Question

If x + y + z = 1, xy + yz + zx = -1 and xyz = -1, find the value of ${x^3} + {y^3} + {z^3}$.

Answer 1

Hint:
The value of the required expression can be found by using the expansion of ${x^3} + {y^3} + {z^3} - 3xyz$. We can obtain the required expression by substituting the values of the terms given in the expansion.

Complete step by step solution:
The expression, ${x^3} + {y^3} + {z^3}$occurs in the expansion of the formula ${x^3} + {y^3} + {z^3} - 3xyz$, so we will use this expansion to find out the value of the required expression.
${x^3} + {y^3} + {z^3} - 3xyz = \left( {x + y + z} \right)\left( {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right)$ …equation (1)
On rearranging the terms in equation (1), we obtain,
${x^3} + {y^3} + {z^3} = \left( {x + y + z} \right)\left( {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right) + 3xyz$ …equation (2)
We now substitute the values of (x + y + z), (xy + yz + zx) and xyz in equation (2), we get,
${x^3} + {y^3} + {z^3} = 1\left[ {{x^2} + {y^2} + {z^2} - \left( { - 1} \right)} \right] + 3\left( { - 1} \right) = {x^2} + {y^2} + {z^2} - 2$
$ \Rightarrow {x^3} + {y^3} + {z^3} = {x^2} + {y^2} + {z^2} - 2$ …equation (3)
But, we need the value of ${x^2} + {y^2} + {z^2}$ in order to obtain the value of the required expression. The term ${x^2} + {y^2} + {z^2}$ can be found in the expansion of ${\left( {x + y + z} \right)^2}$.
${\left( {x + y + z} \right)^2} = {x^2} + {y^2} + {z^2} + 2\left( {xy + yz + zx} \right)$ …equation (4)
On rearranging the terms in equation (4), we obtain,
${x^2} + {y^2} + {z^2} = {\left( {x + y + z} \right)^2} - 2\left( {xy + yz + zx} \right)$ …equation (5)
We find the value of ${x^2} + {y^2} + {z^2}$ by substituting the values of (x + y + z) and (xy + yz + zx) in equation (5)
${x^2} + {y^2} + {z^2} = {\left( 1 \right)^2} - 2\left( { - 1} \right) = 3$
Since we have found the value of ${x^2} + {y^2} + {z^2}$, we can now find the value of the required expression by substituting this value in equation (3),
$ \Rightarrow {x^3} + {y^3} + {z^3} = 3 - 2 = 1$

Hence, the value of ${x^3} + {y^3} + {z^3}$ is 1.

Note:
In questions of this type, we should try using one of such expressions whose expansion contains the required term. It is possible that the expression we need to evaluate is present in more than one expansion. In such cases, we need to use the expansion, which uses the values of the terms given in the question.