let-P-x-x-3-2-x-3-1-determine-the-roots-of-P-2-determine-the-roots-of-P-1-

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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