# 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

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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

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

=>

=> = 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

=>

=> = 0.4 (2.9 - 4 a) --- (5)

then again,

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