SOLUTION: x=3+root8, then (x^6+x^4+x^2+1)/x^3 ans:204 shortcut formula as possible we have to solve this in 2 minutes

Algebra ->  Square-cubic-other-roots -> SOLUTION: x=3+root8, then (x^6+x^4+x^2+1)/x^3 ans:204 shortcut formula as possible we have to solve this in 2 minutes       Log On


   

Question 1042993: x=3+root8, then (x^6+x^4+x^2+1)/x^3
ans:204
shortcut formula as possible
we have to solve this in 2 minutes
Found 3 solutions by ikleyn, MathTherapy, greenestamps:
Answer by ikleyn(54062) About Me  (Show Source):
You can put this solution on YOUR website!
.
x=3+root8, then (x^6+x^4+x^2+1)/x^3
ans:204
shortcut formula as possible
we have to solve this in 2 minutes
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

1.  Let us consider these two numbers: x = 3+%2B+sqrt%288%29 and y = 3-sqrt%288%29.

    It is easy to check that x*y = %283%2Bsqrt%288%29%29%2A%283-sqrt%288%29%29 = 3%5E2+-+%28sqrt%288%29%29%5E2 = 9 - 8 = 1.    (1)

    Thus  y = x%5E%28-1%29;  x  and  y are reciprocal.

    Also notice that x+%2B+1%2Fx = x+%2B+y = %283+%2B+sqrt%288%29%29+%2B+%283-sqrt%288%29%29 = 6.                      (2)

    Also notice that %28x+%2B+1%2Fx%29%5E2 = x%5E2+%2B+2+%2B+1%2Fx%5E2 = 6%5E2 = 36,

         therefore x%5E2+%2B+1%2Fx%5E2 = 36 - 2 = 34.                                      (3)

    Armed with this knowldge, we can make the next step.


2.  Let us regroup

      %28x%5E6%2Bx%5E4%2Bx%5E2%2B1%29%2Fx%5E3 = %28x%5E6+%2B+x%5E4%29%2Fx%5E3 + %28x%5E2+%2B+1%29%2Fx%5E3 = %28%28x%5E6+%2B+x%5E4%29%2Fx%5E5%29%2Ax%5E2 + %28%28%28x%5E2+%2B+1%29%2Fx%29%29%2Fx%5E2 = 

    = %28x%2By%29%2Ax%5E2 + %28x%2By%29%2Fx%5E2 =     ( now substitute x + y = 6, according to (2) )

    = 6x%5E2 + 6%2Fx%5E2 = 6%2A%28x%5E2+%2B+1%2Fx%5E2%29 = 6*34 = 204.   ( I substituted  x%5E2 + 1%2Fx%5E2 = 34, according to (3) )

Answer.  %28x%5E6%2Bx%5E4%2Bx%5E2%2B1%29%2Fx%5E3 = 204.

Solved.


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Answer by MathTherapy(10898) About Me  (Show Source):
You can put this solution on YOUR website!
x=3+root8, then (x^6+x^4+x^2+1)/x^3

ans:204
shortcut formula as possible 
we have to solve this in 2 minutes 
*****************************
%28x%5E6+%2B+x%5E4+%2B+x%5E2+%2B+1%29%2Fx%5E3 = x%5E6%2Fx%5E3+%2B+x%5E4%2Fx%5E3+%2B+x%5E2%2Fx%5E3+%2B+1%2Fx%5E3 ----- Dividing each polynomial by x%5E3
                                = x%5E3+%2B+x+%2B+1%2Fx+%2B+1%2Fx%5E3
                                = x%5E3+%2B+1%2Fx%5E3+%2B+x+%2B+1%2Fx
*****************************
                       %28x+%2B+1%2Fx%29%5E3+=+x%5E3+%2B+1%2Fx%5E3+%2B+3%28x+%2B+1%2Fx%29 --- Cubing x+%2B+1%2Fx
 = x%5E3+%2B+1%2Fx%5E3+%2B+x+%2B+1%2Fx
*****************************
             %28x%5E6+%2B+x%5E4+%2B+x%5E2+%2B+1%29%2Fx%5E3+=+%28x+%2B+1%2Fx%29%5E3+-+2%28x+%2B+1%2Fx%29
*****************************
           %283+%2B+sqrt%288%29%29%283+-+sqrt%288%29%29+=+9+-+8+=+1

                             x+%2B+1%2Fx+=+3+%2B+sqrt%288%29+%2B+1%2F%283+%2B+sqrt%288%29%29 ----- Substituting 3+%2B+sqrt%288%29 for x
                                      =  ----- Substituting %283+%2B+sqrt%288%29%29%283+-+sqrt%288%29%29 for 1
                                      = 
                                      = 3+%2B+sqrt%288%29+%2B+3+-+sqrt%288%29+=+6 

             %28x%5E6+%2B+x%5E4+%2B+x%5E2+%2B+1%29%2Fx%5E3+=+%28x+%2B+1%2Fx%29%5E3+-+2%28x+%2B+1%2Fx%29
             %28x%5E6+%2B+x%5E4+%2B+x%5E2+%2B+1%29%2Fx%5E3+=+%286%29%5E3+-+2%286%29 ---- Substituting 6 for x+%2B+1%2Fx
            highlight%28%28x%5E6+%2B+x%5E4+%2B+x%5E2+%2B+1%29%2Fx%5E3%29+=+216+-+12+=+highlight%28204%29

Answer by greenestamps(13392) About Me  (Show Source):
You can put this solution on YOUR website!




Since x=3%2Bsqrt%288%29,

1%2Fx=1%2F%283%2Bsqrt%288%29%29=%283-sqrt%288%29%29%2F%289-8%29=3-sqrt%288%29

So

x%2B1%2Fx=%283%2Bsqrt%288%29%29%2B%283-sqrt%288%29%29=6

Cubing that...



So

x%5E3%2B1%2Fx%5E3=%28x%2B1%2Fx%29%5E3-3%28x%2B1%2Fx%29=6%5E3-3%286%29=216-18=198

And finally

x%5E3%2Bx%2B1%2Fx%2B1%2Fx%5E3=%28x%5E3%2B1%2Fx%5E3%29%2B%28x%2B1%2Fx%29=198%2B6=204