If $a, b, c$ are the roots of $x^{3} + 2 x^{2} - 3 x - 1 = 0$ , find the value of $a^{- 3} + b^{- 3} + c^{- 3}$ .

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Solution

Given : $a, b, c$ are the roots of $x^{3} + 2 x^{2} - 3 x - 1 = 0$ ....... $(i)$
Writing $\frac{1}{y}$ for $x$ in $(i)$ , the transformed equation is
$y^{3} + 3 y^{2} - 2 y - 1 = 0$
and the given expression is equal to the value of $S_{3}$ in this equation.
Here $S_{1} = a + b + c = - 3$ ; $a b + b c + a c = - 2$
$S_{2} = (a + b + c)^{2} - 2 (a b + b c + a c) = (- 3)^{2} - 2 (- 2) = 13$ ;
and $S_{3} + 3 S_{2} - 2 S_{1} - 3 = 0$ ;
whence we obtain $S_{3} = - 42$ .

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