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

1

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Solution

The correct option is A $1$
$(x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x) = x^{3} + y^{3} + z^{3} - 3 x y z$
Given $x + y + z = 1$ , $x y + y z + z x = - 1$ and $x y z = - 1$
To find $x^{2} + y^{2} + z^{2} = x^{2} + y^{2} + z^{2} + 2 (x y + y z + z x)$
$1 = x^{2} + y^{2} + z^{2} + (2 \times - 1)$
$\Rightarrow$ $x^{2} + y^{2} + z^{2} = 1 + 2 = 3$
$∴$ Substituting all the values in eqn $(i)$ we get
$1 \times (3 - (- 1)) = x^{3} + y^{3} + z^{3} - (3 \times - 1)$
$\Rightarrow 1 \times (3 + 1) = x^{3} + y^{3} + z^{3} + 3$
$\Rightarrow 4 = x^{3} + y^{3} + z^{3} + 3$
$\Rightarrow x^{3} + y^{3} + z^{3} = 1$

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