If x is an irrational number such that x 2 − 2 x and x 3 − 5 x are rational numbers. What does x 3 − 5 x equals to?

Question

If x is an irrational number such that       x    2    −  2  x and       x    3    −  5  x are rational numbers. What does      x    3    −  5  x equals to?

Buckokasg · Accepted Answer

Note that   (  x  −  1      )    2    =  y is a rational number. So   x  =  1  ±      y  . Also,       x    3    −  5  x  =  3  y  −  4  ±      y    (  y  −  2  ) is rational, which is possible only when  y  =  2.Therefore,   x  =  1  ±      2   and       x    3    −  5  x  =  3  y  −  4  =  2.

aniawnua · Accepted Answer

Since   x  ≠  0 and   x  ≠  2 we have                               x          3                −        5        x                              x          2                −        2        x              =                              x          2                −        5                    x        −        2              =  q  ∈      Q  So  2  x  +  r  −  5  =  x  q  −  2  q    ⟹    (  q  −  2  )  x  =  −  2  q  −  r  +  5so if   q  ≠  2 we get  x  =                    −        2        q        −        r        +        5                    q        −        2                  but this can&#039;t be since x is irational and left side is rational so   q  =  2 and   r  =  1:       x    3    −  5  x  =  (      x    2    −  2  x  )  q  =  q  r  =  2

If x is an irrational number such that x 2 </msup> − 2 x and

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