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# What is a coefficient in math?

A coefficient is basically defined as an integer which is multiplied with the variable of a single term or the terms of a polynomial. In other words, it refers to a number or quantity placed with a variable. The variables having no number with them are assumed to have 1 as their coefficient.

## What is a polynomial?

Poly means ‘many’

Nomial means ‘terms’

A mathematical expression having one or more algebraic terms is called a polynomial. Commonly it is expressed as a sum of several terms having different powers of same variable or variables.

Let us understand with few examples.

In an expression: ${ax+bx+z}$

Here, is called variable and “a” and “b” are the coefficients and z is called constant.

In an expression: ${{x}^{2}}+5$ , 1 is the coefficient of ${{x}^{2}}$ .

Now, to identify a coefficient, it should be remembered that it always comes with a variable. In above example of ${ ax+bx+z, x}$ is the variable. Hence, ‘a’ and ‘b’ are the coefficient here.

### Characteristics of a Coefficient:

A coefficient can be a real/imaginary, positive/negative or even in the form of fractions or decimals.

For example:

In an expression:

• 7x, 4.7 is the coefficient of variable x
• In  $\frac{{-1}}{2}y,\,\,\frac{{-1}}{2}$ is the coefficient of variable y.

### Important Points to remember while working on coefficients:

• A coefficient is always attached to a variable, whether variable is a single term or a polynomial.
• In case there is no number or numerical factor in a term, the coefficient of the same is considered a 1.
• The value of the variable is never the same and varies with a situation or according to the question.
• The value of a constant is always fixed and cannot be changed.
• A coefficient can never be zero because when we multiply 0 (as a coefficient) with any variable, the value of the same results into 0.
• While adding or subtracting polynomials, coefficients of the same variables can only participate in these arithmetic operations. But there is no such rule for multiplication and division.

For example :

On adding  $8{{x}^{2}}+12y+9$  and 17x2 – 3y + 4

We get, (8 + 17) x2  + (12 – 3) y + (9 + 4)

The answer becomes 25x2 + 9y + 13

### Examples:

1. Find out the numerical Coefficients in the term : $\frac{{2xy2}}{7}$

Solution: $\frac{2}{7}$  is the coefficient of the above term.

1. Find out the numerical coefficients in the following algebraic expression: 7x2 – 5y + 8.

Solution:  Here we have three algebraic terms. In the term 7x2, the numerical coefficient of the term 7x2 is 7, -5 the coefficient of y, and 8 is a constant. Therefore, the numerical coefficients are 7and -5.

1. Find out the numerical coefficients in the expression: $9{{a}^{2}}-3y+8$

Solution: Amongst these three algebraic terms, the numerical coefficient of the term  $~9{{a}^{2}}$ is 9, -3 is the coefficient of y, and 8 is a constant. Therefore, the numerical coefficients are 9 and -3.

1. Find out the numerical coefficients in the expression: $7{{w}^{2}}+8e+5$

Solution: In the term $7{{w}^{2}}$ the numerical coefficient of the term $7{{w}^{2}}$  is 7, 8 is the coefficient of e, and 5 is a constant. Therefore, the numerical coefficients are 7 and 8.

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