If `alpha, beta and gamma` are the roots of the equation `x^(3) + 3x^(2) - 24x + 1 = 0`
thebn find the value of `(3sqrt(alpha)+3sqrt(beta)+ 3sqrt(gamma)).`

Explanation

`x^(3) + 3x^(2) + 3x + 1 = 27 x `
`(x + 1)^(3) = 27x`
Equation has roots `alpha, beta, and gamma`
`therefore (alpha + 1)^(3) = 27 alpha` (1)
Let `3sqrt(alpha)= y` . Then `alpha = y^(3)`
Now, (1) reduces to
`(y^(3) + 1)^(3) = 27y^(3)`
`rArr y^(3) + 1 = 3y`
`rArr y^(3) - 3y + 1 = 0`
This equation has roots `3sqrt(alpha),3sqrt(beta),3sqrt(gamma)`
`therefore 3sqrt(alpha)+3sqrt(beta)+3sqrt(gamma) = 0`