In algebra, expressions may contain brackets ( ) and like terms.
To make calculations easier, we often need to expand brackets and simplify the resulting expression.

Key words:

• Expand means to remove brackets by multiplying.
• Simplify means to reduce an expression by combining like terms.

Key Concepts

1. Expanding One Bracket

Use the distributive law:
a(b + c) = ab + ac
a(b - c) = ab - ac

Examples:

1. Expand ( 3(x + 5) ) = ( 3x + 15 )
2. Expand ( 4(2a - 3b) ) = ( 8a - 12b )
3. Expand ( -2(3x + 4y) ) = ( -6x - 8y )

2. Expanding Two Brackets

Use the FOIL method (First, Outside, Inside, Last):
(a + b)(c + d) = ac + ad + bc + bd

Examples:

1. Expand ( (x + 3)(x + 2) )
   = x(x + 2) + 3(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6
2. Expand ( (a + 2)(a - 5) )
   = a(a - 5) + 2(a - 5)
   = a^2 - 5a + 2a - 10 = a^2 - 3a - 10

3. Expanding Three or More Brackets

Expand two brackets first, then multiply by the next.

Example:
Expand ( (x + 2)(x + 3)(x + 1) )

Step 1: Expand first two brackets:
( (x + 2)(x + 3) = x^2 + 5x + 6 )

Step 2: Multiply by third bracket:
( (x^2 + 5x + 6)(x + 1) = x^3 + 6x^2 + 11x + 6 )

4. Simplifying Algebraic Expressions

Combine like terms (terms that have the same variables and powers).

Examples:

1. Simplify ( 3x + 5x ) → ( 8x )
2. Simplify ( 7a + 3b - 2a + 4b ) → ( (7a - 2a) + (3b + 4b) = 5a + 7b )
3. Simplify ( 4x^2 + 2x - 3x^2 + 6x ) → ( (4x^2 - 3x^2) + (2x + 6x) = x^2 + 8x )

5. Expanding and Simplifying Together

Example 1:
Simplify ( 2(x + 3) + 4(2x - 1) )
= (2x + 6) + (8x - 4) = 10x + 2

Example 2:
Simplify ( 3(2a - 4) - 2(a + 1) )
= (6a - 12) - (2a + 2) = 4a - 14

Example 3:
Simplify ( (x + 4)(x - 2) )
= x^2 - 2x + 4x - 8 = x^2 + 2x - 8