Question

prove that : (a+b)^3 +(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a) = 2(a^3+b^3+c^3-3abc)

Answer 1

hope helpful budy.......
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Answer 2

HELLO DEAR,

now expand ,

(a+b)³ = a³+b³+3ab(a+b)

(b+c)³ = b³+c³+3bc(b+c)

(c+a)³ = c³+a³+3ca(c+a)



---according to question:---

=> a³+b³+3ab(a+b) + b³+c³+3bc(b+c) + c³+a³+3ca(c+a) -3[(a+b)(b+c)(c+a)]

=> 2a³+2b³+2c² + 3[ab(a+b) + ca(c+a) + bc(b+c)]
- 3[(a+b)(b+c)(c+a)]

=> 2(a³+b³+c³) + 3 [ a²b+ b²a +c²a +a²c +b²c +c²b] - 3[ (ab+ac+b²+bc)(c+a)]

=>2(a³+b³+c³) + 3 [ a²b+ b²a +c²a +a²c +b²c +c²b] - 3[abc + ac² +b²c + bc² + a²b + a²c +ab² +abc]

=> 2(a³+b³+c³) + 3a²b+3b²a +3c²a -3a²c +3b²c +3c²b - 3abc - 3ac² -3b²c - 3bc² - 3a²b - 3a²c -3ab² - 3abc

=> 2(a³+b³+c³) -6abc

=>2(a³+b³+c³-3abc)

I HOPE ITS HELP YOU DEAR,
THANKS