01 · ALGEBRAIC IDENTITIES
Standard Identities
NCERT IDENTITIES
Class 9 Algebraic Formulas
1. (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
2. (x + y)³ = x³ + y³ + 3xy(x + y) = x³ + 3x²y + 3xy² + y³
3. (x - y)³ = x³ - y³ - 3xy(x - y) = x³ - 3x²y + 3xy² - y³
4. x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)
5. Special Case: If x + y + z = 0, then x³ + y³ + z³ = 3xyz.
02 · SOLVED FACTORISATION
Applying Identities
FACTORISATION EXAMPLE
Factorise 8x³ + y³ + 27z³ - 18xyz
Rewrite in standard form (2x)³ + y³ + (3z)³ - 3(2x)(y)(3z):
Using identity x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx):
= (2x + y + 3z)(4x² + y² + 9z² - 2xy - 3yz - 6zx)
= (2x + y + 3z)(4x² + y² + 9z² - 2xy - 3yz - 6zx)