Coefficient
In mathematics, a coefficient is an integer which is multiplied with the variable of a single term or the terms of a polynomial. It is usually a number, but sometimes may be replaced by a letter in an expression. For example, in the expression: ax^{2} + bx + c, x is the variable and 'a' and 'b' are the coefficients.
What is a Coefficient?
A coefficient refers to a number or quantity placed with a variable. It is usually an integer that is multiplied by the variable next to it. The variables which do not have a number with them are assumed to be having 1 as their coefficient. For example, in the expression 3x, 3 is the coefficient. In other words, a coefficient is a multiplicative factor in the terms of a polynomial, a series, or any expression. It is generally a number. Observe the following expression which shows that 5 is the coefficient of x^{2} and 8 is the coefficient of y.
How to Find a Coefficient?
In order to find a coefficient, the most important thing that should be remembered is that it always comes with the variable. Let us understand this with the help of the example: 5x^{2}+2y7. In this expression, we can see that there are three terms: 5x^{2}, 2y, 7. In the first term 5x^{2}, x is the variable and as we know that a coefficient always comes with the variable, the coefficient is 5. In the second term 2y, y is the variable, so the coefficient is 2. In the third term, 7 is known as the constant.
Numerical Coefficient
A numerical coefficient is a number which is the multiplier of the variables in a term. For example, 3 is the numerical coefficient of the term 3mn. The term numerical coefficient is used for the constant multipliers of the variable. A common example is 4xy. Here, the numerical coefficient of xy is 4.
Characteristics of a Coefficient
A coefficient can be positive or negative, real or imaginary, or in the form of decimals or fractions. Another definition of the coefficient says, “Any number with which we multiply a variable." For example, in the term 9.3x, 9.3 is the coefficient of the variable x. In 5z, 5 is the coefficient.
Important Notes
Keep the following points in mind while working with coefficients:
 A coefficient is always attached to a variable.
 Variables without a number have 1 as their coefficient.
 The value of a variable is never the same. It varies according to the question and situation, and therefore, it is called a variable.
 The value of a constant is constant because its value is always fixed, and it cannot be changed.
Think Tank
 Can a variable be a coefficient?
 Can a fraction be a coefficient?
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Solved Examples on Coefficient

Example 1: In the expression ax^{2}+bx+c, spot the coefficient of the term x^{2}.
Solution: In the above expression, we can see that in the term ax^{2}, x is the variable. Thus, the coefficient is 'a'. 
Example 2: Identify the numerical coefficients in the following algebraic expression: 3x^{2}2y+5.
Solution: In the given expression, there are three terms. In the term 3x^{2}, the numerical coefficient of the term 3x^{2} is 3, in 2y the coefficient of y is 2, and 5 is a constant. Therefore, the numerical coefficients are 3 and 2. 
Example 3: Identify the coefficients in the following expression: x^{3}+2x+3.
Solution: In the above expression, there are three terms. In the term x^{3}, the coefficient of x^{3} is 1. In 2x, the coefficient of 2x is 2, and 3 is a constant. Therefore, the coefficients are 1 and 2.
Practice Questions on Coefficient
FAQs on Coefficient
What is a Coefficient in Math?
A coefficient is a number which is multiplied by a variable of a single term or the terms of a polynomial. For example, in the term 7x, 7 is the coefficient.
Can a Coefficient be Negative?
Yes, a coefficient can be negative as well. For example, in the term 9x, 9 is the coefficient.
Can a Fraction be a Coefficient?
Yes, a fraction can also be a coefficient. For example, in the expression: \(\dfrac{3}{4}x\)+2, 3/4 is a fraction and also a coefficient of x.
Can a Coefficient be Zero?
A coefficient cannot be zero because when we multiply 0 (as a coefficient) with any variable, the value of the term results in 0. However, a coefficient can be any natural number, negative number, or a fraction.
What is a Numerical Coefficient in Math?
The term numerical coefficient is used for the constant multipliers of the variable. A common example is 4xy. Here, the numerical coefficient of xy is 4.