Overview
This lecture covers how to factor sums and differences of cubes using standard formulas, with step-by-step examples for both types.
Factoring Sum of Cubes
- The sum of cubes formula is: ( a^3 + b^3 = (a + b)(a^2 - ab + b^2) ).
- To use the formula, identify ( a ) and ( b ) by taking the cube root of each term.
- Example: ( x^3 + 8 ) factors to ( (x + 2)(x^2 - 2x + 4) ).
- Example: ( x^3 + 125 ) factors to ( (x + 5)(x^2 - 5x + 25) ).
- Example: ( 8x^3 + 27 ) factors to ( (2x + 3)(4x^2 - 6x + 9) ).
- Example: ( 27x^3 + 64y^3 ) factors to ( (3x + 4y)(9x^2 - 12xy + 16y^2) ).
Factoring Difference of Cubes
- The difference of cubes formula is: ( a^3 - b^3 = (a - b)(a^2 + ab + b^2) ).
- The sign between ( a ) and ( b ) matches the original expression, but the second factor’s middle term sign flips.
- Example: ( x^3 - 216 ) factors to ( (x - 6)(x^2 + 6x + 36) ).
- Example: ( 64y^3 - 125 ) factors to ( (4y - 5)(16y^2 + 20y + 25) ).
- Example: ( 8y^3 - 27 ) factors to ( (2y - 3)(4y^2 + 6y + 9) ).
- Example: ( x^6 - 64y^9 ) factors to ( (x^2 - 4y^3)(x^4 + 4x^2y^3 + 16y^6) ).
Recognizing Perfect Cubes
- Identify perfect cubes by taking cube roots of coefficients and variables.
- For variables, divide the exponent by 3 for the cube root.
Key Terms & Definitions
- Perfect Cube — A number/term raised to the third power, such as ( a^3 ) or ( 8x^3 ).
- Sum of Cubes — An expression of form ( a^3 + b^3 ).
- Difference of Cubes — An expression of form ( a^3 - b^3 ).
Action Items / Next Steps
- Practice factoring additional sum and difference of cubes problems.
- Review how to identify perfect cubes and compute cube roots for both coefficients and variables.