Prove that this sum of two irrational numbers is rational number 2 − 5 3 + 2 + 5 3

Question

Prove that this sum of two irrational numbers is rational number            2      −              5              3    +            2      +              5              3

barene55d · Accepted Answer

Call the number   x. Then show that       x    3    =  4  −  3  x. Then solve this equation.

Paula Cameron · Accepted Answer

You may notice that   (  2  −      5    )  (  2  +      5    )  =  −  1, hence the given number is   α  −      1    α   with   α being a root of   p  (  x  )  =  (      x    3    −  2      )    2    −  5  =      x    6    −  4      x    3    −  1. On the other hand   p  (  x  )  =  0 implies      x    3    −      1          x      3        =  4which can be written as            (      x      −              1        x            )        3    +  3      (    x    −          1      x        )    −  4  =  0so the given number is a root of   q  (  x  )  =      x    3    +  3  x  −  4  =  (  x  −  1  )  (      x    2    +  x  +  4  ). Since       x    2    +  x  +  4 has a negative discriminant and   α  −      1    α   is real,   α  −      1    α    =      1  .

Prove that this sum of two irrational numbers is rational number root(3)(2-sqrt 5)+root(3)(2+sqrt 5)

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