Overview
This lecture explains how to factor the sum and difference of two cubes, using formulas, the SOAP acronym, and step-by-step examples.
Common Perfect Cubes
- 1³ = 1, 2³ = 8, 10³ = 1000 are common perfect cubes to memorize.
- Recognizing perfect cubes helps identify sum or difference of cubes expressions quickly.
Factoring Formulas & SOAP Acronym
- The sum of cubes formula: a³ + b³ = (a + b)(a² – ab + b²)
- The difference of cubes formula: a³ – b³ = (a – b)(a² + ab + b²)
- SOAP stands for: Same sign, Opposite sign, Always Positive for the quadratic factor's terms.
- Identify a and b by taking the cube root of each term.
Example 1: x³ – 27
- Cube root of x³ is x (a = x); cube root of 27 is 3 (b = 3).
- Apply difference formula: (x – 3)(x² + 3x + 9).
- The resulting trinomial cannot be factored further.
Example 2: 8y³ + 1
- Cube root of 8y³ is 2y (a = 2y); cube root of 1 is 1 (b = 1).
- Apply sum formula: (2y + 1)(4y² – 2y + 1).
Example 3: 64d³ – 125
- Cube root of 64d³ is 4d (a = 4d); cube root of 125 is 5 (b = 5).
- Apply difference formula: (4d – 5)(16d² + 20d + 25).
Example 4: 216c³ + 1000d³
- Cube root of 216c³ is 6c (a = 6c); cube root of 1000d³ is 10d (b = 10d).
- Apply sum formula: (6c + 10d)(36c² – 60cd + 100d²).
Factoring Strategy Tips
- Always check for a greatest common factor (GCF) before applying sum or difference of cubes formulas.
- The quadratic factor in the formula cannot be factored further using real numbers.
Key Terms & Definitions
- Perfect Cube — A number or term raised to the third power (e.g., 8 = 2³).
- Sum of Cubes — An expression of the form a³ + b³.
- Difference of Cubes — An expression of the form a³ – b³.
- SOAP — Acronym for determining the signs in the factorization: Same, Opposite, Always Positive.
Action Items / Next Steps
- Memorize common perfect cubes and the sum/difference of cubes formulas.
- Practice identifying a and b values and use SOAP to apply the formulas.
- Review the decision process for factoring by watching the recommended next video.