A³ + B³ Formula: Expansion and Applications
The algebraic expression a³ + b³ represents the sum of two cubes. This formula is fundamental in mathematics, frequently appearing in algebra, trigonometry, and problem-solving contexts. For Indian students preparing for exams like JEE, NEET, and board mathematics, mastering this formula is essential.
Expansion of a³ + b³
The expanded form of the a³ + b³ formula is:
a³ + b³ = (a + b)(a² - ab + b²)
This identity allows the sum of two cubes to be factored into a product of a linear term and a quadratic term.
Derivation of the Formula
To derive the formula, start with the algebraic identity for the product of two binomials:
(a + b)³ = a³ + 3a²b + 3ab² + b³
Rearranging terms: a³ + b³ = (a + b)³ - 3ab(a + b)
Factor out (a + b): a³ + b³ = (a + b)[(a + b)² - 3ab]
Expand (a + b)²: (a + b)² = a² + 2ab + b²
Substitute back: a³ + b³ = (a + b)(a² + 2ab + b² - 3ab)
Simplify: a³ + b³ = (a + b)(a² - ab + b²)
Practical Examples
Example 1: Simplify 8³ + 27³
Let a = 2 and b = 3 since 2³ = 8 and 3³ = 27.
Apply the formula: 8³ + 27³ = (2 + 3)(2² - 2×3 + 3²) = 5(4 - 6 + 9) = 5×7 = 35
Example 2: Factor x³ + 64
Express 64 as 4³: x³ + 64 = x³ + 4³
Factor using the formula: = (x + 4)(x² - 4x + 16)
Applications in Mathematics
1. Polynomial Factorization
The formula helps factor expressions like x³ + 8 or 27y³ + 1 into simpler terms, aiding in solving equations.
2. Trigonometric Identities
In trigonometry, the formula supports identities involving sin³θ + cos³θ or expressions in cubic equations.
3. Competitive Exams
Indian engineering and medical entrance exams often include questions testing the application of this formula in complex algebraic manipulations.
Common Mistakes to Avoid
- Incorrect Expansion: Forgetting the middle term -ab in (a² - ab + b²).
- Misapplication: Using the formula for differences of cubes (a³ - b³) instead of sums.
Frequently Asked Questions
Q1: What is the difference between a³ + b³ and a³ - b³? A: The formula for a³ - b³ is (a - b)(a² + ab + b²), differing in sign and middle term.
Q2: Can the formula be used for variables with exponents other than 3? A: No, the formula specifically applies to cubes. For other powers, different identities apply.
Q3: How is this formula useful in calculus? A: It simplifies expressions during differentiation or integration of cubic polynomials.
Mastering the a³ + b³ formula enables efficient problem-solving across algebraic, trigonometric, and competitive exam contexts in the Indian education system.
