Prove that 7 + 3 is irrational

Question

Prove that        7    +      3    is irrational

Zackery Harvey · Accepted Answer

Notice that   (      3    +      7        )    2    =  10  +  2      21  , so that   (  (      3    +      7        )    2    −  10      )    2    =  84. This means that the number   α  =      3    +      7   is a root of the polynomial  f  (  X  )  =  (      X    2    −  10      )    2    −  84  =      x    4    −  20      x    2    +  16.You can now use the rational root test to show that   f does not have any rational roots: indeed, it follows from that result that all rational solutions are actually integers which divide   16, and you can easily check that no integer dividing   16 is a root of   f.

glitinosim3 · Accepted Answer

Consider                          a        =                  7                +                  3                                          (        1        )                                  a        (                  7                −                  3                )        =        (                  7                +                  3                )        (                  7                −                  3                )                                  (        2        )                                            7                −                  3                =                  4          a                                          (        3        )                                  2                  7                =        a        +                  4          a                                  summing the reverse of (1) with (3)            If   a is rational, then also       7   is rational.

Prove that <msqrt> 7

Answered question

Answer & Explanation

New Questions in Algebra I