Overview
This lecture explains how to factor sums and differences of two cubes using specific formulas, the importance of recognizing perfect cubes, and provides step-by-step worked examples.
Perfect Cubes
- Common perfect cubes to memorize: ( 1^3 = 1 ), ( 2^3 = 8 ), ( 10^3 = 1000 ), ( 4^3 = 64 ), ( 5^3 = 125 ), ( 6^3 = 216 ).
- Recognizing perfect cubes helps identify when to use sum or difference of cubes factoring formulas.
Factoring Formulas
- The sum of cubes: ( a^3 + b^3 = (a + b)(a^2 - ab + b^2) ).
- The difference of cubes: ( a^3 - b^3 = (a - b)(a^2 + ab + b^2) ).
- Use the SOAP acronym: Same, Opposite, Always Positive (signs for the trinomial terms).
Steps to Factor Sums/Differences of Cubes
- Identify ( a ) and ( b ) by taking cube roots of each term.
- Substitute ( a ) and ( b ) into the appropriate formula.
- Use SOAP to determine the signs in the trinomial.
Worked Examples
- ( x^3 - 27 ): ( a = x ), ( b = 3 ); factors to ( (x - 3)(x^2 + 3x + 9) ).
- ( 8y^3 + 1 ): ( a = 2y ), ( b = 1 ); factors to ( (2y + 1)(4y^2 - 2y + 1) ).
- ( 64d^3 - 125 ): ( a = 4d ), ( b = 5 ); factors to ( (4d - 5)(16d^2 + 20d + 25) ).
- ( 216c^3 + 1000d^3 ): ( a = 6c ), ( b = 10d ); factors to ( (6c + 10d)(36c^2 - 60cd + 100d^2) ).
General Factoring Advice
- Always check for a greatest common factor (GCF) before applying the sum or difference of cubes method.
- Factored trinomials from these formulas can't be simplified further.
Key Terms & Definitions
- Perfect cube — A number that is the result of another number multiplied by itself three times.
- Sum of cubes — An expression in the form ( a^3 + b^3 ).
- Difference of cubes — An expression in the form ( a^3 - b^3 ).
- SOAP acronym — A memory aid for signs in the trinomial: Same, Opposite, Always Positive.
Action Items / Next Steps
- Memorize common perfect cubes for quick recognition.
- Practice factoring expressions using the sum and difference of cubes formulas.
- Always check for a GCF before other factoring methods.