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 find the value of the algebraic expression. We will find the value of the sum of the product of the variables by substituting the given equation in the square of the sum of the three variables' identity. We will substitute all the known values in the algebraic identity to find the value of the given algebraic expression.

Formula Used:
We will use the following formula:
1) Algebraic Identity: \[{\left( {x + y + z} \right)^2} = {x^2} + {y^2} + {z^2} + 2\left( {xy + yz + zx} \right)\]
2) Algebraic Identity: ${{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz=\left( x+y+z \right)\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx \right)$ where $x,y,z$ are three numbers respectively.

Complete step by step solution:
The given equation
$x+y+z=12$……………………………………….\[\left( 1 \right)\]
\[{x^2} + {y^2} + {z^2} = 70\]………………………………….$\left( 2 \right)$
Now, we will square the given equation $\left( 1 \right)$ on both the sides, we get
\[{\left( {x + y + z} \right)^2} = {\left( {12} \right)^2}\]
By using the Algebraic Identity ${{\left( x+y+z \right)}^{2}}={{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2\left( xy+yz+zx \right)$, we get
 \[ \Rightarrow {x^2} + {y^2} + {z^2} + 2\left( {xy + yz + zx} \right) = 144\]………………………………………$\left( 3 \right)$
Now, by substituting the equation $\left( 2 \right)$ in the above equation \[\left( 3 \right)\], we get
\[ \Rightarrow 70 + 2\left( {xy + yz + zx} \right) = 144\]
Subtracting 70 from both the sides, we get
\[ \Rightarrow 2\left( {xy + yz + zx} \right) = 144 - 70\]
$\Rightarrow 2\left( xy+yz+zx \right)=74$
Dividing both sides by 2, we get
\[ \Rightarrow xy + yz + zx = \dfrac{{74}}{2}\]
$\Rightarrow xy+yz+zx=37$ …………………………………………………………….$\left( 4 \right)$
Now, by using the Algebraic Identity ${{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz=\left( x+y+z \right)\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx \right)$, we get
${{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz=\left( x+y+z \right)\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}}-\left( xy+yz+zx \right) \right)$
Substituting the values from equation \[\left( 1 \right),\left( 2 \right)\] and\[\left( 4 \right)\] in above equation, we get
\[ \Rightarrow {x^3} + {y^3} + {z^3} - 3xyz = \left( {12} \right)\left( {70 - 37} \right)\]
Subtracting the terms, we get
$\Rightarrow {{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz=\left( 12 \right)\left( 33 \right)$
Multiplying the equation, we get
\[ \Rightarrow {x^3} + {y^3} + {z^3} - 3xyz = 396\]

Therefore, the value of ${{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz$ is 396.

Note:
We know that an algebraic expression is defined as an expression with a combination of variables, constants, and operators. An algebraic identity is defined as an expression that is true for all the values of the variables. In algebraic expressions, a variable can take any value. If the values of the variables are changed, then the values of the expressions are also changed with respect to the value of the variables. The algebraic identities can be verified by using the method of substitution.