Question

If a+b+c=5 and ab+bc+ca=15,then find the value of (a+b)^3+(b+c)^3+(a+c)^-3(a+b)(b+c)(c+a)

Answer 1

(a + b)3 + (b +c)3 + (c + a)3 - 3(a + b)(b +c)(c + a) = 2a3 + 2b3 +2c3 - 6abc

2a3 + 2b3 + 2c3 - 6abc = 2(a3 + b3 +c3 - 3abc) = 2(a + b+ c)(a2 + b2 +c2 - (ab +bc + ca)) -------- (i)

(a + b +c)2 = a2 +b2 +c2 + 2(ab +bc + ca)

52 = a2 +b2 + c2 + 30

25 - 30 = a2 + b2 + c2

- 5 = a2 +b2 + c2

Putting the values in equation (i) we get

= 2(5)(-5 - 15)

= 2(5)(-20) = - 200

Ans: - 200

Therefore you can conclude to this point that it must be a property that

2a3 + 2b3 + 2c3 - 6abc = 2(a3 + b3 +c3 - 3abc) = 2(a + b+ c)(a2 + b2 +c2 - ab - bc - ca)