Complete Guide to a³ + b³ and a³ – b³ Formulas

Detailed Formula Explanations

1. a³ + b³ Formula (Sum of Cubes)

Formula:

a³ + b³ = (a + b)(a² - ab + b²)

When to Use:

• Factorizing expressions involving the sum of two cubic terms
• Simplifying algebraic fractions
• Solving cubic equations

Example:

8³ + 27 = 512 + 27 = 539
Using formula: (8 + 3)(8² - 8×3 + 3²) = (11)(64 - 24 + 9) = 11 × 49 = 539 ✓

2. a³ – b³ Formula (Difference of Cubes)

Formula:

a³ - b³ = (a - b)(a² + ab + b²)

When to Use:

• Factorizing expressions involving the difference of two cubic terms
• Simplifying complex algebraic expressions
• Solving polynomial equations

Example:

64 - 27 = 37
Using formula: (4 - 3)(4² + 4×3 + 3²) = (1)(16 + 12 + 9) = 1 × 37 = 37 ✓

3. a³ + b³ + c³ – 3abc Formula

Formula:

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

Special Case: When a + b + c = 0, then:

a³ + b³ + c³ = 3abc

Application:

• Solving problems involving three variables
• Competitive exam questions
• Advanced algebraic simplifications

Example: If a = 2, b = 3, c = -5 (where a + b + c = 0):

2³ + 3³ + (-5)³ = 3(2)(3)(-5) = -90
8 + 27 - 125 = -90 ✓

4. (a + b)³ Formula

Formula:

(a + b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b)

Expanded Form:

(a + b)³ = a³ + b³ + 3ab(a + b)

5. (a – b)³ Formula

Formula:

(a - b)³ = a³ - 3a²b + 3ab² - b³ = a³ - b³ - 3ab(a - b)

Expanded Form:

(a - b)³ = a³ - b³ - 3ab(a - b)

Proof of a³ + b³ Formula

Method 1: Algebraic Verification

Starting with the factored form:

(a + b)(a² - ab + b²)

Expanding using distributive property:

= a(a² - ab + b²) + b(a² - ab + b²)
= a³ - a²b + ab² + ba² - ab² + b³
= a³ - a²b + ba² + ab² - ab² + b³
= a³ + b³

Hence proved: a³ + b³ = (a + b)(a² – ab + b²)

Proof of a³ – b³ Formula

Method 1: Algebraic Verification

Starting with the factored form:

(a - b)(a² + ab + b²)

Expanding using distributive property:

= a(a² + ab + b²) - b(a² + ab + b²)
= a³ + a²b + ab² - ba² - ab² - b³
= a³ + a²b - ba² + ab² - ab² - b³
= a³ - b³

Hence proved: a³ – b³ = (a – b)(a² + ab + b²)

Quick Reference Table: Differences Between a³ + b³ and a³ – b³

Practice Problems

Problem 1: Factorize 27x³ + 64y³

Solution:

= (3x)³ + (4y)³
= (3x + 4y)[(3x)² - (3x)(4y) + (4y)²]
= (3x + 4y)(9x² - 12xy + 16y²)

Problem 2: Factorize 125a³ – 8b³

Solution:

= (5a)³ - (2b)³
= (5a - 2b)[(5a)² + (5a)(2b) + (2b)²]
= (5a - 2b)(25a² + 10ab + 4b²)

Frequently Asked Questions (FAQs)

Q1: What is the formula for x³ + y³?

Answer: The formula for x³ + y³ is:

x³ + y³ = (x + y)(x² - xy + y²)

This is the sum of cubes formula, where the expression is factorized into two factors: a linear factor (x + y) and a quadratic factor (x² – xy + y²).

Q2: What is the factorization of a³ – b³?

Answer: The factorization of a³ – b³ is:

a³ - b³ = (a - b)(a² + ab + b²)

This is known as the difference of cubes formula. The first factor is (a – b) and the second factor is a quadratic expression (a² + ab + b²) which cannot be factored further over real numbers.

Q3: How do you factor a³ + b³?

Answer: To factor a³ + b³, follow these steps:

Step 1: Identify the cube roots of each term (a and b)

Step 2: Apply the sum of cubes formula:

a³ + b³ = (a + b)(a² - ab + b²)

Step 3: The first factor is the sum of the cube roots: (a + b)

Step 4: The second factor is: (square of first term) – (product of both terms) + (square of second term)

Example: Factor 8x³ + 27

• Cube roots: 2x and 3
• Formula: (2x + 3)[(2x)² – (2x)(3) + 3²]
• Answer: (2x + 3)(4x² – 6x + 9)

Q4: x³ + y³ का सूत्र क्या है? (What is the formula for x³ + y³ in Hindi?)

उत्तर (Answer): x³ + y³ का सूत्र है:

x³ + y³ = (x + y)(x² - xy + y²)

यह घनों के योग का सूत्र है। इसमें व्यंजक को दो गुणनखंडों में विभाजित किया जाता है: एक रैखिक गुणनखंड (x + y) और एक द्विघात गुणनखंड (x² – xy + y²)।

व्याख्या: जब दो पदों के घनों का योग होता है, तो उसे इस सूत्र की सहायता से गुणनखंड किया जा सकता है।

Q5: a³ – b³ का गुणनखंड क्या है? (What is the factorization of a³ – b³ in Hindi?)

उत्तर (Answer): a³ – b³ का गुणनखंड है:

a³ - b³ = (a - b)(a² + ab + b²)

यह घनों के अंतर का सूत्र है।

विवरण:

• पहला गुणनखंड: (a – b) – यह रैखिक व्यंजक है
• दूसरा गुणनखंड: (a² + ab + b²) – यह द्विघात व्यंजक है जिसे और अधिक गुणनखंड नहीं किया जा सकता

उदाहरण: 64x³ – 27 को गुणनखंड करें

= (4x)³ - 3³
= (4x - 3)[(4x)² + (4x)(3) + 3²]
= (4x - 3)(16x² + 12x + 9)

Important Tips for Students

1. Remember the sign pattern: In a³ + b³, the middle term in the second factor is negative (-ab), while in a³ – b³, it’s positive (+ab)
2. First factor is simple: The first factor always mirrors the operation between a³ and b³
3. Second factor never changes sign of outer terms: In both formulas, a² and b² are always positive
4. Verify your answer: Always expand your factored form to check if you get back the original expression
5. Common mistakes to avoid:
   • Confusing the signs in the quadratic factor
   • Forgetting to identify perfect cubes correctly
   • Not simplifying coefficients when dealing with numerical cubes