The Distributive Property is an algebraic property that is used to multiply a single value and two or more values within a set of parenthesis.

Let’s see a simple example: 2(4 + 3). Since the binomial “4 + 3” is in the parenthesis, according to the order of operations, you have to calculate the answer of 4 + 3 and then multiply it by 2, which gives the resultant value as 14.

Consider the expression: 6(2+4x)

The two values inside the parenthesis cannot be added since they are not like terms, therefore it cannot be simplified any further. We need a different method and this is where Distributive Property can be applied.

If you apply Distributive Property,

6 * 2 + 6 * 4x

The parenthesis no longer exists and every term is multiplied by 6.

Now, you can simplify the multiplication for individual terms.

12 + 24x

The distributive property of multiplication lets you simplify expressions wherein you multiply a number by a sum or difference. According to the property, the product of a sum or difference is equal to the sum or difference of the products.

Distributive Property of Multiplication over Addition

The distributive property of multiplication over addition is applied when you multiply a value by a sum. For example, suppose you want to multiply 5 by the sum of 10 + 3.

According to the property, you can first add the numbers and then multiply by 5.

5(10 + 3) = 5(13) = 65. Or, you can first multiply every addend by 5. This is known as distributing the 5 and then you can add the products.

The multiplication of 5(10) and 5(3) will be performed before you add. 5(10) + 5(3) = 50 + 15 = 65. You can note that the result is same as before.

You probably use this  method without actually knowing that you are using it.

Both the methods are described by the below equations. We have 10 and 3 on the left hand side then multiplied by 5. This expansion is rewritten by applying the distributive property on the right hand side where we distribute 5 then multiply by 5 and add the results. You will see that the resultant is similar in each case.

5(10 + 3) = 5(10) + 5(3)

5(13) = 50 + 15

65 = 65

The distributive properties of addition and subtraction can be utilized to rewrite expressions for different purposes. When you multiply a number by a sum, you may add and multiply. Also, you can first multiply each added  and then add the products. This is applicable to subtraction as well. In every case, you disturb the outer multiplier to every value in the parenthesis, so that multiplication occurs with every value before addition or subtraction.

Examples: Solve the given expression:

(i) $4(2x^{4}+ 7x)$

(ii) $2x(x^{2}+ y)$

(iii) $4(7xy+ 13yx)$

Solution:

(i) $4(2x^{4}+ 28x)$

Using distributive law we have,

$8x^{4}+ 14x$

(ii) $2x(x^{2}+ y)$

Using Distributive property,

$2x^{3}+ 2xy$

(iii) $4(7xy+ 13yx)$

Using distributive property, we have

$4(20xy) = 80 xy$     (As xy = yx)