** **

**Expanding Brackets
**Removing the brackets is known as distributive law. To remove the brackets, multiply each term outside the bracket.

**Example:
**\(3(x + 2)\) Multiply each term inside the bracket by 3.

\(3 (x) + 3 (2) = 3x + 6 \)

**Example 2:** \(2x(x + 2)\)

\( 2x (x) + 2x (2) = 2x^2 + 4x \)

**Pairs of Bracket
**Removing pairs of brackets using FOIL method.

**FOIL Method
**

**F**irst – Multiply the first term in the first bracket with the first term in the second bracket.

**O**utside – Multiply the first term in the first bracket with the second term in the second bracket.

**I**nside – Multiply the second term in the first bracket with first term in the second bracket.

**L**ast – Multiply the second term in the first bracket with the second term in the second bracket.

**Example:**(x + 2) (x + 3)

FOIL method will be:

\( x(x) + x(3) +2(x) + 2 (3) \)

\( x^2 + 5x + 6 \)

**Example 2:** \( (x + 5) (x – 2) \)

\( x (x)+ x (-2) + 5 (x) – 2 (5) \)

\( x^2 + 3x – 10 \)

**Factorising
**Factorising is simplifying a quadratic expression. It is the reverse of expanding the bracket. To factorise, look for two numbers that have the sum of the second term and product of the third term.

**Example:** \(x^2 + 9x + 12 \)

Look for two numbers whose sum is 9 and if multiplied the answer is 12.

The factor pair of 12 are the following:

a. 12 and 1

b. 4 and 3

c. 6 and 2

In the pair of a factor, the product of 12 and 1, and 4 and 3 is 12 but the sum is not 9. The pair of the factor that suits the condition is 6 and 2. Factor of \( x^2 + 9x + 12 \) is \( (x + 6) and (x + 2)\) .

\( x^2 + 9x + 12 = (x + 6) (x + 2) \)

**Example 2:** \( x^2 + x – 30 \)

To factor the expression, think of two numbers that the product is -30 and added up is 1. Since the product is a negative, the two numbers must have a positive and a negative number.

These are all the factor of -30:

a. 15 and -5

b. -15 and 5

c. 10 and -3

d. -10 and 3

e. 6 and -5

f. -6 and 5

Only 6 and -5 have the sum of 1.

\( x^2 + x – 30 = (x + 6) (x – 5) \)

**Factorising the Difference of Two Squares**

**Example:** \( x^2 – 36 \)

Notice that the only term is \( x^2 \) and a number. Some quadratics do not have the second term. Therefore we will look for a factor whose sum is 0.

The pair of factor of 36.

a. 36 and – 1

b. -36 and 1

c. 18 and -2

d. -18 and 2

e. 12 and -3

f. -12 and 3

g. 9 and -4

h. -9 and 4

i. 6 and -6

\( x^2 – 36 \) can be factor by 6 and -6. These examples of factoring are called the sum and difference of square or difference of two squares.

However, not all without a second term can be factorised.

**Example 2:** \(x^2 – 30 \)

Here are all the factor of -30.

a. 15 and -5

b. -15 and 5

c. 10 and -3

d. -10 and 3

e. 6 and -5

f. -6 and 5

From the pair of factor, it does not have the sum equal to zero. So, \( x^2 – 30 \) does not have a pair of factor. Therefore, it cannot be factorized.