Prove the sum of 3 + 4 3 is Irrational number

Question

Prove the sum of       3    +      4    3   is Irrational number

Daniel Valdez · Accepted Answer

Let   x  =      3    +      4    3   so   0  =  (  x  −      3        )    3    −  4  =      x    3    −  3      x    2        3    +  9  x  −  3      3    −  4. If   x  ∈      Q  ,       3    =                    x        3            +      9      x      −      4              3      (              x        2            +      1      )        ∈      Q  , a contradiction.

Dania Mueller · Accepted Answer

Every number of the form   α  =  a  +  b      3   with   a  ,  b  ∈      Q   is a root of a quadratic polynomial because   (  a  +  b      3        )    2    =      a    2    +  3      b    2    +  2  a  b      3   and so   α is a solution of   0  =  (      X    3    −  4  )  −  (  X  +  2  a  )  ⋅  (      X    2    −  2  a  X  +      a    2    −  3      b    2    )  =  (  …  )  ⋅  X  +  (  …  )and therefore rational - contradiction. (Verify that a least one of the &quot;(…)&quot; is non-zero!)

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