x³ + y³ and x³ − y³ Algebraic Identities

Identity 1: Sum of Cubes x³ + y³ = (x + y)(x² − xy + y²)

Identity 2: Difference of Cubes x³ − y³ = (x − y)(x² + xy + y²)

Memory trick: For sum of cubes (x³+y³), the second bracket has a minus sign in the middle (x²−xy+y²). For difference of cubes (x³−y³), the second bracket has a plus sign (x²+xy+y²). The SOAP method: Same sign, Opposite sign, Always Positive.

Multiply (x + y)(x² − xy + y²): = x(x² − xy + y²) + y(x² − xy + y²) = x³ − x²y + xy² + x²y − xy² + y³ = x³ − x²y + x²y + xy² − xy² + y³ = x³ + y³ ✓

Multiply (x − y)(x² + xy + y²): = x(x² + xy + y²) − y(x² + xy + y²) = x³ + x²y + xy² − x²y − xy² − y³ = x³ + x²y − x²y + xy² − xy² − y³ = x³ − y³ ✓

Example 1: Factorise 8a³ + 27b³ = (2a)³ + (3b)³ = (2a + 3b)[(2a)² − (2a)(3b) + (3b)²] = (2a + 3b)(4a² − 6ab + 9b²)

Example 2: Factorise 125x³ − 64y³ = (5x)³ − (4y)³ = (5x − 4y)[(5x)² + (5x)(4y) + (4y)²] = (5x − 4y)(25x² + 20xy + 16y²)

Example 3: Evaluate 99³ + 1 (using x=99, y=1) = (99+1)(99²−99+1) = 100 × (9801−99+1) = 100 × 9703 = 970300

• (x+y)² = x² + 2xy + y²
• (x−y)² = x² − 2xy + y²
• x²−y² = (x+y)(x−y)
• (x+y)³ = x³ + 3x²y + 3xy² + y³ = x³ + y³ + 3xy(x+y)
• (x−y)³ = x³ − 3x²y + 3xy² − y³ = x³ − y³ − 3xy(x−y)