Question

If α, b, g are the roots of the equation 5x3─8x2 + 7x + 6 = 0Find the equation whose roots areα2 + b2 + bα , b2 + g2 + g b, g2 + α2 + gα

Answer 1

a, b, g are roots of equation: 
     5 x³ - 8 x² + 7 x + 6 = 0
     5 [ x³ - (a+b+g) x²  + (ab + bg + ag) x - abg ]  = 0

=>   a + b + g = 8/5   --- (1)
       a b + b g + a g = 7/5    --- (2)
       a b g = - 6 / 5  ---(3)

from (1),  g = 1.6 - a - b
   (2)  =>    a b + (a+b) (1.6 - a- b) = 1.4
                 (a+b) 1.6 - (a² + b²+ ab) = 1.4
  =>   a² + b² + ab = 1.6 (a + b) - 1.4
  =>          = 1.6 (1.6 - g) - 1.4 =  1.16 - 1.6 g = 0.4 (2.9 - 4 g)    --- (4)

from (1),  a = 1.6 - b - g
  (2) =>    a (b + g) + b g = 1.4
               (b+g) (1.6 - b - g) + b g = 1.4
       =>    b² + g² + b g = 1.6 (b+g) - 1.4
       =>                      =  0.4 (2.9 - 4 a)    --- (5)

then again,  g² + a² + ag = 0.4 (2.9 - 4 b) - 1.4        --- (6)

new roots are :  u = 0.4 (2.9 - 4 g) ,    v =  0.4 (2.9 - 4 a)  and,  w = 0.4(2.9 - 4 b)

  =>    u + v + w = 3.48 - 1.6 (a+b+g) =  3.48 - 1.6 * 1.6 =  0.92

 =>    u v + v w + u w = 3*1.16² + 2.56 (ab+bg+ag) - 1.6*2.32(a+b+g)
                                =  1.6816 

=>     u v w  =  7.1888

the cubic polynomial wanted:  x³ - 0.92 x² + 1.6816 x - 7.1888