Phương trình (z-1/i)(z-1/i^2)...(z-1/i^20) = 0 có bao nhiêu

* Hướng dẫn giải

Đáp án D

$$\begin{gathered} \mathrm{PT} \mathrm{có} \mathrm{nghiệm}: \\ \left\{\begin{array}{l}\mathrm{z}=\frac{1}{\mathrm{i}}\\ \mathrm{z}=\frac{1}{{\mathrm{i}}^2}\\ ...\\ \mathrm{z}=\frac{1}{{\mathrm{i}}^{20}}\end{array}\right. \\ \mathrm{Ta} \mathrm{thấy}: {\mathrm{i}}^{2\mathrm{n}+2}={(-1)}^{\mathrm{n}+1} \left(\mathrm{lũy} \mathrm{thừa} \mathrm{bậc} \mathrm{chẵn} \mathrm{của} \mathrm{i} \mathrm{luôn} \mathrm{bằng} \pm 1\right) \\ \mathrm{và} {\mathrm{i}}^{2\mathrm{n}+1}=\mathrm{i}.{(-1)}^{\mathrm{n}} \left(\mathrm{lũy} \mathrm{thừa} \mathrm{bậc} \mathrm{lẻ} \mathrm{của} \mathrm{i} \mathrm{luôn} \mathrm{bằng} \pm \mathrm{i} \right) \\ \Rightarrow \mathrm{PT}\hspace{0.17em}\mathrm{luôn} \mathrm{chỉ} \mathrm{có} 4 \mathrm{nghiệm} \mathrm{là} \mathrm{z}=\pm 1 \mathrm{và} \mathrm{z}=\pm \frac{1}{\mathrm{i}} \end{gathered}$$