Assuming 0 &lt; ϵ ≪ 1 find all the roots of the polynomial ϵ 2 x 3 + x + 1which are O ( 1 ) up to a precision of O ( ϵ 2 )

Question

Assuming   0  &amp;lt;  ϵ  ≪  1 find all the roots of the polynomial      ϵ          2            x          3        +  x  +  1which are   O  (  1  ) up to a precision of   O  (      ϵ          2        )

Jewel Brooks · Accepted Answer

First, assume that   x  (  ϵ  )  =  O  (  1  ). Then we can say   x  (  ϵ  )  =      x    0    +  f  (  ϵ  ), where   f  (  ϵ  )  =  o  (  1  ) is some function in   ϵ, and       x    0   is constant in   ϵ. Filling this in, and using that   x  (  ϵ  ) is a root of the equation, we get      ϵ    2    (      x    0    +  f  (  ϵ  )      )    3    +  (      x    0    +  f  (  ϵ  )  )  +  1  =  0.Expanding the cubic, we get      (          ϵ      2              x      0      3        +    3          ϵ      2              x      0      2        f    (    ϵ    )    +    3          ϵ      2              x      0        f    (    ϵ          )      2        +          ϵ      2        f    (    ϵ          )      3        )    +      (          x      0        +    f    (    ϵ    )    )    +  1  =  0.Now, note that   f  (  ϵ  )  =  o  (  1  ), so also   f  (  ϵ      )    2    =  o  (  1  ) and   f  (  ϵ      )    3    =  o  (  1  ). Further note that       x    0    =  O  (  1  ), so also       x    0    2    =  O  (  1  ) and       x    0    3    =  O  (  1  ). So the second, third and fourth terms above are all   o  (      ϵ    2    ). So if we combine all these   o  (      ϵ    2    )-terms into one   o  (      ϵ    2    )-term, we get      ϵ    2        x    0    3    +  o  (      ϵ    2    )  +      x    0    +  f  (  ϵ  )  +  1  =  0.Rewriting this slightly, to group the terms of the same order, we get  (      x    0    +  1  )  +      (    f    (    ϵ    )    +          x      0      3              ϵ      2        +    o    (          ϵ      2        )    )    =  0.Equating the proper order terms on both sides, we must therefore have       x    0    =  −  1 and   f  (  ϵ  )  =  −      x    0    3        ϵ    2    +  o  (      ϵ    2    )  =      ϵ    2    +  o  (      ϵ    2    ), from which we may conclude that   x  (  ϵ  )  =  −  1  +      ϵ    2    +  o  (      ϵ    2    )If we do not assume   x  (  ϵ  )  =  O  (  1  ), then we get more solutions. In particular, the two other solutions are of the form                    x        (        ϵ        )                            =                              ±            i                    ϵ                +        O        (        1        )            For these solutions, we cannot say that   x  (  ϵ  )  =      x    0    +  o  (  1  ) for some constant       x    0  , so we did not find these solutions with the above method.

Assuming 0 < epsilon << 1 find all the roots of the polynomial epsilon^2x^3+x+1 which are O(1) up to a precision of O(epsilon^2)

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