jmc

Let $r_1,r_2,\dots ,r_{2018}$ be the roots. By Vieta's formulas, $r_1+r_2+\cdots +r_{2018}=0.$ To get the squared terms we want, we square both sides, giving $$(r_1^2+r_2^2+\cdots +r_{2018}^2)+2(r_1{r}_2+r_1{r}_3+\cdots )=0,$$where the second term on the left-hand side is the sum of all terms $r_i{r}_j,$ where $i<j.$ By Vieta's formulas, this also equals $0,$ so $$r_1^2+r_2^2+\cdots +r_{2018}^2=\boxed{0}.$$