# Distributive Property of Multiplication over Addition and Subtraction and Combine Like Terms

In this lesson, I will show you how to apply distributive property of multiplication over both operations addition and subtraction and simplifying the result using combine like terms rule.

## Simple rule of distributive property of multiplication

Let’s take the rectangle below and calculate its area using two different ways.  ### Method 1:

We take the rectangle on the right (the two colored rectangles together). Its length is and its width is , so its area is:

### Method 2:

The second method consists to calculate the area of the two rectangles separated at left in the figure above.

The dimensions of the green rectangle are and , then this area is

The dimensions of the purple rectangle are and , then this area is

These two rectangles are the same parts as the rectangle on the right.

So, the area of the rectangle on the right is equal to the sum of the areas of the two rectangles on the left. We can express this result in the following expression:

This result we got is called the distributive property of multiplication over addition.

### Note:

Let and real numbers:

By similarity, we can achieve to the distributive property of multiplication over subtraction.

### Note:

Let and real numbers:

So how do we mathematically understand the distributive property?

Simply put, we multiply the number outside the parentheses by all the numbers inside the parentheses (distributive).

In case we have more than two numbers in parentheses, we apply the same rule as shown in the next note.

### Note:

Let and real numbers:

When we factor, we get the same number of terms as inside the parentheses.

#### Example1:

Distribute then simplify the algebraic expression below:

#### Solution:

This example is very simple, we multiply 2 by and .

#### Example2:

Distribute then simplify the algebraic expression below:

#### Solution:

by similarity, we multiply 3 by and and 5.

#### Example3:

Distribute then simplify the algebraic expression below:

#### Solution:

In this case, we need to pay attention to the sign of the number outside the bracket.

Here we multiply -5 by , and 1

### Note:

We don't apply the distributive property if what is inside the parentheses is a product.

## Generalized rule of distributive property of multiplication

In this part I will try to show you another case of distributive in a simplified way. It is more general than the first property.

We start from the next rule:

### Note:

Let and real numbers.

So as you can see, distributive property in this case is basically based on the first rule and applying it twice.

Let's take some exemples.

#### Example1:

Distributive then simplify the algebraic expression below:

#### Solution:

After removing the parentheses and simplifying each term separately, we now combine like terms.

#### Example2:

Distributive then simplify the algebraic expression below:

#### Solution:

After removing the parentheses and simplifying each term separately, we now combine like terms.

#### Example3:

Distributive then simplify the algebraic expression below:

#### Solution:

After removing the parentheses and simplifying each term separately, we now combine like terms.

#### Example4:

Distributive then simplify the algebraic expression below:

#### Solution:

After removing the parentheses and simplifying each term separately, we now combine like terms.