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