let P_α (x) =x^3 +2α x −3 1) determine the roots of P_α 2) determine the roots of P


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Question Number 39636 by math khazana by abdo last updated on 09/Jul/18

$l e t P_{α} (x) = x^{3} + 2 α x - 3$ $1) d e t e r m i n e t h e r o o t s o f P_{α}$ $2) d e t e r m i n e t h e r o o t s o f P_{- 1}$
	Answered by ajfour last updated on 09/Jul/18

	$l e t x = u + v$ $\Rightarrow {(u + v)}^{3} + 2 α (u + v) - 3 = 0$ $\Rightarrow u^{3} + v^{3} + (u + v) (3 u v + 2 α) - 3 = 0$ $F u r t h e r l e t (w e c a n . .) t h a t$ $3 u v + 2 α = 0$ $\Rightarrow u^{3} v^{3} = - \frac{8 α^{3}}{27}$ $T h e n u^{3} + v^{3} = 3$ $u^{3}, v^{3} a r e t h e n r o o t s o f e q .$ $z^{2} - 3 z - \frac{8 α^{3}}{27} = 0$ $\Rightarrow u^{3}, v^{3} = \frac{3 \pm \sqrt{9 + \frac{32 α^{3}}{27}}}{2}$ $A s x = u + v$ $\Rightarrow x (α) = \sqrt[3]{\frac{3}{2} + \sqrt{\frac{9}{4} + \frac{8 α^{3}}{27}}}$ $+ \sqrt[3]{\frac{3}{2} - \sqrt{\frac{9}{4} + \frac{8 α^{3}}{27}}}$ $x ∣_{α = - 1} = \sqrt[3]{\frac{3}{2} + \sqrt{\frac{211}{108}}}$ $+ \sqrt[3]{\frac{3}{2} - \sqrt{\frac{211}{108}}} .$
	Commented by MrW3 last updated on 09/Jul/18

	$A j f o u r s i r i s n o w a l s o a s p e c a l i s t f o r$ $c u b i c e q u a t i o n s, o n s i d e o f M J S s i r .$
	Commented by ajfour last updated on 09/Jul/18

	$Y o u t a u g h t m e h o w, s o m e t i m e b a c k . .$
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