Just a century

first off, one rule all should remember is that if $s_1=s_2=...=s_n=0$ , then $p_1=p_2=...=p_n=0$ . this is true since in newtons sum $p_n$ only contains terms upto $s_n$ .proved here . so we see by vietas that $s_1=s_2=...=s_{98}=0$ we see that the sum translates to $\sum(x^0)+\sum (x^{99})+\sum (x^{100})$ now $x^{100}=-x-1$ $\sum (x^{100})=-\sum(x)-\sum(1)=0-100=-100$ $x^{99}=-1-\dfrac{1}{x}$ $\sum(x^{99})=-\sum(1)-\sum(\dfrac{1}{x})=-100-\dfrac{s_{99}}{s_{100}}=-100+1=-99$ so $100-100-99=\boxed{-99}$