Question

(a+b+c) ^3, find with proof

Answer 1

Algebra,

We have,
(a+b+c)³

Have to proof,
a³+ b³+ c³+ 3(a+b)(b+c)(c+a)

Now,
(a+b+c)³

= {a+(b+c)}³

= a³+(b+c)³+3a(b+c){a+(b+c)}

= a³+b³+c³+3bc(b+c)+3ab{a(b+c)}+3ac{a(b+c)}

= a³+b³+c³+3bc(b+c)+3a²b+3ab(b+c)+3a²c+3ac(b+c)

= a³+b³+c³+3bc(b+c)+3ab(b+c)+3ac(b+c)+3a²b+3a²c

= a³+b³+c³+3bc(b+c)+3ab(b+c)+3ac(b+c)+3a²(b+c)

= a³+b³+c³+3(b+c)(bc+ab+ac+a²)

= a³+b³+c³+3(b+c){b(a+c)+a(a+c)}

= a³+b³+c³+3(b+c)(a+c)(b+a)

= a³+b³+c³+3(a+b)(b+c)(c+a) [proved]

That's it
Hope it helped (≧▽≦)