Câu 3.1: Rút gọn phân thức :
a. \({{{x^4} – {y^4}} \over {{y^3} – {x^3}}}\)
b. \({{\left( {2x – 4} \right)\left( {x – 3} \right)} \over {\left( {x – 2} \right)\left( {3{x^2} – 27} \right)}}\)
c. \({{2{x^3} + {x^2} – 2x – 1} \over {{x^3} + 2{x^2} – x – 2}}\)
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a. \({{{x^4} – {y^4}} \over {{y^3} – {x^3}}}\) \( = {{\left( {{x^2} + {y^2}} \right)\left( {{x^2} – {y^2}} \right)} \over {\left( {y – x} \right)\left( {{y^2} + xy + {x^2}} \right)}} = {{\left( {{x^2} + {y^2}} \right)\left( {x + y} \right)\left( {x – y} \right)} \over {\left( {y – x} \right)\left( {{y^2} + xy + {x^2}} \right)}}\)
\( = – {{\left( {{x^2} + {y^2}} \right)\left( {x + y} \right)\left( {x – y} \right)} \over {\left( {x – y} \right)\left( {{x^2} + xy + {y^2}} \right)}} = {{\left( {{x^2} + {y^2}} \right)\left( {x + y} \right)} \over {{x^2} + xy + {y^2}}}\)
b. \({{\left( {2x – 4} \right)\left( {x – 3} \right)} \over {\left( {x – 2} \right)\left( {3{x^2} – 27} \right)}}\) \( = {{2\left( {x – 2} \right)\left( {x + 3} \right)} \over {\left( {x – 2} \right)3\left( {{x^2} – 9} \right)}} = {{2\left( {x + 3} \right)} \over {3\left( {x + 3} \right)\left( {x – 3} \right)}} = {2 \over {3\left( {x – 3} \right)}}\)
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c. \({{2{x^3} + {x^2} – 2x – 1} \over {{x^3} + 2{x^2} – x – 2}}\)\( = {{2x\left( {{x^2} – 1} \right) + \left( {{x^2} – 1} \right)} \over {x\left( {{x^2} – 1} \right) + 2\left( {{x^2} – 1} \right)}} = {{\left( {{x^2} – 1} \right)\left( {2x + 1} \right)} \over {\left( {{x^2} – 1} \right)\left( {x + 2} \right)}} = {{2x + 1} \over {x + 2}}\)
Câu 3.2: 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\( = {{{x^{10}} – {x^8} – {x^7} + {x^6} + {x^5} + {x^4} – {x^3} – {x^2} + 1} \over {{x^{30}} + {x^{24}} + {x^{18}} + {x^{12}} + {x^6} + 1}}\)
\(\eqalign{ & = {{\left( {{x^{10}} – {x^8} + {x^6}} \right) – \left( {{x^7} – {x^5} + {x^3}} \right) + \left( {{x^4} – {x^2} + 1} \right)} \over {\left( {{x^{30}} + {x^{24}} + {x^{18}}} \right) + \left( {{x^{12}} + {x^6} + 1} \right)}} \cr & = {{{x^6}\left( {{x^4} – {x^2} + 1} \right) – {x^3}\left( {{x^4} – {x^2} + 1} \right) + \left( {{x^4} – {x^2} + 1} \right)} \over {{x^{18}}\left( {{x^{12}} + {x^6} + 1} \right) + \left( {{x^{12}} + {x^6} + 1} \right)}} \cr & = {{\left( {{x^4} – {x^2} + 1} \right)\left( {{x^6} – {x^3} + 1} \right)} \over {\left( {{x^{12}} + {x^6} + 1} \right)\left( {{x^{18}} + 1} \right)}} = {{\left( {{x^4} – {x^2} + 1} \right)\left( {{x^6} – {x^3} + 1} \right)} \over {\left( {{x^{12}} + 2{x^6} + 1 – {x^6}} \right)\left[ {{{\left( {{x^6}} \right)}^3} + 1} \right]}} \cr & = {{\left( {{x^4} – {x^2} + 1} \right)\left( {{x^6} – {x^3} + 1} \right)} \over {\left[ {{{\left( {{x^6} + 1} \right)}^2} – {{\left( {{x^3}} \right)}^2}} \right]\left( {{x^6} + 1} \right)\left( {{x^{12}} – {x^6} + 1} \right)}} \cr & = {{\left( {{x^4} – {x^2} + 1} \right)\left( {{x^6} – {x^3} + 1} \right)} \over {\left( {{x^6} + {x^3} + 1} \right)\left( {{x^6} + 1 – {x^3}} \right)\left( {{x^6} + 1} \right)\left( {{x^{12}} – {x^6} + 1} \right)}} \cr & = {{{x^4} – {x^2} + 1} \over {\left( {{x^6} + {x^3} + 1} \right)\left( {{x^2} + 1} \right)\left( {{x^4} – {x^2} + 1} \right)\left( {{x^{12}} – {x^6} + 1} \right)}} \cr & = {1 \over {\left( {{x^6} + {x^3} + 1} \right)\left( {{x^2} + 1} \right)\left( {{x^{12}} – {x^6} + 1} \right)}} \cr} \)
