respectively. 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. Coefficients are located directly in front of the variable in a term. A leading coefficient is the coefficient of the term containing the highest power of x. In a polynomial equation, the highest power of x usually comes first which is why its coefficient is called leading. It is important to note that all variables will have a coefficient.