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Cộng, trừ nhiều phân thức đại số

Cộng, trừ nhiều phân thức khác mẫu như thế nào? Phép cộng nhiều phân thức đại số có tính chất gì?
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1. Lý thuyết

- Quy tắc cộng, trừ nhiều phân thức: Muốn cộn꧋g, trừ nhiều phân thức khác mẫu thức, ta thực hiện các bước:

- Quy đồng mẫu thức;- Cộng, trừ các phân thức có cùng mẫu thức vừa tìm được.

- Tính chất phép cộng phân thức đại số: Phép cộng ꦐcác phân th🎐ức cũng có các tính chất giao hoán, kết hợp:

+ Giao hoán: \(\frac{A}{B} + \frac{C}{D} = \frac{C}{D} + \frac{A}{B}; \)+ Kết hợp: \(\left( {\frac{A}{B} + \frac{C}{D}} \right) + \frac{E}{F} = \frac{A}{B} + \left( {\frac{C}{D} + \frac{E}{F}} \right);\)+ Cộng với 0: \(\frac{A}{B} + 0 = 0 + \frac{A}{B} = \frac{A}{B}.\)

Chú ý: Nhờ tính chất kết hợp nên trong một dãy phép cộng nhiều p🅺hân thức, ta có thể không cần đặt dấu𝐆 ngoặc.

2. Ví dụ minh họa

\(\begin{array}{l}\frac{x}{{x + y}} + \frac{{2xy}}{{{x^2} - {y^2}}} - \frac{y}{{x + y}} = \frac{{x(x - y)}}{{(x + y)(x - y)}} + \frac{{2xy}}{{(x + y)(x - y)}} - \frac{{y(x - y)}}{{(x + y)(x - y)}}\\\frac{{{x^2} - xy + 2xy - xy + {y^2}}}{{(x + y)(x - y)}} = \frac{{{x^2} + {y^2}}}{{{x^2} - {y^2}}}\end{array}\)

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