Cho hai đa thức A = 6x³ - 4x² - 12x - 7 và B = 2x² - 7.

Thực hiện phép nhân A . B bằng hai cách.

Cách 1: Khai triển tích:

A . B = (6x³ - 4x² - 12x - 7) . (2x² - 7)

= 6x³ . 2x² - 4x² . 2x² - 12x . 2x² - 7 . 2x² – 7 . 6x³ + 7 . 4x² + 7 . 12x + 7 . 7

= 12x⁵ - 8x⁴ - 24x³ -14x² - 42x³ + 28x² + 84x + 49

= 12x⁵ - 8x⁴ - 66x³ + 14x² + 84x + 49.

Cách 2: Đặt tính nhân:

\[\begin{array}{l} \times \underline {\begin{array}{*{20}{c}}\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,6{{\rm{x}}^3}\\\end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l} - \\\end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l}4{{\rm{x}}^2}\\\end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l} - \\\end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l}12{\rm{x}}\\2{{\rm{x}}^2}\end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l} - \\ - \end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l}7\\7\end{array}\end{array}\,\,\,\,\,\,\,} \,\,\,\,\,\\\underline { + \begin{array}{*{20}{c}}\begin{array}{l}\\\begin{array}{*{20}{c}}{12{{\rm{x}}^5}}\end{array}\end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l}\\ - \end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l}\\8{{\rm{x}}^4}\end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l} - \\ - \end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l}42{{\rm{x}}^3}\\24{{\rm{x}}^3}\end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l} + \\ - \end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l}28{{\rm{x}}^2}\\14{{\rm{x}}^2}\end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l} + \\\begin{array}{*{20}{c}}{}\end{array}\end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l}84{\rm{x}}\\\end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l} + \\\end{array}\end{array}\begin{array}{*{20}{c}}\begin{array}{l}49\,\,\,\\\end{array}\end{array}} \\\,\,\,12{{\rm{x}}^5} - 8{{\rm{x}}^4} - 66{{\rm{x}}^3} + 14{{\rm{x}}^2} + 84{\rm{x}} + 49\end{array}\]