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Thực hiện các phép tính:
a)\({{x + 1} \over {x – 3}} – {{1 – x} \over {x + 3}} – {{2x\left( {1 – x} \right)} \over {9 – {x^2}}}\)
b)\({{3x + 1} \over {{{\left( {x – 1} \right)}^2}}} – {1 \over {x + 1}} + {{x + 3} \over {1 – {x^2}}}\)
Hướng dẫn làm bài:
a)\({{x + 1} \over {x – 3}} – {{1 – x} \over {x + 3}} – {{2x\left( {1 – x} \right)} \over {9 – {x^2}}} = {{x + 1} \over {x – 3}} + {{ – \left( {1 – x} \right)} \over {x + 3}} + {{2x\left( {1 – x} \right)} \over { – \left( {9 – {x^2}} \right)}}\)
\( = {{x + 1} \over {x – 3}} + {{x – 1} \over {x + 3}} + {{2x\left( {1 – x} \right)} \over {{x^2} – 9}} = {{x + 1} \over {x – 3}} + {{x – 1} \over {x + 3}} + {{2x – 2{x^2}} \over {\left( {x – 3} \right)\left( {x + 3} \right)}}\)
\( = {{\left( {x + 1} \right)\left( {x + 3} \right) + \left( {x – 1} \right)\left( {x – 3} \right) + 2x – 2{x^2}} \over {\left( {x – 3} \right)\left( {x + 3} \right)}}\)
\( = {{{x^2} + 4x + 3 + {x^2} – 4x + 3 + 2x – 2{x^2}} \over {\left( {x – 3} \right)\left( {x + 3} \right)}}\)
\( = {{2x + 6} \over {\left( {x – 3} \right)\left( {x + 3} \right)}} = {{2\left( {x + 3} \right)} \over {\left( {x – 3} \right)\left( {x + 3} \right)}} = {2 \over {x – 3}}\)
b)\({{3x + 1} \over {{{\left( {x – 1} \right)}^2}}} – {1 \over {x + 1}} + {{x + 3} \over {1 – {x^2}}} = {{3x + 1} \over {{{\left( {x – 1} \right)}^2}}} + {{ – 1} \over {x + 1}} + {{ – \left( {x + 3} \right)} \over { – \left( {1 – {x^2}} \right)}}\)
\( = {{3x + 1} \over {{{\left( {x – 1} \right)}^2}}} + {{ – 1} \over {x + 1}} + {{ – \left( {x + 3} \right)} \over {{x^2} – 1}} = {{3x + 1} \over {{{\left( {x – 1} \right)}^2}}} + {{ – 1} \over {x + 1}} + {{ – \left( {x + 3} \right)} \over {\left( {x – 1} \right)\left( {x + 1} \right)}}\)
\( = {{\left( {3x + 1} \right)\left( {x + 1} \right) – {{\left( {x – 1} \right)}^2} – \left( {x + 3} \right)\left( {x – 1} \right)} \over {{{\left( {x – 1} \right)}^2}\left( {x + 1} \right)}}\)
\( = {{3{x^2} + 4x + 1 – \left( {{x^2} – 2x + 1} \right) – \left( {{x^2} + 2x – 3} \right)} \over {{{\left( {x – 1} \right)}^2}\left( {x + 1} \right)}}\)
\( = {{3{x^2} + 4x + 1 – {x^2} + 2x – 1 – {x^2} – 2x + 3} \over {{{\left( {x – 1} \right)}^2}\left( {x + 1} \right)}}\)
\( = {{{x^2} + 4x + 3} \over {{{\left( {x – 1} \right)}^2}\left( {x + 1} \right)}} = {{{x^2} + x + 3x + 3} \over {{{\left( {x – 1} \right)}^2}\left( {x + 1} \right)}}\)
\( = {{x\left( {x + 1} \right) + 3\left( {x + 1} \right)} \over {{{\left( {x – 1} \right)}^2}\left( {x + 1} \right)}} = {{\left( {x + 1} \right)\left( {x + 3} \right)} \over {{{\left( {x – 1} \right)}^2}\left( {x + 1} \right)}} = {{x + 3} \over {{{\left( {x – 1} \right)}^2}}}\)