If `alpha, beta and gamma` the roots of the equation `x^(3) + 3x^(2) - 4x - 2 = 0.`
then find the values of the following expressions:
(i) `alpha ^(2) + beta^(2) + gamma^(2)`
(ii)`alpha ^(3) + beta^(3) + gamma^(3)`
(iii) `(1)/(alpha)+(1)/(beta)+(1)/(gamma)`

Explanation

Given that `alpha,beta and gamma` are the roots of the equation
`x^(3) + 3x^(2) - 4x - 2 = 0`
Therefore, `alpha + beta + gamma = -(3)/(1) = -3, alpha beta + beta gamma + gamma alpha = (-4)/(1) = -4`,
and `alpha beta gamma = ((-2))/(1) = 2.`
`(alpha + beta + gamma)^(2) = alpha^(2) + beta^(2) + gamma^(2) + 2alphabeta + 2beta gamma + 2gammaalpha` ltbgt `rArr alpha^(2) + beta^(2) + gamma^(2) = (alpha + beta + gamma)^(2) - 2(alphabeta + beta gamma + gammaalpha)`
`= (-3)^(2) - 2(-4) = 9+ 8 = 17`
(ii) `alpha^(3) + beta^(3) + gamma^(3) -3alpha beta gamma = (alpha + beta + gamma) (alpha^(2) + beta^(2) + gamma^(2)-alphabeta - beta gamma - gammaalpha)`
`rArr alpha^(3) + beta^(3) + gamma^(3)-3(2) = (-3)(17-(-4))`
`rArr alpha^(3) + beta^(3) + gamma^(3)-63 + 6 = -57`
(iii) `(1)/(alpha)+(1)/(beta)+(1)/(gamma)=(alphabeta + beta gamma + gammaalpha)/(alphabetagamma)=(-4)/(2) = -2`.