If a + b + c = 0 , then find the value of a 2 b c + b 2 c a + c 2 a b

Given that, a + b + c = 0 . We know that, a 3 + b 3 + c 3 - 3 a b c = a + b + c a 2 + b 2 + c 2 - a b - b c - c a a 3 + b 3 + c 3 - 3 a b c = 0 a + b + c = 0 a 3 + b 3 + c 3 = 3 a b c Now, a 2 b c + b 2 a c + c 2 a b = a 3 + b 3 + c 3 a b c = 3 a b c a b c = 3 Hence, a 2 b c + b 2 a c + c 2 a b = 3 .

Using suitable identity , find the value of :- 86 × 86 × 86 + 14 × 14 × 14 86 × 86 − 86 × 14 + 14 × 14

We know that a 3 + b 3 = ( a + b ) ( a 2 - a b + b 2 ) a 3 + b 3 a 2 − a b + b 3 = a + b Putting a = 86 a n d b = 14 , we get 86 3 + 14 3 86 2 − 86 × 14 + 14 3 = 86 + 14 86 × 86 + 14 × 14 86 × 86 − 86 × 14 + 14 × 14 = 86 + 14 = 100 86 × 86 + 14 × 14 86 × 86 − 86 × 14 + 14 × 14 = 100 " Therefore, the required solution is obtained".