Bài 35 trang 50 SGK Toán 8 tập 1

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

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

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