Let S be the set of all non-real complex numbers z satisfying z^4 = z (|z|^2 - 3|z| + 3) . Then the value of _ z S (3 - z) is equal to _____ .

Step-by-step solution

Given the equation z^4 = z (|z|^2 - 3|z| + 3) . Taking the modulus on both sides: |z^4| = | z | | |z|^2 - 3|z| + 3 | Since |z|^2 - 3|z| + 3 = (|z| - 3 2 )^2 + 3 4 > 0 , we can drop the absolute value on the real quadratic: |z|^4 = |z| (|z|^2 - 3|z| + 3) Since z is non-real, z 0 , so we can divide by |z| : |z|^3 = |z|^2 - 3|z| + 3 |z|^3 - |z|^2 + 3|z| - 3 = 0 Factoring by grouping: |z|^2(|z| - 1) + 3(|z| - 1) = 0 (|z| - 1)(|z|^2 + 3) = 0 Since |z| is a real number, |z|^2 + 3 0 . Thus, |z| = 1 . Substitute |z| = 1 ba