[tex]|x-1|+2|x^2-1|+|x^3-1| = 0\\ \\ |x-1|+2|(x-1)(x+1)|+|(x-1)(x^2+x+1)| = 0 \\ \\ |x-1|+2|x-1||x+1|+|x-1||x^2+x+1| = 0\\ \\ |x-1|\cdot \big(1+2|x+1|+|x^2+x+1|\big) = 0\\ \\\\\boxed{1}\quad |x-1| = 0 \Rightarrow x = 1\\\\\boxed{2}\quad 1+\underbrace{2|x+1|}_{\geq 0}+\underbrace{|x^2+x+1|}_{\geq 0}= 0\,\,\, (F) \Rightarrow x\in \emptyset \\ \\ \\\text{Din }\boxed{1}\,\cup\,\boxed{2}\Rightarrow \boxed{S = \{1\}}[/tex]
