if-the-root-of-equation-x-9-10x-1-0-are-x-1-and-x-2-x-1-x-2-1-find-x-1-x-2-

Question Number 164155 by HongKing last updated on 14/Jan/22

if the root of equation

x^{9} - 10 x + 1 = 0 are x_{1} and x_{2}

x_{1} x_{2} = 1

find : x_{1} + x_{2} = ?

Commented by MJS_new last updated on 14/Jan/22

{there}^{'} s no pair of roots of the given equation

with x_{1} x_{2} = 1

Commented by mr W last updated on 15/Jan/22

y o u a r e r i g h t s i r .

b u t, i s t h e r e a n e a s y w a y t o s e e t h i s ?

Answered by mr W last updated on 15/Jan/22

a s s u m e t h e r e e x i s t s u c h t w o r o o t s .

i t m e a n s t h e r e e x i s t s a α, s u c h t h a t

α a n d \frac{1}{α} a r e t h e r o o t s o f t h e e q n .

α \neq 0, α \neq 1 .

α^{9} - 10 α + 1 = 0

\frac{1}{α^{9}} - \frac{10}{α} + 1 = 0 \Rightarrow 1 - 10 α^{8} + α^{9} = 0

\Rightarrow 10 α^{8} - 10 α = 0

\Rightarrow 10 α^{8} - 10 α = 0 \Rightarrow α^{7} = 1 \Rightarrow α = e^{\frac{2 k π i}{7}} \dots (I)

\Rightarrow α^{2} - 10 α + 1 = 0 \Rightarrow α = 5 \pm 2 \sqrt{6} \dots (I I)

(I) a n d (I I) a r e c o n t r a d i c t i o n .

t h a t m e a n s x^{9} - 10 x + 1 = 0 c a n n o t

h a v e t w o r o o t s x_{1} a n d x_{2} w i t h x_{1} x_{2} = 1 .

Commented by HongKing last updated on 15/Jan/22

thank you very much my dear Sir