Question

(a+b)³+(b+c)³+(c+a)³-3(a+b)(b+c)(c+a)

Answer 1

Hello friends!!

Here is your answer :

(a+b)³+(b+c)³+(c+a)³-3(a+b)(b+c)(c+a)


Using identity : a³ + b³ + c³ - 3abc = ( a + b + c) ( a² + b² + c² - ab - bc - ca)

= [ (a+b) + (b+c) + (c+a)] [ (a+b)² + (b+c)² + (c+a)² - {(a+b)(b+c)} - {(b+c) (c+a)} - {(c+a)(a+b)}

Using identity : ( x + y )² = x² + y² + 2xy

= [ a + b + b + c + c + a] [ { (a)² + (b)² + 2(a)(b)} + { (b)² + (c)² + 2(b)(c)} + { (c)² + (a)² + 2(c)(a)} - { a(b+c) +b(b+c)} - { b(c+a) +c(c+a)} - { c(a+b) +a(a+b)}

= [ 2a + 2b +2c] [ a² + b² + 2ab + b² + c² + 2bc + c² + a² + 2ca - { ab + ca + b² + bc } - { bc + ab + c² + ca } - { ca + bc + a² + ab }

= [ 2a + 2b + 2c] [ 2a² + 2b² + 2c² + 2ab +2bc + 2ca - ab - ca - b² - bc - bc - ab - c² - ca - ca - bc - a² - ab]

= [ 2a + 2b + 2c] [ 2a² + 2b² + 2c² + 2ab +2bc + 2ca - 3ab - 3ca - 3bc - a² - b² - c² ]

=[ 2a + 2b + 2c] [ a² + b² + c² - ab - bc - ca]


Hope it helps you... ☺️☺️

Answer 2

I hope this will help you


-by ABHAY