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Phép nhân đa thức với đa thức

Nhân đa thức với đa thức như thế nào?
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1. Lý thuyết

- Quy tắc nhân hai đa thức:

Muốn nhân một đa thức với một đa thức, ta nhân mỗi hạng tử của đa thức này với từng hạng tử của đa thức kia rồi cộng các tích với nhau.\(\left( {A + B} \right)\left( {C + D} \right) = AC + AD + BC + BD\)

- Tính chất:

+ Giao hoán: A.B = B.A+ Kết hợp: (A.B).C = A.(B.C)+ Phân phối của phép nhân đối với phép cộng: (A + B).C = AB + AC

2. Ví dụ minh họa

- Nhân hai đa thức \(xy + 1\) và \(xy - 3\) ta được:\(\begin{array}{l}(xy + 1)(xy - 3)\\ = (xy).\left( {xy} \right) + xy - 3xy - 3\\ = {x^2}{y^2} - 2xy - 3\end{array}\)- Nhân hai đa thức \(2x + y\) và \(3{x^2} + xy - {y^2}\) ta được:\(\begin{array}{l}(2x + y)(3{x^2} + xy - {y^2})\\ = 2x.3{x^2} + y.3{x^2} + 2x.xy + y.xy + 2x.\left( { - {y^2}} \right) + y.\left( { - {y^2}} \right)\\ = 6{x^3} + 3{x^2}y + 2{x^2}y + x{y^2} - 2x{y^2} - {y^3}\\ = 6{x^3} + 5{x^2}y - x{y^2} - {y^3}\end{array}\)

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