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Question Number 90709 by Cynosure last updated on 25/Apr/20

α, β a n d γ a r e t h e r o o t s o f x^{3} - 9 x + 9 = 0

f i n d t h e v a l u e o f (1) α^{- 3} + β^{- 3} + γ^{- 3}

(2) α^{- 5} + β^{- 5} + γ^{- 5}

Commented by jagoll last updated on 25/Apr/20

l e t α^{3} = t \Rightarrow α = \sqrt[3]{t}

\Rightarrow {(\sqrt[3]{t})}^{3} + 9 = 9 \sqrt[3]{t}

{(t + 9)}^{3} = 729 t \Rightarrow t^{3} + 27 t^{2} + 243 t + 729 = 729 t

t^{3} + 27 t^{2} - 486 t + 729 = 0, h a s r o o t s

α^{3}, β^{3} & γ^{3}

f i n d t h e p o l y n o m i a l w i t h r o o t s

\frac{1}{α^{3}}, \frac{1}{β^{3}} & \frac{1}{γ^{3}} \Rightarrow 729 x^{3} - 486 x^{2} + 27 x + 1 = 0

b y v i e t a r u l e \frac{1}{α^{3}} + \frac{1}{β^{3}} + \frac{1}{γ^{3}} = \frac{486}{729} = \frac{162}{243}

= \frac{54}{81} = \frac{2}{3}

Commented by Cynosure last updated on 25/Apr/20

w h a t o f t h e s e c o n d o n e s i r ?