Expanding brackets is a method used in algebra to simplify expressions that are inside brackets by multiplying them out. It means taking each term inside the bracket and multiplying it by the term outside the bracket.
For example, if you have: 2(x + 3)
To expand the brackets, you multiply 2 by both x and 3 separately:
2 × x = 2x and 2 × 3 = 6
So, the expanded form of 2(x + 3) is: 2x + 6
Expanding Single Brackets
General Rule: If having an expression like a(b + c), expand it as: a(b + c) = ab + ac
Example 1: 3(x + 5)
3(x + 5) = 3 × x + 3 × 5 = 3x + 15
Example 2: -2(4x - 7)
-2(4x - 7) = -2 × 4x + (-2) × (-7) = -8x + 14
FOIL Method for Expanding Double Brackets
Foil method is the technique used to expand the product of two binomials such as (a + b)(c + d). Acronym Foil stands for First, Outer, Inner, and Last that describes the order in which to multiply terms:
- First (“first” terms of each expression are multiplied together)
- Outer (“outside” terms are multiplied—that is, the first term of the first expression and the second term of the second expression)
- Inner (“inside” terms of the expressions are multiplied—second term of the first expression and first term of the second expression)
- Last (“last” terms of each expression are multiplied)
Let's consider another example for better understanding.
Example: (x + 3)(x + 4)
Solution:
- First: Multiply first terms of each brackets: x × x = x2
- Outer: Multiply outer terms of each brackets: x × 4 = 4x
- Inner: Multiply inner terms: 3 × x = 3x
- Last: Multiply last terms of each brackets: 3 × 4 = 12
After performing above Multiplications now combine like terms:
x2 + 4x + 3x + 12 = x2 + 7x + 12
Special Cases
- Perfect Squares: when expanding binomial squared (a + b)2 or (a -b)2 use formula: (a + b)2 = a2 + 2ab + b2 and (a - b)2 = a2 + 2ab - b2
Example:
- (x + 5)2 = x2 + 2(5)(x) + 52 = x2 + 10x + 25
- (x - 3)2 = x2 + 2(x)(3) - 32 = x2 + 6x - 9
- Difference of Squares: For expression like (a - b)(a + b), the result is difference of squares: (a - b)(a + b) = a2 -b2
Example: (x - 3)(x + 3) = x2 - 9
Expanding More than Two Brackets
For three brackets we can either use formula: (x + a)(x + b)(x + c) = x3 + (a + b + c)x2 + (ac + bc + ab)x + abc
or we can expand this step by step as following example:
Example: (x + 2)(x - 1)(x + 3)
First expand 1st two brackets:
(x + 2)(x - 1) = x2 - x + 2x - 2 = x2 + x - 2
Next multiply by (x + 3):
(x2 + x - 2)(x + 3) = x2(x + 3) + x(x + 3) - 2(x + 3) = x3 + 3x2 + x2 + 3x - 2x - 6 = x3 + 4x2 + x - 6
Conclusion
Expanding the brackets is an essential skill in algebra which ensures simplifying expressions and solving the equations efficiently. Whether dealing with single, double, or multiple brackets, understanding methods and practicing will ensure accuracy and confidence in solving algebraic problems.
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