Question

If the addition of three quantities a, band c is $a + b + c = 9$and${a^2} + {b^2} + {c^2} = 35$, then find the value of ${a^3} + {b^3} + {c^3} - 3abc$
$
  (a){\text{ 52}} \\
  (b){\text{ 108}} \\
  (c){\text{ 216}} \\
  (d){\text{ 182}} \\
$

Answer 1

Hint: In this question the two separate equations depicting the relation between a, b, and c are given to us. Use the basic algebraic identity ${\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2\left( {ab + bc + ca} \right)$to start with the solution. Simplify further by substituting the value to find out a relation that will help in evaluating the required equation.

Complete step-by-step answer:

Given data
$a + b + c = 9$……………………. (1)
And ${a^2} + {b^2} + {c^2} = 35$………………….. (2)
Then we have to find the value of ${a^3} + {b^3} + {c^3} - 3abc$
Now as we know that
${\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2\left( {ab + bc + ca} \right)$
So from equation (1) and (2) we have,
${\left( 9 \right)^2} = 35 + 2\left( {ab + bc + ca} \right) = 81$
So simplify the above equation we have,
$ \Rightarrow 2\left( {ab + bc + ca} \right) = 81 - 35 = 46$
Now divide by 2 throughout we have,
$ \Rightarrow \left( {ab + bc + ca} \right) = 23$……………………. (3)
Now it is a known fact that
$\left[ {{{\left( {a + b + c} \right)}^3} - 3abc} \right] = \left( {a + b + c} \right)\left( {{a^2} + {b^2} + {c^2} - \left( {ab + bc + ca} \right)} \right)$
So from equation (1), (2) and (3) we have,
$\left[ {{{\left( {a + b + c} \right)}^3} - 3abc} \right] = 9\left( {35 - 23} \right) = 9\left( {12} \right) = 108$
So this is the required answer.
Hence option (B) is correct.

Note: Whenever we face such types of problems the key concept is simply to have a good understanding of the basic algebraic identities, some of them are being mentioned above. This concept will help you get on the right track to reach the answer.