f-x-1-x-x-5-1-x-5-f-3-

Question Number 181447 by Socracious last updated on 25/Nov/22

f (x + \frac{1}{x}) = x^{5} + \frac{1}{x^{5}}

f (3) = ?

Answered by Frix last updated on 25/Nov/22

y = x + \frac{1}{x} \Rightarrow

x = \frac{y \pm \sqrt{y^{2} - 4}}{2} \land x^{- 1} = \frac{y \mp \sqrt{y^{2} - 4}}{2} \Rightarrow

f (y) = y^{5} - 5 y^{3} + 5

f (3) = 123

Answered by manxsol last updated on 25/Nov/22

Answered by Rasheed.Sindhi last updated on 25/Nov/22

x + \frac{1}{x} = y

x^{2} + \frac{1}{x^{2}} = y^{2} - 2

x^{3} + \frac{1}{x^{3}} = y^{3} - 3 y

(x^{2} + \frac{1}{x^{2}}) (x^{3} + \frac{1}{x^{3}}) = (y^{2} - 2) (y^{3} - 3 y)

x^{5} + \frac{1}{x^{5}} + \frac{1}{x} + x = (y^{2} - 2) (y^{3} - 3 y)

x^{5} + \frac{1}{x^{5}} = (y^{2} - 2) (y^{3} - 3 y) - y

f (x) = (x^{2} - 2) (x^{3} - 3 x) - x

f (3) = (9 - 2) (27 - 9) - 3

= 7 (18) - 3 = 123

Answered by Rasheed.Sindhi last updated on 25/Nov/22

f (x + \frac{1}{x}) = x^{5} + \frac{1}{x^{5}}; f (3) = ?

x + \frac{1}{x} = 3; x^{5} + \frac{1}{x^{5}} = ?

{(x + \frac{1}{x})}^{2} = 3^{2} = 9

x^{2} + \frac{1}{x^{2}} = 9 - 2 = 7

{(x + \frac{1}{x})}^{3} = 3^{3} = 27

x^{3} + \frac{1}{x^{3}} = 27 - 3 (3) = 18

(x^{2} + \frac{1}{x^{2}}) (x^{3} + \frac{1}{x^{3}}) = (7) (18)

x^{5} + \frac{1}{x^{5}} + x + \frac{1}{x} = 126

x^{5} + \frac{1}{x^{5}} + 3 = 126

x^{5} + \frac{1}{x^{5}} = 126 - 3 = 123

f (x + \frac{1}{x}) = x^{5} + \frac{1}{x^{5}}

f (3) = 123

Commented by manxsol last updated on 25/Nov/22

f i r s t : g r a n d i d e a (x^{3} + \frac{1}{x^{3}}) (x^{2} + \frac{1}{x^{2}}) = (x^{5} + \frac{1}{x^{5}}) + (x + \frac{1}{x})

s e c o n d : d e s a r r o l l o

Commented by Rasheed.Sindhi last updated on 26/Nov/22

W h a t i s m e a n t b y^{'} {d e s a r r o l l o}^{'} s i r ?