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Prior Knowledge - Expanding Brackets
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Prior Knowledge - Expanding Brackets
Prior Knowledge - Expanding Brackets
Basic Principles
- Expanding brackets involves multiplying every term inside the bracket by the factor outside the bracket.
- It is based on the distributive property in algebra which states that a*(b+c) = ab + ac.
- When dealing with double brackets, apply the FOIL method (First, Outer, Inner, Last) for expansion.
- Brackets can also consist of more than two terms. Each term must be multiplied by the term outside the bracket.
Different Kinds of Brackets
- Linear brackets e.g. 3(x + 2) result to a linear expression.
- Quadratic brackets (two brackets with linear expressions) i.e. (x + a)(x + b) transforms into a quadratic expression.
- Cubic brackets i.e. (x + a)(x + b)(x + c) expand into a cubic expression.
Special Cases
- Square of a binomial: (a + b)² = a² + 2ab + b²; and (a - b)² = a² - 2ab + b².
- Difference of squares: a² - b² = (a + b)(a - b). It is crucial to memorise this identity.
Practical Applications
- Expanding brackets is critical in simplifying algebraic expressions and equations.
- They are heavily used in factorising, completing the square and solving quadratic equations.
- Polynomial division or long division is also an area where expanded brackets come into play.
- It's useful in calculus, specifically in finding the derivative using the Product Rule.
Common Pitfalls
- Sign errors are common when expanding brackets. Pay attention to negatives.
- Missing terms can occur during the `FOIL' step for new learners.
- Mistakes can happen with squared terms, remember (x + a)² ≠ x² + a².
- Take care with the order of operations to avoid miscalculations.
Key Takeaways
- Mastering expanding brackets provides a foundation for more complex topics in A-Level Mathematics.
- Remember key identities such as the square of a binomial and difference of squares.
- Always check your work for possible sign errors to ensure accuracy.
