Question

Find the value of \[{x^3} + {y^3} + {z^3} - 3xyz\], if \[x + y + z = 12\] and \[{x^2} + {y^2} + {z^2} = 70\].

Answer 1

Hint: Here we have to simplify the given equation to find its value. We will first modify the equation whose value is to be found in terms of the given equation so that we can substitute their values in the equation. We will further simplify the equation to get the value of the main equation.

Complete step by step solution:
We will modify the equation \[{x^3} + {y^3} + {z^3} - 3xyz\].
Rewriting the equation, we get
\[ \Rightarrow {x^3} + {y^3} + {z^3} - 3xyz = \left( {x + y + z} \right)\left( {{x^2} + {y^2} + {z^2} - \left( {xy + yz + zx} \right)} \right)\]
Substituting the values \[x + y + z = 12\] and \[{x^2} + {y^2} + {z^2} = 70\] in the above equation, we get
\[ \Rightarrow {x^3} + {y^3} + {z^3} - 3xyz = \left( {12} \right)\left( {70 - \left( {xy + yz + zx} \right)} \right)\] …………… (1)
Now we have to find the value of the expression \[xy + yz + zx\] . We know that \[{\left( {x + y + z} \right)^2} = {x^2} + {y^2} + {z^2} + 2\left( {xy + yz + zx} \right)\].
So by this we will find the value of \[xy + yz + zx\] and then put it in the equation (1). So, we get
\[ \Rightarrow {\left( {12} \right)^2} = 70 + 2\left( {xy + yz + zx} \right)\]
\[ \Rightarrow 144 = 70 + 2\left( {xy + yz + zx} \right)\]
Adding and subtracting the like terms, we get
\[ \Rightarrow \left( {xy + yz + zx} \right) = \dfrac{{144 - 70}}{2}\]
Simplifying the above equation, we get
\[ \Rightarrow \left( {xy + yz + zx} \right) = \dfrac{{74}}{2} = 37\]
Now we will substitute the value of \[xy + yz + zx\] in the equation (1). Therefore, we get
\[\begin{array}{l} \Rightarrow {x^3} + {y^3} + {z^3} - 3xyz = \left( {12} \right)\left( {70 - \left( {xy + yz + zx} \right)} \right) = 12(70 - 37) = 12 \times 33\\ \Rightarrow {x^3} + {y^3} + {z^3} - 3xyz = \left( {12} \right)\left( {70 - \left( {xy + yz + zx} \right)} \right) = 12 \times 33\end{array}\]
Multiplying the terms, we get
\[ \Rightarrow {x^3} + {y^3} + {z^3} - 3xyz = 396\]
Hence, 396 is the value of the equation \[{x^3} + {y^3} + {z^3} - 3xyz\].

Note: Here the important thing to solve the question is that we need to rewrite the equation whose values to be found out such that the modified equation has the terms whose value is already given. Also, we need to know the different algebraic identities to simplify the equation easily. Algebraic identities are equations where the value of the left-hand side of the equation is identically equal to the value of the right-hand side of the equation. An algebraic identity is an equality that holds for any values of its variables like the identity we used in our question i.e. \[ \Rightarrow {\left( {x + y + z} \right)^2} = {x^2} + {y^2} + {z^2} + 2\left( {xy + yz + zx} \right)\].