Expanding brackets, also known as removing parentheses, in mathematics, particularly at the Year 10 level, refers to the process of multiplying a term (either a number, a variable, or a combination of both) outside a set of brackets by each term inside the brackets. This simplifies the expression by removing the brackets.
Understanding Expanding Brackets
The fundamental principle behind expanding brackets is the distributive property. This property states that a term multiplied by a sum (or difference) is equal to the sum (or difference) of the term multiplied by each individual element within the brackets.
According to the reference, expanding brackets means multiplying everything inside the bracket by the letter or number outside the bracket.
Examples and Explanation
Let's break this down with examples:
Simple Numerical Example
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Consider the expression: 3(m + 7)
To expand this, we multiply 3 by both 'm' and '7':
3(m + 7) = (3 m) + (3 7) = 3m + 21
Algebraic Example
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Consider the expression: a(b + c)
To expand this, we multiply 'a' by both 'b' and 'c':
a(b + c) = ab + ac
More Complex Example
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Consider the expression: 2x(3x - 4y + 5)
To expand this, we multiply 2x by each term inside:
2x(3x - 4y + 5) = (2x 3x) - (2x 4y) + (2x * 5) = 6x2 - 8xy + 10x
Common Mistakes to Avoid
- Forgetting to multiply every term inside the bracket: Ensure you multiply the term outside the bracket by each term inside.
- Incorrectly applying the sign: Pay close attention to signs, especially when dealing with negative numbers.
- Combining unlike terms: After expanding, only combine like terms (e.g., you can combine 3x and 5x, but not 3x and 5x2).
Expanding Double Brackets
In Year 10, you'll also likely encounter expanding double brackets. This involves applying the distributive property twice. A common method used is the FOIL method:
- First: Multiply the first terms in each bracket.
- Outer: Multiply the outer terms.
- Inner: Multiply the inner terms.
- Last: Multiply the last terms.
Example:
- (x + 2)(x + 3) = (x x) + (x 3) + (2 x) + (2 3) = x2 + 3x + 2x + 6 = x2 + 5x + 6
Table Summary
| Concept | Description | Example |
|---|---|---|
| Expanding Brackets | Multiplying the term outside the bracket by each term inside it. | 4(x + 2) = 4x + 8 |
| Distributive Property | a(b + c) = ab + ac | 2(y - 3) = 2y - 6 |
| Double Brackets | Using FOIL (First, Outer, Inner, Last) to expand. | (a + 1)(a - 2) = a2 - a - 2 |
