Bài 58 trang 62 sgk toán 8 tập 1

Bài 58 trang 62 sgk toán 8 tập 1

Thực hiện các phép tính sau:

Thực hiện các phép tính sau:

a) (left( {{{2x + 1} over {2x - 1}} - {{2x - 1} over {2x + 1}}} ight):{{4x} over {10x - 5}})                     

b) (left( {{1 over {{x^2} + x}} - {{2 - x} over {x + 1}}} ight):left( {{1 over x} + x - 2} ight);)

c) ({1 over {x - 1}} - {{{x^3} - x} over {{x^2} + 1}}.left( {{1 over {{x^2} - 2x + 1}} + {1 over {1 - {x^2}}}} ight).)

Hướng dẫn làm bài:

a) (left( {{{2x + 1} over {2x - 1}} - {{2x - 1} over {2x + 1}}} ight):{{4x} over {10x - 5}} = {{{{left( {2x + 1} ight)}^2} - {{left( {2x - 1} ight)}^2}} over {left( {2x - 1} ight)left( {2x + 1} ight)}}.{{10x + 5} over {4x}})

=({{4{x^2} + 4x + 1 - 4{x^2} + 4x - 1} over {left( {2x - 1} ight)left( {2x + 1} ight)}}.{{5left( {2x + 1} ight)} over {4x}})

=({{8x.5left( {2x + 1} ight)} over {left( {2x - 1} ight)left( {2x + 1} ight).4x}} = {{10} over {2x - 1}})

b) (left( {{1 over {{x^2} + x}} - {{2 - x} over {x + 1}}} ight):left( {{1 over x} + x - 2} ight))

=(left( {{1 over {xleft( {x + 1} ight)}} + {{x - 2} over {x + 1}}} ight):{{1 + {x^2} - 2x} over x})

=({{1 + xleft( {x - 2} ight)} over {xleft( {x + 1} ight)}}.{x over {{x^2} - 2x + 1}})

=({{left( {{x^2} - 2x + 1} ight)x} over {xleft( {x + 1} ight)left( {{x^2} - 2x + 1} ight)}} = {1 over {x + 1}})

c) ({1 over {x - 1}} - {{{x^3} - x} over {{x^2} + 1}}.left( {{1 over {{x^2} - 2x + 1}} + {1 over {1 - {x^2}}}} ight))

=({1 over {x - 1}} - {{{x^3} - x} over {{x^2} + 1}}.left[ {{1 over {{{left( {x - 1} ight)}^2}}} - {1 over {left( {x - 1} ight)left( {x + 1} ight)}}} ight])

=({1 over {x - 1}} - {{xleft( {{x^2} - 1} ight)} over {{x^2} + 1}}.{{x + 1 - left( {x - 1} ight)} over {{{left( {x - 1} ight)}^2}.left( {x + 1} ight)}})

=({1 over {x - 1}} - {{xleft( {x - 1} ight)left( {x + 1} ight)} over {{x^2} + 1}}.{{x + 1 - x + 1} over {{{left( {x - 1} ight)}^2}left( {x + 1} ight)}})

=({1 over {x - 1}} - {{xleft( {x - 1} ight)left( {x + 1} ight).2} over {left( {{x^2} + 1} ight){{left( {x - 1} ight)}^2}left( {x + 1} ight)}} = {1 over {x - 1}} - {{2x} over {left( {{x^2} + 1} ight)left( {x - 1} ight)}})

=({{{x^2} + 1 - 2x} over {left( {{x^2} + 1} ight)left( {x - 1} ight)}} = {{{{left( {x - 1} ight)}^2}} over {left( {{x^2} + 1} ight)left( {x - 1} ight)}} = {{x - 1} over {{x^2} + 1}})