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Cộng các phân thức:
a. \({1 \over {\left( {x – y} \right)\left( {y – z} \right)}} + {1 \over {\left( {y – z} \right)\left( {z – x} \right)}} + {1 \over {\left( {z – x} \right)\left( {x – y} \right)}}\)
b. \({4 \over {\left( {y – x} \right)\left( {z – x} \right)}} + {3 \over {\left( {y – x} \right)\left( {y – z} \right)}} + {3 \over {\left( {y – z} \right)\left( {x – z} \right)}}\)
c. \({1 \over {x\left( {x – y} \right)\left( {x – z} \right)}} + {1 \over {y\left( {y – z} \right)\left( {y – x} \right)}} + {1 \over {z\left( {z – x} \right)\left( {z – y} \right)}}\)
a. \({1 \over {\left( {x – y} \right)\left( {y – z} \right)}} + {1 \over {\left( {y – z} \right)\left( {z – x} \right)}} + {1 \over {\left( {z – x} \right)\left( {x – y} \right)}}\)
\(\eqalign{ & = {{z – x} \over {\left( {x – y} \right)\left( {y – z} \right)\left( {z – x} \right)}} + {{x – y} \over {\left( {x – y} \right)\left( {y – z} \right)\left( {z – x} \right)}} + {{y – z} \over {\left( {x – y} \right)\left( {y – z} \right)\left( {z – x} \right)}} \cr & = {{z – x + x – y + y – z} \over {\left( {x – y} \right)\left( {y – z} \right)\left( {z – x} \right)}} = 0 \cr} \)
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b. \({4 \over {\left( {y – x} \right)\left( {z – x} \right)}} + {3 \over {\left( {y – x} \right)\left( {y – z} \right)}} + {3 \over {\left( {y – z} \right)\left( {x – z} \right)}}\)
\(\eqalign{ & = {{ – 4} \over {\left( {y – x} \right)\left( {x – z} \right)}} + {3 \over {\left( {y – x} \right)\left( {y – z} \right)}} + {3 \over {\left( {y – z} \right)\left( {x – z} \right)}} \cr & = {{ – 4\left( {y – z} \right)} \over {\left( {x – z} \right)\left( {y – z} \right)\left( {y – x} \right)}} + {{3\left( {x – z} \right)} \over {\left( {x – z} \right)\left( {y – z} \right)\left( {y – x} \right)}} + {{3\left( {y – x} \right)} \over {\left( {x – z} \right)\left( {y – z} \right)\left( {y – x} \right)}} \cr & = {{ – 4y + 4z + 3x – 3z + 3y – 3x} \over {\left( {x – z} \right)\left( {y – z} \right)\left( {y – x} \right)}} = {{z – y} \over {\left( {x – z} \right)\left( {y – z} \right)\left( {y – x} \right)}} \cr & = {{ – \left( {y – z} \right)} \over {\left( {x – z} \right)\left( {y – z} \right)\left( {y – x} \right)}} = {{ – 1} \over {\left( {x – z} \right)\left( {y – x} \right)}} = {1 \over {\left( {x – z} \right)\left( {x – y} \right)}} \cr} \)
c. \({1 \over {x\left( {x – y} \right)\left( {x – z} \right)}} + {1 \over {y\left( {y – z} \right)\left( {y – x} \right)}} + {1 \over {z\left( {z – x} \right)\left( {z – y} \right)}}\)
\(\eqalign{ & = {1 \over {x\left( {x – y} \right)\left( {x – z} \right)}} + {1 \over {y\left( {x – y} \right)\left( {y – z} \right)}} + {1 \over {z\left( {x – z} \right)\left( {y – z} \right)}} \cr & = {{yz\left( {y – z} \right)} \over {xyz\left( {x – y} \right)\left( {x – z} \right)\left( {y – z} \right)}} + {{ – xz\left( {x – z} \right)} \over {xyz\left( {x – y} \right)\left( {x – z} \right)\left( {y – z} \right)}} + {{xy\left( {x – y} \right)} \over {xyz\left( {x – y} \right)\left( {x – z} \right)\left( {y – z} \right)}} \cr & = {{{y^2}z – y{z^2} – {x^2}z + x{z^2} + {x^2}y – x{y^2}} \over {xyz\left( {x – y} \right)\left( {x – z} \right)\left( {y – z} \right)}} = {{{z^2}\left( {x – y} \right) + xy\left( {x – y} \right) – z\left( {x – y} \right)\left( {x + y} \right)} \over {xyz\left( {x – y} \right)\left( {x – z} \right)\left( {y – z} \right)}} \cr & = {{\left( {x – y} \right)\left( {{z^2} + xy – xz – yz} \right)} \over {xyz\left( {x – y} \right)\left( {x – z} \right)\left( {y – z} \right)}} = {{\left( {x – y} \right)\left[ {x\left( {y – z} \right) – z\left( {y – z} \right)} \right]} \over {xyz\left( {x – y} \right)\left( {x – z} \right)\left( {y – z} \right)}} \cr & = {{\left( {x – y} \right)\left( {y – z} \right)\left( {x – z} \right)} \over {xyz\left( {x – y} \right)\left( {x – z} \right)\left( {y – z} \right)}} = {1 \over {xyz}} \cr} \)