Answer:
[tex]\large\boxed{x=1\ \vee\ x=-3i\ \vee\ x=3i}[/tex]
Step-by-step explanation:
[tex]\text{The zeros:}\\\\x^3-x^2+9x-9=0\\\\x^2(x-1)+9(x-1)=0\\\\(x-1)(x^2+9)=0\iff x-1=0\ \vee\ x^2+9=0\\\\x-1=0\qquad\text{add 1 to both sides}\\x-1+1=0+1\\\boxed{x=1}\\\\x^2+9=0\qquad\text{subtract 9 from both sides}\\x^2=-9<0\qquad\text{no real roots}\\\text{In the set of complex numbers:}\\x^2=-9\to x=\pm\sqrt{-9}\\\boxed{x=-3i\ \vee\ x=3i}[/tex]
