Question

If a + b + c = 9 and a²+ b² + c² =35, find the value of a³ + b³ + c³ −3abc

Answer 1

In the given problem, we have to find value of  a³ + b³ + c³ −3abc

Given  a + b + c = 9 , a²+ b² + c² =35

We shall use the identity

`(a+b+c)^2 = a^2 +b^2 + c^2 + 2 (ab+bc+ ca)`

`(a+b+c)^2 =35 + 2 (ab+bc+ ca)`

              `(9)^2 =35 + 2 (ab+bc+ ca)`

       `81 - 35 = 2 (ab+bc+ ca)`

               `46/ 2 =  (ab+bc+ ca)`

                 `23 = (ab+bc+ ca)`

We know that 

`a^3 + b^3 + c^3- 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - bc -ca)`

`a^3 + b^3 + c^3- 3abc = (a+b+c)[(a^2 + b^2 + c^2) - (ab + bc +ca)`

Here substituting  `a+b+c = 9,a^2  +b^2 + c^2 = 35 , ab +bc + ca = 23` we get

`a^3 +b^3 + c^3 - 3abc = 9 [(35 - 23)]`

                                      ` =9 xx 12`

                                      ` = 108`

Hence the value of  a³ + b³ + c³ −3abc is 108.