Câu 18 trang 28 Sách bài tập (SBT) Toán 8 tập 1

Cộng các phân thức khác mẫu thức:

a. \({5 \over {6{x^2}y}} + {7 \over {12x{y^2}}} + {{11} \over {18xy}}\)

b. \({{4x + 2} \over {15{x^3}y}} + {{5y - 3} \over {9{x^2}y}} + {{x + 1} \over {5x{y^3}}}\)

c. \({3 \over {2x}} + {{3x - 3} \over {2x - 1}} + {{2{x^2} + 1} \over {4{x^2} - 2x}}\)

d. \({{{x^3} + 2x} \over {{x^3} + 1}} + {{2x} \over {{x^2} - x + 1}} + {1 \over {x + 1}}\)

Giải:

a. \({5 \over {6{x^2}y}} + {7 \over {12x{y^2}}} + {{11} \over {18xy}}\)\( = {{30y} \over {36{x^2}{y^2}}} + {{21x} \over {36{x^2}{y^2}}} + {{22xy} \over {36{x^2}{y^2}}} = {{30y + 21x + 22xy} \over {36{x^2}{y^2}}}\)

b. \({{4x + 2} \over {15{x^3}y}} + {{5y - 3} \over {9{x^2}y}} + {{x + 1} \over {5x{y^3}}}\)

\(\eqalign{  &  = {{3{y^2}\left( {4x + 2} \right)} \over {45{x^3}{y^3}}} + {{5x{y^2}\left( {5y - 3} \right)} \over {45{x^3}{y^3}}} + {{9{x^2}\left( {x + 1} \right)} \over {45{x^3}{y^3}}}  \cr  &  = {{12x{y^2} + 6{y^2} + 25x{y^3} - 15x{y^2} + 9{x^3} + 9{x^2}} \over {45{x^3}{y^3}}} \cr&= {{6{y^2} + 25x{y^3} - 3x{y^2} + 9{x^3} + 9{x^2}} \over {45{x^3}{y^3}}} \cr} \)

c. \({3 \over {2x}} + {{3x - 3} \over {2x - 1}} + {{2{x^2} + 1} \over {4{x^2} - 2x}}\)

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

\(\eqalign{  &  = {{3\left( {2x - 1} \right)} \over {2x\left( {2x - 1} \right)}} + {{2x\left( {3x - 3} \right)} \over {2x\left( {2x - 1} \right)}} + {{2{x^2} + 1} \over {2x\left( {2x - 1} \right)}}\cr& = {{6x - 3 + 6{x^2} - 6x + 2{x^2} + 1} \over {2x\left( {2x - 1} \right)}}  \cr  &  = {{8{x^2} - 2} \over {2x\left( {2x - 1} \right)}} \cr&= {{2\left( {4{x^2} - 1} \right)} \over {2x\left( {2x - 1} \right)}} = {{\left( {2x + 1} \right)\left( {2x - 1} \right)} \over {x\left( {2x - 1} \right)}} = {{2x + 1} \over x} \cr} \)

d. \({{{x^3} + 2x} \over {{x^3} + 1}} + {{2x} \over {{x^2} - x + 1}} + {1 \over {x + 1}}\)\( = {{{x^3} + 2x} \over {\left( {x + 1} \right)\left( {{x^2} - x + 1} \right)}} + {{2x} \over {{x^2} - x + 1}} + {1 \over {x + 1}}\)

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

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