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If \(x+\frac1x=-5\), what is \(x^3+\frac1{x^3}\) ?

 Aug 27, 2017
 #1
avatar+100513 
+1

 

x  + 1/x  = -5    (1)  

 

square  both sides  and we have

 

x^2 + 2 + (1/x)^2  = 25    →  x^2 + (1/x)^2  = 23    (2)

 

x^3  +  (1/x)^3      can be factored as a sum of cubes  thusly

 

(x + 1/x)  ( x^2  - 1  + (1/x)^2  )  =

 

(x + 1/x) ( x^2 + (1/x)^2  - 1 )    sub (1)  and (2)  into this

 

(-5) ( 23  -  1)  =

 

(-5) (22)  = 

 

-110

 

 

cool cool cool

 Aug 27, 2017

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