if a,b and c root of the x^3 −16x^2 −57x+1=0 thi find thd volue of a^(1/5) +b^(1/5) +c^(1/5) =?


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Question Number 190520 by mathlove last updated on 04/Apr/23

$i f a, b a n d c r o o t o f t h e$ $x^{3} - 16 x^{2} - 57 x + 1 = 0$ $t h i f i n d t h d v o l u e o f$ $a^{\frac{1}{5}} + b^{\frac{1}{5}} + c^{\frac{1}{5}} = ?$
	Answered by Frix last updated on 05/Apr/23

	$If the solution is \in Z we need to find a$ $polynomial factor of degree 3 of$ $t^{15} - 16 t^{10} - 57 t^{5} + 1$ $I used software to find the factor$ $t^{3} - 1 t^{2} - 2 t + 1$ $The other factor of degree 12 is prime$ $\Rightarrow answer is 1$ $[I {don}^{'} t know if {there}^{'} s an analytical$ $method to solve this]$
	Answered by behi834171 last updated on 05/Apr/23

	$x^{3} = 16 x^{2} + 57 x - 1$ $x^{4} = 16 x^{3} + 57 x^{2} - x = 16 (16 x^{2} + 57 x - 1) + 57 x^{2} - x =$ $= 313 x^{2} + 911 x - 16$ $x^{5} = 313 x^{3} + 911 x^{2} - 16 x = 313 (16 x^{2} + 57 x - 1) + 911 x^{2} - 16 x =$ $= 5919 x^{2} + 17852 x - 313$ $\Rightarrow x^{5} - 5919 x^{2} - 17852 x + 313 = 0$ $y = \frac{1}{x} \Rightarrow \frac{1}{y^{5}} - \frac{5919}{y^{2}} - \frac{17852}{y} + 313 = 0$ $\Rightarrow 313 y^{5} - 17852 y^{4} - 5919 y^{3} + 1 = 0$ $\Rightarrow Σ a^{\frac{1}{5}} = - \frac{- 17852}{313} = \frac{17852}{313} . ◼$
	Commented by mr W last updated on 07/Apr/23

	$t h a n k s s i r!$ $y o u r m e t h o d s e e m s t o b e a g o o d$ $i d e a, b u t i t h i n k i t i s n o t c o r r e c t .$ $w h e n y o u t r a n s f o r m t h e e q u a t i o n$ $t o$ $313 y^{5} - 17852 y^{4} - 5919 y^{3} + 1 = 0,$ $i t h a s 5 r o o t s . s o y o u g e t w i t h$ $Σ y = Σ \frac{1}{x} = \frac{17852}{313} t h e s u m o f a l l i t s 5$ $r o o t s,$ $i . e . \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e} = \frac{17852}{313} .$ $b u t t h i s i s n o t t h a t w h a t t h e o r i g i n a l$ $q u e s t i o n h a s r e q u e s t e d :$ $a^{\frac{1}{5}} + b^{\frac{1}{5}} + c^{\frac{1}{5}} = ?$
	Commented by behi834171 last updated on 08/Apr/23

	$h e l l o m y d e a r m a s t e r .$ $y o u a r e r i g h t, a n d, i h a v e a t e r r i b l e t y p o$ $i d o n t d e l e t e m y w a y, b u t w i l l c o r r e c t i t$ $a n d i w i l l p o s t t h a t .$ $t h a n k s f o r p o i n t i n g t h i s .$
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