Đề bài

Thực hiện các phép tính:

a) \(\dfrac{{x + 1}}{{x - 3}} - \dfrac{{1 - x}}{{x + 3}} - \dfrac{{2x\left( {1 - x} \right)}}{{9 - {x^2}}}\)

b) \(\dfrac{{3x + 1}}{{{{\left( {x - 1} \right)}^2}}} - \dfrac{1}{{x + 1}} + \dfrac{{x + 3}}{{1 - {x^2}}}\)

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Phương pháp giải - Xem chi tiết

Áp dụng:

- Quy đồng phân thức

- Quy tắc trừ hai phân thức: \( \dfrac{A}{B}-\dfrac{C}{D}=\dfrac{A}{B}+\left( { - \dfrac{C}{D}} \right)\)

- Quy tắc đổi dấu: \(  -\dfrac{A}{B} = \dfrac{{ A}}{-B} \).

Lời giải chi tiết

\(\eqalign{
& a)\;{{x + 1} \over {x - 3}} - {{1 - x} \over {x + 3}} - {{2x\left( {1 - x} \right)} \over {9 - {x^2}}} (x \ne 3;x \ne -3 )\cr 
& = {{x + 1} \over {x - 3}} + {{ - \left( {1 - x} \right)} \over {x + 3}} + {{2x\left( {1 - x} \right)} \over { - \left( {9 - {x^2}} \right)}} \cr
& = {{x + 1} \over {x - 3}} + {{x - 1} \over {x + 3}} + {{2x\left( {1 - x} \right)} \over {{x^2} - 9}} \cr
& = {{x + 1} \over {x - 3}} + {{x - 1} \over {x + 3}} + {{2x - 2{x^2}} \over {\left( {x - 3} \right)\left( {x + 3} \right)}} \cr & = {{(x + 1)(x+3)} \over {(x - 3)(x+3)}} + {{(x - 1)(x-3)} \over {(x + 3)(x-3)}} + {{2x - 2{x^2}} \over {\left( {x - 3} \right)\left( {x + 3} \right)}} \cr 
& = {{\left( {x + 1} \right)\left( {x + 3} \right) + \left( {x - 1} \right)\left( {x - 3} \right) + 2x - 2{x^2}} \over {\left( {x - 3} \right)\left( {x + 3} \right)}} \cr
& = {{{x^2} + 3x + x + 3 + {x^2} - 3x - x + 3 + 2x - 2{x^2}} \over {\left( {x - 3} \right)\left( {x + 3} \right)}} \cr
& = {{2x + 6} \over {\left( {x - 3} \right)\left( {x + 3} \right)}}\cr& = {{2\left( {x + 3} \right)} \over {\left( {x - 3} \right)\left( {x + 3} \right)}} = {2 \over {x - 3}} \cr} \)

\(\eqalign{
& b)\,\,{{3x + 1} \over {{{\left( {x - 1} \right)}^2}}} - {1 \over {x + 1}} + {{x + 3} \over {1 - {x^2}}} (x \ne 1; x \ne -1 )\cr
& = {{3x + 1} \over {{{\left( {x - 1} \right)}^2}}} + {{ - 1} \over {x + 1}} + {{ - \left( {x + 3} \right)} \over { - \left( {1 - {x^2}} \right)}} \cr
& = {{3x + 1} \over {{{\left( {x - 1} \right)}^2}}} + {{ - 1} \over {x + 1}} + {{ - \left( {x + 3} \right)} \over {{x^2} - 1}} \cr
& = {{3x + 1} \over {{{\left( {x - 1} \right)}^2}}} + {{ - 1} \over {x + 1}} + {{ - \left( {x + 3} \right)} \over {\left( {x - 1} \right)\left( {x + 1} \right)}} \cr & = {{(3x + 1)(x+1)} \over {{{\left( {x - 1} \right)}^2(x+1)}}} + {{ - (x-1)^2} \over {(x + 1)(x-1)^2}} + {{ - \left( {x + 3} \right)(x-1)} \over {\left( {x - 1} \right)^2\left( {x + 1} \right)}} \cr 
& = {{\left( {3x + 1} \right)\left( {x + 1} \right) - {{\left( {x - 1} \right)}^2} - \left( {x + 3} \right)\left( {x - 1} \right)} \over {{{\left( {x - 1} \right)}^2}\left( {x + 1} \right)}} \cr
& = {{3{x^2} + 4x + 1 - \left( {{x^2} - 2x + 1} \right) - \left( {{x^2} + 2x - 3} \right)} \over {{{\left( {x - 1} \right)}^2}\left( {x + 1} \right)}} \cr
& = {{3{x^2} + 4x + 1 - {x^2} + 2x - 1 - {x^2} - 2x + 3} \over {{{\left( {x - 1} \right)}^2}\left( {x + 1} \right)}} \cr
& = {{{x^2} + 4x + 3} \over {{{\left( {x - 1} \right)}^2}\left( {x + 1} \right)}} = {{{x^2} + x + 3x + 3} \over {{{\left( {x - 1} \right)}^2}\left( {x + 1} \right)}} \cr
& = {{\left( {{x^2} + x} \right) + \left( {3x + 3} \right)} \over {{{\left( {x - 1} \right)}^2}\left( {x + 1} \right)}} \cr
& = {{x\left( {x + 1} \right) + 3\left( {x + 1} \right)} \over {{{\left( {x - 1} \right)}^2}\left( {x + 1} \right)}} \cr
& = {{\left( {x + 1} \right)\left( {x + 3} \right)} \over {{{\left( {x - 1} \right)}^2}\left( {x + 1} \right)}} = {{x + 3} \over {{{\left( {x - 1} \right)}^2}}} \cr} \)

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