Expansion means multiplying out brackets to remove them and express the result as a sum or difference of terms.
- Expansion of a Single Bracket
When a term is multiplied by a bracket, multiply the term by each term inside the bracket.
Example 1:
Expand ( 3(x + 4) )
Solution:
( 3 \times x + 3 \times 4 = 3x + 12 )
- Expansion of Brackets with Negative Signs
Be careful with negative signs when expanding.
Example 2:
Expand ( -2(x - 5) )
Solution:
( -2 \times x + (-2) \times (-5) = -2x + 10 )
- Expansion of Two Brackets (Double Brackets)
To expand two brackets, multiply every term in the first bracket by every term in the second.
Example 3:
Expand ( (x + 3)(x + 2) )
Solution:
(x + 3)(x + 2) = x(x + 2) + 3(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6
- Expansion with a Negative Bracket
Example 4:
Expand ( (x + 4)(x - 3) )
Solution:
x(x - 3) + 4(x - 3) = x^2 - 3x + 4x - 12 = x^2 + x - 12
- Expansion Involving Coefficients
Example 5:
Expand ( 2(x + 5)(x - 3) )
Solution:
First expand the brackets:
( (x + 5)(x - 3) = x^2 + 2x - 15 )
Now multiply by 2:
( 2(x^2 + 2x - 15) = 2x^2 + 4x - 30 )
- Expansion of Three Terms
Example 6:
Expand ( (x + 2)(x^2 + 3x + 4) )
Solution:
Multiply ( x ) and ( 2 ) by each term:
x(x^2 + 3x + 4) + 2(x^2 + 3x + 4) = x^3 + 3x^2 + 4x + 2x^2 + 6x + 8
Simplify:
x^3 + 5x^2 + 10x + 8