Rút gọn phân thức: Q= x^10 - x^8 - x^7 + x^6 + x^5 + x^4 - x^3 -x^2 +1

* Hướng dẫn giải

$\begin{array}{l}\mathrm{Q}=\frac{{\mathrm{x}}^{10}-{\mathrm{x}}^8+{\mathrm{x}}^7+{\mathrm{x}}^6+{\mathrm{x}}^5+{\mathrm{x}}^4-{\mathrm{x}}^3-{\mathrm{x}}^2+1}{{\mathrm{x}}^{30}+{\mathrm{x}}^{24}+{\mathrm{x}}^{18}+{\mathrm{x}}^{12}+{\mathrm{x}}^6+1}\\ =\frac{\left({\mathrm{x}}^{10}-{\mathrm{x}}^8+{\mathrm{x}}^6\right)-\left({\mathrm{x}}^7-{\mathrm{x}}^5+{\mathrm{x}}^3\right)+\left({\mathrm{x}}^4-{\mathrm{x}}^2+1\right)}{{\mathrm{x}}^{18}\left({\mathrm{x}}^{12}+{\mathrm{x}}^6+1\right)+{\mathrm{x}}^{12}+{\mathrm{x}}^6+1}\\ =\frac{\left({\mathrm{x}}^4-{\mathrm{x}}^2+1\right)\left({\mathrm{x}}^6-{\mathrm{x}}^3+1\right)}{\left({\mathrm{x}}^{12}+{\mathrm{x}}^6+1\right)\left({\mathrm{x}}^{18}+1\right)}=\frac{\left({\mathrm{x}}^4-{\mathrm{x}}^2+1\right)\left({\mathrm{x}}^6-{\mathrm{x}}^3+1\right)}{\left({\mathrm{x}}^{12}+2{\mathrm{x}}^6+1-{\mathrm{x}}^6\right)\left({\mathrm{x}}^6+1\right)\left({\mathrm{x}}^{12}-{\mathrm{x}}^6+1\right)}\end{array}$

$=\frac{\left({\mathrm{x}}^4-{\mathrm{x}}^2+1\right)\left({\mathrm{x}}^6-{\mathrm{x}}^3+1\right)}{\left[\left({\mathrm{x}}^6+1\right)-{\mathrm{x}}^6\right]\left({\mathrm{x}}^2+1\right)\left({\mathrm{x}}^4-{\mathrm{x}}^2+1\right)\left({\mathrm{x}}^{12}-{\mathrm{x}}^6+1\right)}$

$\begin{array}{l}=\frac{{\mathrm{x}}^6-{\mathrm{x}}^3+1}{\left({\mathrm{x}}^6+{\mathrm{x}}^3+1\right)\left({\mathrm{x}}^6-{\mathrm{x}}^3+1\right)\left({\mathrm{x}}^2+1\right)\left({\mathrm{x}}^{12}-{\mathrm{x}}^6+1\right)}\\ =\frac{1}{\left({\mathrm{x}}^6+{\mathrm{x}}^3+1\right)\left({\mathrm{x}}^2+1\right)\left({\mathrm{x}}^{12}-{\mathrm{x}}^6+1\right)}\end{array}$