+0  
 
0
167
2
avatar

If (x+1/x)^2=3, what is x^3+1/x^3

Guest May 15, 2017
Sort: 

2+0 Answers

 #1
avatar+79894 
+1

(x+1/x)^2 = 3      expand the left side

 

x^2 + 2 +  1/x^2  = 3         subtract 3 from both sides

 

x^2 -  1  +  1/x^2   =  0         (1)

 

Now      x^3   +  1/x^3    is   a sum of cubes  which can be factored as

 

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

 

( x + 1/x)   (  0 )     =

 

0

 

 

cool cool cool

CPhill  May 15, 2017
 #2
avatar+18777 
+1

If (x+1/x)^2=3, what is x^3+1/x^3

 

\(\begin{array}{|rcll|} \hline \left(x+\frac{1}{x}\right)^2 &=&3 \\ x+\frac{1}{x} &=& \sqrt{3} \\ \hline \end{array} \)

 

\(\small{ \begin{array}{|rclcll|} \hline (x+\frac{1}{x})^3 &=& \left(x+\frac{1}{x}\right)^2 \cdot \left(x+\frac{1}{x}\right) &=& 3\cdot \sqrt{3} \\ x^3+3x^2\cdot \frac{1}{x} +3x\cdot \frac{1}{x^2}+\frac{1}{x^3} &=& && 3\cdot \sqrt{3} \\ x^3+3x +3\cdot \frac{1}{x}+\frac{1}{x^3} &=& && 3\cdot \sqrt{3} \\ x^3+\frac{1}{x^3}+ 3x +3\cdot \frac{1}{x} &=& && 3\cdot \sqrt{3} \\ x^3+\frac{1}{x^3}+ 3\cdot(x + \frac{1}{x}) &=& && 3\cdot \sqrt{3} \quad & | \quad x + \frac{1}{x}=\sqrt{3} \\ x^3+\frac{1}{x^3}+ 3\cdot\sqrt{3} &=& && 3\cdot \sqrt{3} \\ x^3+\frac{1}{x^3} &=& && 3\cdot \sqrt{3}- 3\cdot\sqrt{3} \\ \mathbf{x^3+\frac{1}{x^3}} &\mathbf{=} & &&\mathbf{ 0 } \\ \hline \end{array} }\)

 

laugh

heureka  May 16, 2017

14 Online Users

avatar
avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details