Overview
This lecture explains how to identify perfect cubes, evaluate cube expressions, and factor both the sum and difference of two cubes using specific algebraic formulas.
Identifying Perfect Cubes
- A perfect cube is a number that can be written as an integer raised to the third power, e.g., ( n^3 ).
- Examples of perfect cubes: 8 ((2^3)), 27 ((3^3)), 64 ((4^3)), 125 ((5^3)), 216 ((6^3)), 343 ((7^3)).
- Non-examples: 25, 40, 60, 72 are not perfect cubes.
Evaluating Exponential Notation (Cubes)
- (2^3 = 8)
- (3^3 = 27)
- (4^3 = 64)
- (5^3 = 125)
- (6^3 = 216)
- (7^3 = 343)
Factoring the Sum of Two Cubes
- The sum of two cubes is written as (x^3 + y^3).
- Factoring formula: (x^3 + y^3 = (x + y)(x^2 - xy + y^2)).
- Example: (a^3 + 64 = (a + 4)(a^2 - 4a + 16)).
- Example: (8b^3 + 27c^3 = (2b + 3c)(4b^2 - 6bc + 9c^2)).
Factoring the Difference of Two Cubes
- The difference of two cubes is written as (x^3 - y^3).
- Factoring formula: (x^3 - y^3 = (x - y)(x^2 + xy + y^2)).
- Example: (27c^3 - d^3 = (3c - d)(9c^2 + 3cd + d^2)).
- Example: (8e^3f^6 - 125g^3 = (2ef^2 - 5g)(4e^2f^4 + 10ef^2g + 25g^2)).
- Example: (64 - p^6 = (4 - p^2)(16 + 4p^2 + p^4)).
Key Terms & Definitions
- Perfect Cube — a number that results from multiplying an integer by itself three times.
- Factoring — rewriting an expression as a product of its factors.
- Sum of Cubes — expression in form (x^3 + y^3), factored as ((x + y)(x^2 - xy + y^2)).
- Difference of Cubes — expression in form (x^3 - y^3), factored as ((x - y)(x^2 + xy + y^2)).
Action Items / Next Steps
- Practice identifying perfect cubes.
- Use the formulas to factor both sum and difference of cubes in given exercises.