Question

If α, β, γ are the cube roots of p (p < 0), then for any x, y and z, `(xalpha + "y"beta + "z"gamma)/(xbeta + "y"gamma + "z"alpha)` = ______.

Options

• `1/2 (- 1 - "i"sqrt3)`

• `1/2 (1 + "i"sqrt3)`

• `1/2 (1 - "i"sqrt3)`

• `1/2(- 1 + "i"sqrt3)`

Answer 1

If α, β, γ are the cube roots of p (p < 0), then for any x, y and z, `(xalpha + "y"beta + "z"gamma)/(xbeta + "y"gamma + "z"alpha)` = `underline(1/2 (- 1 - "i"sqrt3))`.

Explanation:

Since, p < 0, Let p = - q, where q is positive.

Therefore `"p"^(1/3) = - "q"^(1/3) (1)^(1/3)`

Hence `alpha = - "q"^(1/3), beta = - "q"^(1/3) omega and  gamma = - "q"^(1/3) omega^2`

The given expression = `(x + "y"omega + "z"omega)/(xomega + "y"omega^2 + "z")`

`= 1/omega * (xomega + "y"omega^2 + "z")/(xomega + "y"omega^2 + "z")`

`= 1/omega * 1 = 1/omega * omega^3`

= ω² 

`= 1/2 (- 1 - "i"sqrt3)`