# Coefficient in Math

In this chapter we will discuss about the concept of coefficient with examples.

In order to understand the chapter you should have basic knowledge about constants, variables and terms in mathematics.

## What are coefficients?

Coefficients are the constant numbers that are present in front of the variables.

These coefficients are also sometimes called numerical coefficient.

Note that the coefficient can be integer, fraction or decimal number. The important point is that the number should be a constant.

### Examples of coefficient

(a) 5x

Here x is a variable.
Number 5 is a constant present in front of variable, hence is called coefficient.

(b) 10x + 7y

There are two coefficient in the above equation.

First entity ⟹10x
Number 10 is the coefficient

Second entity ⟹ 7y
Number 7 is the coefficient.

(c) \mathtt{21\ x^{2} y\ -\ 16\ \frac{y^{2}}{z}}

There are two entities here.

First entity ⟹ \mathtt{21\ x^{2} y}
Number 21 is the coefficient

Second entity ⟹ \mathtt{-\ 16\ \frac{y^{2}}{z}}
-16 is the coefficient of this entity.

(d) \mathtt{\frac{9}{2} \ x\ +2.5\ y\ +\ \frac{1}{7} z}

Here we are provided with three entities.

First entity ⟹ \mathtt{\frac{9}{2} \ x\ }
Fraction 9/2 is the coefficient.

Second entity ⟹ \mathtt{2.5\ y}
Decimal 2.5 is the coefficient

Third entity ⟹ \mathtt{\frac{1}{7} z}
Fraction 1/7 is the coefficient.

(e) x + 2.7 yz

Here two entities are provided.

First entity ⟹ x
Here number 1 is the coefficient

Second entity ⟹ 2.7 yz
Here fraction 2.7 is the coefficient.

### What does the coefficient tell?

The coefficient tells the number of times the variable present in the entity.

For Example;
Consider the entity ⟹ 5m

Here 5 is a constant and m is a variable.
Both 5 & m are multiplied to produce 5m.

⟹ 5 x m
⟹ 5m

It means that the variable m is present 5 times in the equation.

Let us consider another example;
Algebraic equation ⟹ 5x + 6yz

It means that, the variable x is present 5 times and variable yz is present 6 time.

This is an important insight as it will help you solve algebraic equation.

## Addition/Subtraction of coefficient in algebraic equation

Note that the addition or subtraction of entities is only possible among like terms.

Like terms are the entities with same variables but may have different coefficient.

In Addition/Subtraction of like terms, only the coefficient of the entities will change, the variable part will remain the same.

Let us look at some of the examples:

(a) 9x + 10y + 3x

In the questions three entities are present; 9x, 10y & 3x.

Here 9x and 3x are the like terms, so the addition can be done between the two.

(b) \mathtt{6x^{2} y^{2} \ +\ 3xy\ +\ 2x^{2} y^{2}}

Combining entity under the bracket

\mathtt{\Longrightarrow \left( 6x^{2} y^{2} \ +\ 2x^{2} y^{2}\right) +\ 3xy}\\\ \\ \mathtt{\Longrightarrow \ 8x^{2} y^{2} +\ 3xy}

(c) \mathtt{3\ +\ 6xy\ +\ 9x^{2} y\ +\ 21xy}

Rearranging the entities with same variable under a bracket.

\mathtt{\Longrightarrow \ 3\ +\ ( 6xy\ +\ 21xy) +\ 9x^{2} y}\\\ \\ \mathtt{\Longrightarrow \ 3\ +\ 27xy\ +\ 9x^{2} y}

(d) 9x + 9y + 2x +14xy + 7y

Solution
Arranging the like terms in bracket.
⟹ ( 9x + 2x ) + (9y + 7y) +14xy
⟹ 11x + 16y + 14xy

(e) \mathtt{10x^{3} y\ +\ 5xy^{3} +\ 7x^{3} y\ +\ 2xy}

Arranging the like term in a bracket.

\mathtt{\Longrightarrow \ \left( 10x^{3} y\ +\ 7x^{3} y\right) \ +\ 2xy\ +\ 5xy^{3}}\\\ \\ \mathtt{\Longrightarrow \ 17x^{3} y\ +\ 2xy\ +\ 5xy^{3}}

(f) \mathtt{13xy^{2} \ +\ 2xy\ +\ 2xy^{2} \ +\ 3xy}

First combine the like terms.

\mathtt{\Longrightarrow \ \left( 13xy^{2} +\ 2xy^{2}\right) \ +\ ( 2xy\ +\ 3xy)}\\\ \\ \mathtt{\Longrightarrow \ 15xy^{2} +\ 5xy}

## Frequently asked Questions – Coefficient

(01) What is the coefficient of variable x?

The coefficient of variable without a number is 1.

Hence, the coefficient of variable x is 1.

(02) Can a fraction or decimal number be a coefficient?

Yes!!
If a fraction or decimal number is attached to a variable then it becomes the coefficient.

Given below are some examples of fraction or decimal coefficient:
⟹ 2.5 x
⟹ 3.7 y
⟹ (1/2) xy
⟹ (3/4) zx

(03) Can you explain importance of coefficient in polynomial?

Coefficient plays important role in polynomial.
The change in coefficient value can significantly change the property of the polynomial.

Let’s consider the general equation of simple polynomial:
y = mx + c

Where, m is the coefficient of variable x.

If we change the value of coefficient m, the whole polynomial will show different property.

For Example, let’s consider two polynomial;
y = 2x + 3
y = -2x + 3

Observe the above graph.
Note that simply changing the coefficient value from 2 to -2 has completely changed the line graph.

The polynomial equation is subject of higher mathematics.
I just want to give an idea that how change in coefficient can completely change the polynomial property.

(04) Identify the coefficient in below expression:
⟹ 10 + 9y – 11 – x – 3

Solution
There are two variables in above expression; 9y and -x.

So the coefficients are:
9y ⟹ 9 is the coefficient
-x ⟹ -1 is the coefficient