What is x 10 + x 8 + x 2 + 1 x 10 + x 6 + x 4 + 1 given x 2 + x − 1 = 0?

Question

What is                     x                  10                    +              x        8            +              x        2            +      1                      x                  10                    +              x        6            +              x        4            +      1       given       x    2    +  x  −  1  =  0?

popman14ee · Accepted Answer

HINT: Note that            x      2        =    −    x    +    1    .By multiplying the given expression by   x  , we can obtained            x      3        =    2    x    −    1    .Again by multiplying   x we have            x      4        =    −    3    x    +    2  and so on..

veirarer · Accepted Answer

x    2    =  1  −  x      x    4    =  (  1  −  x      )    2    =      x    2    −  2  x  +  1  =  1  −  x  −  2  x  +  1  =  2  −  3  x      x    6    =  (  1  −  x  )  (  2  −  3  x  )  =  3      x    2    −  5  x  +  2  =  3  (  1  −  x  )  −  5  x  +  2  =  5  −  8  x      x    8    =  (  2  −  3  x      )    2    =  9      x    2    −  12  x  +  4  =  9  (  1  −  x  )  −  12  x  +  4  =  13  −  21  x      x          10        =  (  1  −  x  )  (  13  −  21  x  )  =  21      x    2    −  34  x  +  13  =  21  (  1  −  x  )  −  34  x  +  13  =  34  −  55  xTherefore,      x          10        +      x    8    +      x    2    +  1  =  34  −  55  x  +  13  −  21  x  +  1  −  x  +  1  =  49  −  77  x      x          10        +      x    6    +      x    4    +  1  =  34  −  55  x  +  5  −  8  x  +  2  −  3  x  +  1  =  42  −  66  xYour equation =            49      −      77      x              42      −      66      x        =      7    6

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