Hướng dẫn trả lời:
a) `{1}/{x - y} + {3xy}/{y^3 - x^3} + {x - y}/{x^2 + xy + y^2}` (ĐKXĐ: `x ne y`)
`= {1}/{x - y} - {3xy}/{x^3 - y^3} + {x - y}/{x^2 + xy + y^2}`
`= {1}/{x - y} - {3xy}/{(x - y)cdot(x^2 + xy + y^2)} + {x - y}/{x^2 + xy + y^2}`
`= {x^2 + xy + y^2}/{(x - y)cdot(x^2 + xy + y^2)} - {3xy}/{(x - y)cdot(x^2 + xy + y^2)} + {(x - y)^2}/{(x - y)cdot(x^2 + xy + y^2)}`
`= {(x^2 + xy + y^2) - 3xy + (x - y)^2}/{(x - y)cdot(x^2 + xy + y^2)}`
`= {x^2 + xy + y^2 - 3xy + x^2 - 2xy + y^2}/{(x - y)cdot(x^2 + xy + y^2)}`
`= {(x^2 + x^2) + (xy - 3xy - 2xy) + (y^2 + y^2)}/{(x - y)cdot(x^2 + xy + y^2)}`
`= {2x^2 - 4xy + 2y^2}/{(x - y)cdot(x^2 + xy + y^2)}`
`= {2cdot(x^2 - 2xy + y^2)}/{(x - y)cdot(x^2 + xy + y^2)}`
`= {2cdot(x - y)^2}/{(x - y)cdot(x^2 + xy + y^2)}`
`= {2cdot(x - y)}/{x^2 + xy + y^2}`
b) `{3}/{x^2 + 2xy + y^2} + {4}/{2xy - x^2 - y^2} + {5}/{x^2 - y^2}` (ĐKXĐ: `x ne y`)
`= {3}/{(x + y)^2} - {4}/{x^2 - 2xy + y^2} + {5}/{(x + y)cdot(x - y)}`
`= {3}/{(x + y)^2} - {4}/{(x - y)^2} + {5}/{(x + y)cdot(x - y)}`
`= {3cdot(x - y)^2}/{(x + y)^2cdot(x - y)^2} - {4cdot(x + y)^2}/{(x + y)^2cdot(x - y)^2} + {5cdot(x + y)cdot(x - y)}/{(x + y)^2cdot(x - y)^2}`
`= {3cdot(x^2 - 2xy + y^2) - 4cdot(x^2 + 2xy + y^2) + 5cdot(x^2 - y^2)}/{(x + y)^2cdot(x - y)^2}`
`= {3x^2 - 6xy + 3y^2 - 4x^2 - 8xy - 4y^2 + 5x^2 - 5y^2}/{(x + y)^2cdot(x - y)^2}`
`= {(3x^2 - 4x^2 + 5x^2) + (- 6xy - 8xy) + (3y^2 - 4y^2 - 5y^2)}/{(x + y)^2cdot(x - y)^2}`
`= {4x^2 - 14xy - 6y^2}/{(x + y)^2cdot(x - y)^2}`
`= {2cdot(2x^2 - 7xy - 3y^2)}/{(x + y)^2cdot(x - y)^2}`
