SOLUTION: if (x+1/x)^2 = 3 then find the value of x^206 + x^200 + x^90 + x^84 + x^18 + x^12 + x^6 + 1

Question 1036823: if (x+1/x)^2 = 3 then find the value of x^206 + x^200 + x^90 + x^84 + x^18 + x^12 + x^6 + 1
Answer by ikleyn(52781) (Show Source): You can put this solution on YOUR website!
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if (x+1/x)^2 = 3 then find the value of x^206 + x^200 + x^90 + x^84 + x^18 + x^12 + x^6 + 1
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Answer. If = then the value of is zero.

Solution

If   =   then

 +  +  = ,

 +  = ,

 = ,   ( <--- Introduce new variable u =  )  ---> 

 = ,   ( <---multiply both sides by u)  ---> 

 = ,    ( <--- solve by using the quadratic formula )  ---> 

 =  = ,   --->

 = .   ( since x =  )

In other words, if   =   then x is the primitive root of the degree 12 of 1.

Then

 = ,  so   = ;

 = ,  so   = ;

 = ,  so   = ,  and also

 = ,  so   = .

Hence,  the value of  is zero.

Solved.

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