Overview
This lecture covers how to multiply brackets in algebra, focusing on expanding and simplifying expressions like (x - 5)(2x² - 3x + 6).
Multiplying Brackets (Expanding)
- Multiplying brackets means distributing each term in the first bracket across all terms in the second bracket.
- Example: (x - 5)(2x² - 3x + 6).
Step-by-Step Method ("Rainbow" Approach)
- Multiply x by each term: x × 2x² = 2x³, x × -3x = -3x², x × 6 = 6x.
- Multiply -5 by each term: -5 × 2x² = -10x², -5 × -3x = +15x, -5 × 6 = -30.
- Combine all results: 2x³ - 3x² + 6x - 10x² + 15x - 30.
Combining Like Terms
- Group and add/subtract like terms for simplification:
- Cubed term: 2x³
- x² terms: -3x² and -10x² = -13x²
- x terms: 6x and 15x = 21x
- Constant: -30
- Final simplified answer: 2x³ - 13x² + 21x - 30
Alternative Method (Breaking Into Two Parts)
- Separate the expression:
- x(2x² - 3x + 6) = 2x³ - 3x² + 6x
- -5(2x² - 3x + 6) = -10x² + 15x - 30
- Add both results and simplify as before for the same answer.
Key Terms & Definitions
- Expanding Brackets — Multiplying out brackets to remove them and combine all terms into a single expression.
- Like Terms — Terms with the same variable(s) and exponent(s) that can be combined by addition or subtraction.
- Distributive Property — Rule stating a(b + c) = ab + ac, used to expand brackets.
Action Items / Next Steps
- Practice multiplying and expanding brackets using both methods with different examples.
- Review class notes on distributive property for further understanding.