# Coefficient of a Term

## Definition of Coefficient

The Coefficient of a term may be defined as the constant or numerical value that is multiplied with a variable or multiple variables in an Algebraic Expression or Algebraic Equation.

We are aware that the Term of an Algebraic Expression consists of one or more factors. These factors may be constants or variables. If a term has a constant multiplied by a variable, then the constant is called the Numerical Coefficient or simply the Coefficient of the term.

## Example of Coefficient of a Term

To understand the concept of Coefficient of a Term, let us consider $2x$ as a term of any algebraic expression. The Term $2x$ is formed by multiplying the factors $2$ and $x$. Here, $2$ is a constant and $x$ is a variable. Therefore, $2$ is called the Coefficient of the term $2x$.

When the Coefficient of a term is $+1$, the coefficient is not written in the term or usually omitted.

For example, the term $xy$ is formed by multiplying the variables $x$ and $y$. However, the constant, $+1$ is also multiplied with the term and is not written. In this case, the Coefficient of the term $xy$ is $+1$.

Let us take another example of a term $-x^2$. In this case, the term is formed by multiplying $x$ and $x$ with $-1$. Here, $-1$ is not written and the negative ($"–"$) sign represents that $-1$ is multiplied with the term. Therefore, the Coefficient of the the term $-x^2$ will be $-1$.

Sometimes, the word Coefficient is used in a more general way. For example, let’s take the term $3xy$. This term has three factors which are $3, x$, and $y$.

Now, it can also be said that $x$ is a coefficient of $3y$.

Similarly, $y$ is a coefficient of $3x$.

Therefore, it may be concluded that a Coefficient is not always a constant or a numerical value. It may be either a numerical value, or an algebraic factor having a variable, or a product of two or more algebraic factors with constants and variables.

## How to find the Coefficient of a term?

The Coefficient of a term can be found out by checking the factors of the term.

• Case-I: If the factors of the term consist of only Constants, then there is no Coefficient of the term.
• Case II: If the factors of the term consist of both Constants and Variables, then the Constant that is multiplied with the Variable or the Variables is the Coefficient of the term.
• Case III: If the term has only Variables and there is no negative $(-)$ sign in front of the Variables, then the Coefficient of the term is $1$.
• Case IV: If the term has only Variables and there is a negative sign $(-)$ in front of the Variables, then the Coefficient of the term is $-1$.

## Solved Examples

• Find the coefficient of $x$ in the algebraic expression $4x+3y$?

Solution: The coefficient of $x$ in the algebraic expression $4x+3y$ is 4 as it is the constant that is multiplied with $x$.

• Identify the numerical coefficient of $y$ in the algebraic expression $3{x^4} - 2{y^3} + 5$?

Solution: The numerical coefficient of $y$ in the algebraic expression $3{x^4} - 2{y^3} + 5$ is -2.

• Find the coefficient of $x$ in the algebraic expression $-x+3y$?

Solution: The coefficient of $x$ in the algebraic expression $-x+3y$ is -1.