Rút gọn phân thức:
Q\( = {{{x^{10}} – {x^8} – {x^7} + {x^6} + {x^5} + {x^4} – {x^3} – {x^2} + 1} \over {{x^{30}} + {x^{24}} + {x^{18}} + {x^{12}} + {x^6} + 1}}\)
Q\( = {{{x^{10}} – {x^8} – {x^7} + {x^6} + {x^5} + {x^4} – {x^3} – {x^2} + 1} \over {{x^{30}} + {x^{24}} + {x^{18}} + {x^{12}} + {x^6} + 1}}\)
\(\eqalign{ & = {{\left( {{x^{10}} – {x^8} + {x^6}} \right) – \left( {{x^7} – {x^5} + {x^3}} \right) + \left( {{x^4} – {x^2} + 1} \right)} \over {\left( {{x^{30}} + {x^{24}} + {x^{18}}} \right) + \left( {{x^{12}} + {x^6} + 1} \right)}} \cr & = {{{x^6}\left( {{x^4} – {x^2} + 1} \right) – {x^3}\left( {{x^4} – {x^2} + 1} \right) + \left( {{x^4} – {x^2} + 1} \right)} \over {{x^{18}}\left( {{x^{12}} + {x^6} + 1} \right) + \left( {{x^{12}} + {x^6} + 1} \right)}} \cr & = {{\left( {{x^4} – {x^2} + 1} \right)\left( {{x^6} – {x^3} + 1} \right)} \over {\left( {{x^{12}} + {x^6} + 1} \right)\left( {{x^{18}} + 1} \right)}} = {{\left( {{x^4} – {x^2} + 1} \right)\left( {{x^6} – {x^3} + 1} \right)} \over {\left( {{x^{12}} + 2{x^6} + 1 – {x^6}} \right)\left[ {{{\left( {{x^6}} \right)}^3} + 1} \right]}} \cr & = {{\left( {{x^4} – {x^2} + 1} \right)\left( {{x^6} – {x^3} + 1} \right)} \over {\left[ {{{\left( {{x^6} + 1} \right)}^2} – {{\left( {{x^3}} \right)}^2}} \right]\left( {{x^6} + 1} \right)\left( {{x^{12}} – {x^6} + 1} \right)}} \cr & = {{\left( {{x^4} – {x^2} + 1} \right)\left( {{x^6} – {x^3} + 1} \right)} \over {\left( {{x^6} + {x^3} + 1} \right)\left( {{x^6} + 1 – {x^3}} \right)\left( {{x^6} + 1} \right)\left( {{x^{12}} – {x^6} + 1} \right)}} \cr & = {{{x^4} – {x^2} + 1} \over {\left( {{x^6} + {x^3} + 1} \right)\left( {{x^2} + 1} \right)\left( {{x^4} – {x^2} + 1} \right)\left( {{x^{12}} – {x^6} + 1} \right)}} \cr & = {1 \over {\left( {{x^6} + {x^3} + 1} \right)\left( {{x^2} + 1} \right)\left( {{x^{12}} – {x^6} + 1} \right)}} \cr} \)