Contoh Soal 4 Polinom

$\begin{array}{l}\\ 16.& \text{Diketahui akar-akar polinom}\\ & x^{2017}+x^{2016}+x^{2015}+...+x^2+x+1=0\\ & \text{adalah}{x}_1,x_2,x_3,...,x_{2017}\\ & \text{Tentukan nilai dari}\\ & {\displaystyle \frac{1}{1-x_1}+\frac{1}{1-x_2}+\frac{1}{1-x_3}+...+{\displaystyle \frac{1}{1-x_{2017}}}}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}{\displaystyle \frac{x^{2018}-1}{x-1}}& =x^{2017}+x^{2016}+x^{2015}+...+x^2+x+1=0\\ \text{perlu dii}& \text{ngat bahwa kondisi ini mensyaratkan}x\ne 1,\\ & \text{sehingga}\\ \\ x^{2018}-1& =0\\ x^{2018}& =1\\ x& =\pm 1,\text{pilih}x=-1\\ \text{maka}& \text{nilai dari}\\ {\displaystyle \frac{1}{1-x_1}+}& \frac{1}{1-x_2}+\frac{1}{1-x_3}+...+{\displaystyle \frac{1}{1-x_{2017}}}\\ & =\underset{\text{sebanyak 2017}}{\underbrace{{\displaystyle \frac{1}{1-(-1)}+\frac{1}{1-(-1)}+\frac{1}{1-(-1)}+...+\frac{1}{1-(-1)}}}}\\ & =\underset{\text{sebanyak 2017}}{\underbrace{{\displaystyle \frac{1}{2}+\frac{1}{2}+\frac{1}{2}+...+{\displaystyle \frac{1}{2}}}}}\\ & ={\displaystyle \frac{2017}{2}}\end{array}\end{array}$