a³ + b³ and a³ - b³ Identities

Definition: The sum and difference of cubes identities are:

a³ + b³ = (a + b)(a² − ab + b²)

a³ − b³ = (a − b)(a² + ab + b²)

Where:

• a and b are any real numbers or algebraic expressions
• The first factor is a linear binomial (a + b) or (a − b)
• The second factor is a quadratic trinomial with three terms

Important:

• In the sum of cubes, the quadratic factor has a negative middle term (−ab).
• In the difference of cubes, the quadratic factor has a positive middle term (+ab).
• The sign pattern follows: same, opposite, always positive — the first sign matches the original, the middle sign is opposite, and the last term is always positive.

Derivation of a³ + b³ by expanding the right-hand side:

To verify: (a + b)(a² − ab + b²) = a³ + b³

Step-by-step expansion:

1. Multiply a with each term in the trinomial:
   a × a² = a³
   a × (−ab) = −a²b
   a × b² = ab²
2. Multiply b with each term in the trinomial:
   b × a² = a²b
   b × (−ab) = −ab²
   b × b² = b³
3. Combine all terms:
   a³ − a²b + ab² + a²b − ab² + b³
4. Cancel the middle terms:
   −a²b + a²b = 0 and ab² − ab² = 0
5. Result: a³ + b³

Derivation of a³ − b³ by expanding the right-hand side:

To verify: (a − b)(a² + ab + b²) = a³ − b³

Step-by-step expansion:

1. Multiply a with each term:
   a × a² = a³
   a × ab = a²b
   a × b² = ab²
2. Multiply −b with each term:
   −b × a² = −a²b
   −b × ab = −ab²
   −b × b² = −b³
3. Combine all terms:
   a³ + a²b + ab² − a²b − ab² − b³
4. Cancel the middle terms:
   a²b − a²b = 0 and ab² − ab² = 0
5. Result: a³ − b³

Common Forms and Patterns:

1. Direct Substitution

• Replace a and b with numbers or simple variables.
• Example: x³ + 8 = x³ + 2³ = (x + 2)(x² − 2x + 4)

2. Expressions with Coefficients

• Rewrite terms as perfect cubes before applying the identity.
• Example: 27x³ − 64 = (3x)³ − 4³ = (3x − 4)(9x² + 12x + 16)

3. Numerical Simplification

• Used to compute sums or differences of cubes quickly.
• Example: 53³ + 47³ = (53 + 47)(53² − 53 × 47 + 47²) = 100 × (2809 − 2491 + 2209) = 100 × 2527

4. Factorising Polynomial Expressions

• Higher-degree polynomials may contain sum/difference of cubes as a factor.
• Example: x⁶ − 1 = (x³)² − 1² = (x³ − 1)(x³ + 1), then apply both identities.

5. Combined with Other Identities

• a³ + b³ can also be written as (a + b)³ − 3ab(a + b).
• This form is useful when a + b and ab are known but individual values are not.

Applications of a³ + b³ and a³ − b³ Identities:

• Factorisation of Polynomials: These identities are the primary tool for factorising cubic expressions into linear and quadratic factors. They appear directly in NCERT exercises on polynomial factorisation.
• Simplifying Algebraic Fractions: When a³ + b³ or a³ − b³ appears in the numerator or denominator, factorising allows cancellation of common factors.
• Solving Cubic Equations: Equations of the form x³ = k can be rearranged to x³ − k = 0 and factorised using the difference of cubes identity to find all roots.
• Number Theory and Mental Arithmetic: Quick evaluation of expressions like 99³ + 1³ or 50³ − 30³ without computing individual cubes.
• Physics and Engineering: Volume difference calculations for hollow cubes and cubic containers use these identities for efficient computation.
• Higher Mathematics: These identities extend to sum/difference of nth powers and form the basis for factoring patterns in abstract algebra.