Câu 20 trang 29 Sách bài tập (SBT) Toán 8 tập 1
Cộng các phân thức: ...
Cộng các phân thức:
Cộng các phân thức:
a. ({1 over {left( {x - y} ight)left( {y - z} ight)}} + {1 over {left( {y - z} ight)left( {z - x} ight)}} + {1 over {left( {z - x} ight)left( {x - y} ight)}})
b. ({4 over {left( {y - x} ight)left( {z - x} ight)}} + {3 over {left( {y - x} ight)left( {y - z} ight)}} + {3 over {left( {y - z} ight)left( {x - z} ight)}})
c. ({1 over {xleft( {x - y} ight)left( {x - z} ight)}} + {1 over {yleft( {y - z} ight)left( {y - x} ight)}} + {1 over {zleft( {z - x} ight)left( {z - y} ight)}})
Giải:
a. ({1 over {left( {x - y} ight)left( {y - z} ight)}} + {1 over {left( {y - z} ight)left( {z - x} ight)}} + {1 over {left( {z - x} ight)left( {x - y} ight)}})
(eqalign{ & = {{z - x} over {left( {x - y} ight)left( {y - z} ight)left( {z - x} ight)}} + {{x - y} over {left( {x - y} ight)left( {y - z} ight)left( {z - x} ight)}} + {{y - z} over {left( {x - y} ight)left( {y - z} ight)left( {z - x} ight)}} cr & = {{z - x + x - y + y - z} over {left( {x - y} ight)left( {y - z} ight)left( {z - x} ight)}} = 0 cr} )
b. ({4 over {left( {y - x} ight)left( {z - x} ight)}} + {3 over {left( {y - x} ight)left( {y - z} ight)}} + {3 over {left( {y - z} ight)left( {x - z} ight)}})
(eqalign{ & = {{ - 4} over {left( {y - x} ight)left( {x - z} ight)}} + {3 over {left( {y - x} ight)left( {y - z} ight)}} + {3 over {left( {y - z} ight)left( {x - z} ight)}} cr & = {{ - 4left( {y - z} ight)} over {left( {x - z} ight)left( {y - z} ight)left( {y - x} ight)}} + {{3left( {x - z} ight)} over {left( {x - z} ight)left( {y - z} ight)left( {y - x} ight)}} + {{3left( {y - x} ight)} over {left( {x - z} ight)left( {y - z} ight)left( {y - x} ight)}} cr & = {{ - 4y + 4z + 3x - 3z + 3y - 3x} over {left( {x - z} ight)left( {y - z} ight)left( {y - x} ight)}} = {{z - y} over {left( {x - z} ight)left( {y - z} ight)left( {y - x} ight)}} cr & = {{ - left( {y - z} ight)} over {left( {x - z} ight)left( {y - z} ight)left( {y - x} ight)}} = {{ - 1} over {left( {x - z} ight)left( {y - x} ight)}} = {1 over {left( {x - z} ight)left( {x - y} ight)}} cr} )
c. ({1 over {xleft( {x - y} ight)left( {x - z} ight)}} + {1 over {yleft( {y - z} ight)left( {y - x} ight)}} + {1 over {zleft( {z - x} ight)left( {z - y} ight)}})
(eqalign{ & = {1 over {xleft( {x - y} ight)left( {x - z} ight)}} + {1 over {yleft( {x - y} ight)left( {y - z} ight)}} + {1 over {zleft( {x - z} ight)left( {y - z} ight)}} cr & = {{yzleft( {y - z} ight)} over {xyzleft( {x - y} ight)left( {x - z} ight)left( {y - z} ight)}} + {{ - xzleft( {x - z} ight)} over {xyzleft( {x - y} ight)left( {x - z} ight)left( {y - z} ight)}} + {{xyleft( {x - y} ight)} over {xyzleft( {x - y} ight)left( {x - z} ight)left( {y - z} ight)}} cr & = {{{y^2}z - y{z^2} - {x^2}z + x{z^2} + {x^2}y - x{y^2}} over {xyzleft( {x - y} ight)left( {x - z} ight)left( {y - z} ight)}} = {{{z^2}left( {x - y} ight) + xyleft( {x - y} ight) - zleft( {x - y} ight)left( {x + y} ight)} over {xyzleft( {x - y} ight)left( {x - z} ight)left( {y - z} ight)}} cr & = {{left( {x - y} ight)left( {{z^2} + xy - xz - yz} ight)} over {xyzleft( {x - y} ight)left( {x - z} ight)left( {y - z} ight)}} = {{left( {x - y} ight)left[ {xleft( {y - z} ight) - zleft( {y - z} ight)} ight]} over {xyzleft( {x - y} ight)left( {x - z} ight)left( {y - z} ight)}} cr & = {{left( {x - y} ight)left( {y - z} ight)left( {x - z} ight)} over {xyzleft( {x - y} ight)left( {x - z} ight)left( {y - z} ight)}} = {1 over {xyz}} cr} )
