SOLUTION: The roots of the polynomial equation 2x^3 - 8x^2 + 3x + 5 = 0 are alpha, beta and gamma. Find the polynomial equation with roots alpha^2, beta^2, gamma^2 Any help is so much ap

The given equation

     = 0        (1)

is equivalent to

     = 0     (2)  (all the coefficients of (1) are divided by 2)


Equation (2) has the same roots  ,    and    as equation (1).  Therefore, 

     = ,                       (3)

and, according to Vieta's theorem

     = 4,   = 1.5,   = -2.5.      (4)


Now, an equation with the roots  ,    and    is

     = 0.                     (5)


By the Vieta's theorem (or by applying FOIL directly), the coefficients of the left side polynomial are

      at  x^2;                          (6)

      at x;   and                 (7)

      as the constant term.                (8)


So, my task now is to express the coefficient (6), (7) and (8)  via  the coefficients (4) of the equation (2).


Regarding   ,  it is easy:

     =  =  = 16-3 = 13.


So, the coefficient at x^2 of the polynomial (5)  is   = -13.


Regarding  ,  it is easy, too :

     =  =  = 6.25.


So, the constant term of the polynomial (5)  is   = -6.25.


Regarding  , it is slightly more long way :

     = 1.5  of (4)  implies (squaring both sides)

    2.25 =  = 

         =  +  = substituting the known values from (4) = 

         =  + 2*(-2.5)*4,

which implies

     = 2.25 + 20 = 22.25.


Thus we know all three coefficients of the polynomial (5)

     = -13  at  x^2;          

     = 22.25 at x;   and 

     = -6.25 as the constant term.   


Answer.  The polynomial equation under the question is   = 0.