Loading Document( a + b )² = a² + 2ab + b²
( a - b )² = a² - 2ab + b²
a² - b² = (a - b)(a + b)
a² + b² = ( a + b )² - 2ab
=( a - b )² + 2ab
(x + a )(x + b ) = x² + (a + b)x + ab
ab=(2a+b)−(2a−b)
(a + b )³ = a³ + 3a²b + 3ab² + b³
= a³ + b³ + 3ab(a + b)
(a - b )³ = a³ - 3a²b + 3ab² - b³
= a³ - b³ - 3ab(a - b)
a³ + b³ = ( a + b )( a² -ab + b² )
= ( a + b )³ - 3ab( a + b )
a³ - b³ = ( a - b )( a² + ab + b² )
= ( a - b )³ + 3ab( a - b )
( a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ca
a³ + b³ + c³ - 3abc = ( a + b + c )( a² + b² + c² - ab - bc - ca )
=21 ( a + b + c )[ ( a - b )² + ( b - c )² + ( c - a )²
⇒a³ + b³ + c³ = 3abc provided ( a + b + c ) = 0
The above-mentioned identities are sufficient. Several other identities can be obtained by the above identities.