Overview
This lecture covers key algebraic identities for expression expansions, their derivations, applications to both variable and numerical problems, and strategies for simplification and problem-solving based on the ICSE Class 9 Maths syllabus.
Introduction to Expansions
- Expansion transforms a compact algebraic expression into a longer, equivalent form.
- Identities are derived using multiplication and can be rederived if forgotten.
Key Quadratic Identities
- The four primary ICSE identities are:
- (a + b)² = a² + 2ab + b²
- (a – b)² = a² – 2ab + b²
- (x + a)(x + b) = x² + (a + b)x + ab
- (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Applying Quadratic Identities
- Always write the identity before solving to aid memorization.
- Use identities to expand expressions like (2x + 7y)².
- For numbers, such as 101², use a = 100, b = 1: (100 + 1)² = 10201.
- For (a – b)², carefully handle negative terms.
Cubic and Other Advanced Identities
- (a + b)³ = a³ + b³ + 3ab(a + b)
- (a – b)³ = a³ – b³ – 3ab(a – b)
- Key special case: if a + b + c = 0, then a³ + b³ + c³ = 3abc.
Strategies for Simplification
- Convert subtraction to the addition of negative numbers for easier expansion.
- Use identities to break down and expand or simplify calculations.
- When a + b + c = 0, use cube identities directly for efficiency.
Combining Identities and Problem Solving
- (a + b)² + (a – b)² = 2(a² + b²)
- (a + b)(a – b) = a² – b² (difference of squares)
- Use combinations and manipulations of identities for algebraic problem solving.
Examples & Applications
- Expand expressions like (x + 2)³ and (2a – b)³ using relevant identities.
- Use given values (e.g., x + y = 4, xy = 5) to find x² + y², etc., by applying and rearranging identities.
Finding Coefficients
- To find the coefficient of a term, first fully expand using identities, then identify the required term's coefficient.
Key Terms & Definitions
- Expansion — Rewriting an algebraic expression as an equivalent, longer expression.
- Identity — An equation true for all variable values (e.g., (a + b)² = a² + 2ab + b²).
- Coefficient — The numerical factor of a variable term in an expression.
- Difference of Squares — (a² – b²) = (a + b)(a – b).
Action Items / Next Steps
- Practice expanding expressions with all key identities.
- Write the identity before each exercise for memorization.
- Solve homework involving sum and product of numbers and their cubes.
- Review breaking down numbers for squaring and cubing without direct calculation.