Algebraic Formulae

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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]