Difference of Two Cubes Formula: Proof, Chart & Examples

Algebra is a vital branch of mathematics that simplifies complex problems using variables such as x, y, and z. It often involves evaluating statements asserting the equality of two expressions constructed using variables, coefficients, and constants. These algebraic equations are frequently represented in the form P=0, where P is a polynomial. To tackle these problems efficiently, understanding foundational identities like the difference of two cubes is essential for students preparing for competitive exams or general mathematics.

Understanding Algebraic Equations and Formulas

An algebraic formula is a standardized mathematical rule expressed as an equation. It describes the relationship between variables, making it easier to solve complex expressions. Algebra itself is a core branch of mathematics that represents real-world situations and problems using variables and operational symbols like addition and subtraction.

a3 b3 Formula Chart

Quick reference charts are invaluable when dealing with polynomial expansions. For further exploration of standard identities, you can consult resources on the Algebra Formula.

• a³-b³ = (a - b) (a²+ab+b²)
• a³ + b³ = (a + b) (a² – ab + b²)
• (a + b)³ = a³ + b³ + 3ab(a+b)
• (a – b)³ = a³ – b³ – 3ab (a-b)
• a³ + b³ + c³ – 3abc = (a + b + c) (a²+b²+c²–ab–bc–ca)

Steps for the Proof and Derivation of a³-b³ Formula

1. Start from the known cube expansion formula: (a-b)³ = a³ - b³ - 3ab(a-b)
2. Rearrange the terms: a³ - b³ = (a-b)³ + 3ab(a-b)
3. Factor out (a-b): a³ - b³ = (a-b) [(a-b)² + 3ab]
4. Expand the term within the square brackets on the Right Hand Side (RHS): a³ - b³ = (a - b) (a² + b² - 2ab + 3ab)
5. Combine like terms: a³ - b³ = (a - b) (a² + b² + ab)
6. Reach the final factored form: a³ - b³ = (a - b) (a² + ab + b²)

Solved Examples and Applications

Mastery of this identity allows candidates appearing for competitive exams such as NTSE, NDA, AFCAT, SSC, and Railway to quickly simplify expressions:

• Example 1: Factorizing 125a³ - 27b³ gives (5a - 3b)(25a² + 15ab + 9b²).
• Example 2: Factorizing (3a + b)³ - (2a + b)³ gives a(19a² + 15ab + 3b²).
• Example 3: The value of 5³ - 2³ calculated using the formula is 117.

After practicing these expansions, candidates should systematically review related polynomial equations and practice timed problem sets to ensure swift execution during high-stakes testing scenarios.