**9.1 Distributive property**:

**9.1.1 Introduction**: Distributive property or distributive law states that,

**(a + b) x c = a x b + b x c**

When given a set of numbers, the answer remains the same when

i) you add up the numbers first and then perform multiplication

ii) you multiply each number first and then perform addition.

**For Example: **(2+6) x 3 = 2 x 3 +6 x 3

In the above equation, on left hand side 3 is multiplied to the sum of 2 and 6. Whereas in the right hand side 3 is multiplied to each number and then the products are added. But answer remains same on both sides. And we say that multiplication is distributive over addition.

It can further be categorized in two ways:

- i) Where multiplication is left distributed over addition. that is,

c x (a + b) = c x a + c x b

- ii) Where multiplication is right distributive over addition. that is,

(a + b) x c = a x b + b x c

**9.1.2 Examples:**

i) For this 6 x (4+7) = 6 x 11 = 66

Answer is same as this 6 x 4 + 6 x 7 = 24 + 42 = 66

ii) For this (5 +2) x 3 = 7 x 3 = 21

Answer is same as this 5 x 3 + 2 x 3 = 15 + 6 = 21

**9.1.3 Uses**

i) Easy to break up a difficult multiplication.

511 x 3 = 500 x 3 +11 x 3 = 1500 + 33 = 1533

ii) Easy to combine a difficult multiplication.

4 x 21 + 3 x 21 = 21 x (4+3) = 21 x 7 = 91

**9.2 Associative Property**

**9.2.1 Introduction**: Associative property or associative law states that

(**a + b) + c = a + (b + c)**

OR

**(a x b) x c = a x (b x c)**

It means that while addition or multiplication grouping of number is not of much concern. That is rearrangement of parenthesis will not change the value.

**9.2.2 Examples:**

(4 + 5) +7 = 4 + 5 + 7 = 16

Is same as 4 + (5 + 7) = 4 + 5 + 7 = 16

(5 x 8) x 3 = 5 x 8 x 3 = 120

Is same as 5 x (8 x 3) = 5 x 8 x 3 = 120

**9.2.3 Uses**

i) Sometimes it makes addition or multiplication a little easier when arranged in different order.

54 +75 + 5 = 54 + (75 + 5) = 54 + 80 =134

7 x 15 x 2 = 7 x (15 x 2) = 7 x 30 = 210

**9.3 Commutative property**

**9.3.1 Introduction:**Commutative property or commutative law sates that

a + b = b + a

a x b =b x a

it means that when you add or multiply, even after swapping numbers over you get the same answer.

**9.3.2 Examples**:

3 +5 = 8

Is same as 5 + 3= 8

7 x 4 = 28

Is same as 4 x 7 = 28

**9.4 FOIL**

**9.4.1 Introduction**: FOIL is a method used in algebra to solve two binomials. Earlier in this chapter we have studied about the Distributive property for multiplying a monomial by a binomial. That is

**a x (b + d x)**

but for multiplying a binomial with another binomial FOIL method is used.

FOIL stands for

**F**irst- Multiply **First** term on each set of parenthesis.

**O**uter- Multiply **Outer** term of each set of parenthesis.

**I**nner- Multiply **Inner** term of each set of parenthesis.

**L**ast - Multiply **Last** term of each set of parenthesis.

Let us explain it well with an example

Take (7 x 3* x***)(**4 x 5*x***)**

As mentioned above multiply the first terms of each parenthesis, 7 and 4

7 x 4 = 28

Now the next step that is multiplying the outer terms of each parenthesis, 7 and 5x

Therefore,

7 x 5*x*** = **35*x*

28 + 35*x*

Now multiply the inner terms of parenthesis, 3* x* and 4

3*x* **x **4 = 12*x*

28 + 35*x* + 12*x*

And finally multiply the last terms of each parenthesis, 5*x* and 3*x***.**

5*x* x 3*x*** = **15x^{2}

Now the final step is to combine these terms

28 + 47*x* + 15x^{2}= 0

Or 15x^{2}+ 47*x*** + **28 = 0

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