Let z be a non-zero complex number satisfying the equation z^3 + i z = 0 . If z is expressed in polar form as r e^ i where 0 < 2 , then the sum of all possible values of is :

Step-by-step solution

Let z = r e^ i . Then z = r e^ -i . Substituting into the given equation: r^3 e^ i3 + i r e^ -i = 0 Since z is non-zero, r 0 . Dividing by r e^ -i : r^2 e^ i4 = -i Taking the modulus of both sides: r^2 = |-i| = 1 r = 1 Now, writing -i in polar form: e^ i4 = e^ i ( 3 2 + 2k ) Equating the arguments: 4 = 3 2 + 2k = 3 8 + k 2 For [0, 2 ) , we substitute k = 0, 1, 2, 3 : ₁ = 3 8 ₂ = 7 8 ₃ = 11 8 ₄ = 15 8 The sum of all possible values of is: 3 8 + 7 8 + 11 8 + 15 8 = 36 8 = 9 2 Answer: 9 2