10.3.11 Operations with complex numbers

Problem: A polynomial and one of its roots are given: \[ P(z)=a z^{4}+b z^{3}+c z^{2}+d z+e, \quad z_{1}=\frac{3+3 i \sqrt{3}}{2} . \] 1) Find all the other roots of the polynomial \( P(z) \). 2) Expand \( P(z) \) into irreducible factors in \( \mathbb{C}, \mathbb{R} \). \[ a=1, \quad b=-2, \quad c=8, \quad d=3, \quad e=18 . \]