Formulae For two Index, two term
(a) (x + y)² = x² + y² + 2xy = (x–y)² + 4xy(b) (x–y)² = x² + y² – 2xy = (x² + y)² – 4xy
(c) (x+y)² + (x–y)² = 2 (x²+y²) (d) (x+y)² – (x–y)² = 4xy
Formulae For Tree Index, two terms
(a) (a+b)³ = a³ + 3a² b + 3ab² + b³ = a³ + b³ + 3ab (a+b)(b) (a–b)³ = a³ – 3a² b + 3ab² –b³ = a³ – b³ – 3ab (a–b)
(c) (a+b)³ + (a–b)³ = 2(a³ + 3ab²) = 2a (a² +3b²) (d) (a+b)² – (a–b)³ = 6a² b + 2b³ = 2b (3a²+b²)
Formulae For Four & Five Index, two terms
(a) (a + b)⁴ = a⁴ + 4a³ + 6a² b² + 4ab³ + b⁴
(b) (a – b)⁴ = a⁴ – 4a³b + 6a² b² – 4ab³ + b⁴
(c) (a + b)⁵ = a⁵ + 5a⁴ b + 10a³ b² + 10a² b³ + 5ab⁴ + b⁵
(d) (𝐚 – 𝐛)𝟓 = a5– 5𝑎21b+10a3b2–10a2b3 +5ab4–b5
Ordinary Factors
(a) a² – b² = (a + b) (a – b)
(b) a³ + b³ = (a + b) (a² – ab + b²) = (a + b)³ – 3ab (a + b)
(c) a³ – b³ = (a– b) (a² + ab + b²) = (a– b)³ + 3ab (a – b)
(d) a⁴ – b⁴ = (a² + b²) (a² – b²) = (a – b) (a³ + a²b + ab² + b³)
Special Factors
(a) (a + b + c) (bc + ca + ab) – abc = (b + c) (c + a) (a + b)
(b) (a + b + c)³ – a³ – b³ – c³ = 3 (a + b) (b + c) (c + a)
(c) a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – bc – ca) ➢ If a³ + b³ + c³ = 3abc.
(d) a³ + b³ + c³ – 3abc = 1 2 (a + b + c) [(a – b)² + (b – c)² + (c – a)²]
(e) a² + b² + c² –ab – bc – ca = 1 2 {(a – b)² + (b– c)² + (c – a)²]
