Let us also memorize expansion formulas for cubic expressions.
Cubic Multiplication Formulas
- (a + b)³ = a³ + 3a²b + 3ab² + b³
- (a − b)³ = a³ − 3a²b + 3ab² − b³
- (a + b)(a² − ab + b²) = a³ + b³
- (a − b)(a² + ab + b²) = a³ − b³
Example ①
(1) (x + 2)³ (2) (2x − 1)³
Answer:(1) x³ + 6x² + 12x + 8 (2) 8x³ − 12x² + 6x − 1
(1) (x + 2)³ (2) (2x − 1)³
Answer:(1) x³ + 6x² + 12x + 8 (2) 8x³ − 12x² + 6x − 1
Example ②
Expand (x + 3)(x² − 3x + 9).
Answer:Using the formula (a+b)(a²−ab+b²) = a³+b³, x³ + 27
Expand (x + 3)(x² − 3x + 9).
Answer:Using the formula (a+b)(a²−ab+b²) = a³+b³, x³ + 27
Key Points
- The coefficients of (a±b)³ are 1, 3, 3, 1
- Memorize the factorization forms of a³ ± b³
- Cubic formulas frequently appear in factorization
Practice Problems
- Expand (x − 4)³.
- Expand (x + 5)(x² − 5x + 25).
- Expand (2a − 3)³.
Answers and Explanations
- Answer:x³ − 12x² + 48x − 64
Explanation: x³ − 3x²·4 + 3x·16 − 64. - Answer:x³ + 125
Explanation: Using the formula for a³ + b³, x³ + 5³. - Answer:8a³ − 36a² + 54a − 27
Explanation: (2a)³ − 3(2a)²·3 + 3·2a·9 − 27.