Поделить многочлен \(\displaystyle 18x^{\,11}y^{\,18}z^{\,14}-12x^{\,9}y^{\,11}z^{\,7}-9x^{\,7}y^{\,5}z^{\,8}\) на \(\displaystyle -3x^{\,3}y^{\,2}z^{\,4}\) означает поделить каждое его слагаемое на \(\displaystyle -3x^{\,3}y^{\,2}z^{\,4}{\small .}\)

Поэтому

\(\displaystyle \begin{array}{l} (18x^{\,11}y^{\,18}z^{\,14}-12x^{\,9}y^{\,11}z^{\,7}-9x^{\,7}y^{\,5}z^{\,8}): (-3x^{\,3}y^{\,2}z^{\,4})=\\ \kern{2em} =(18x^{\,11}y^{\,18}z^{\,14}): (-3x^{\,3}y^{\,2}z^{\,4})-(12x^{\,9}y^{\,11}z^{\,7}): (-3x^{\,3}y^{\,2}z^{\,4})-(9x^{\,7}y^{\,5}z^{\,8}): (-3x^{\,3}y^{\,2}z^{\,4}) {\small .}\end{array}\)

Заменим знак деления \(\displaystyle ":"\) на черту дроби:

\(\displaystyle \begin{array}{l} (18x^{\,11}y^{\,18}z^{\,14}): (-3x^{\,3}y^{\,2}z^{\,4})-(12x^{\,9}y^{\,11}z^{\,7}): (-3x^{\,3}y^{\,2}z^{\,4})-(9x^{\,7}y^{\,5}z^{\,8}): (-3x^{\,3}y^{\,2}z^{\,4})=\\[10px] \kern{20em} =\frac{18x^{\,11}y^{\,18}z^{\,14}}{-3x^{\,3}y^{\,2}z^{\,4}}-\frac{12x^{\,9}y^{\,11}z^{\,7}}{-3x^{\,3}y^{\,2}z^{\,4}}-\frac{9x^{\,7}y^{\,5}z^{\,8}}{-3x^{\,3}y^{\,2}z^{\,4}} {\small .}\end{array}\)

Сократим каждую дробь на \(\displaystyle -3\) и применим к степеням формулу частного степеней:

\(\displaystyle \begin{array}{l} \frac{18x^{\,11}y^{\,18}z^{\,14}}{-3x^{\,3}y^{\,2}z^{\,4}}-\frac{12x^{\,9}y^{\,11}z^{\,7}}{-3x^{\,3}y^{\,2}z^{\,4}}-\frac{9x^{\,7}y^{\,5}z^{\,8}}{-3x^{\,3}y^{\,2}z^{\,4}}=\\[10px] \kern{10em} =\frac{18}{-3}\frac{x^{\,11}y^{\,18}z^{\,14}}{x^{\,3}y^{\,2}z^{\,4}}-\frac{12}{-3}\frac{x^{\,9}y^{\,11}z^{\,7}}{x^{\,3}y^{\,2}z^{\,4}}-\frac{9}{-3}\frac{x^{\,7}y^{\,5}z^{\,8}}{x^{\,3}y^{\,2}z^{\,4}}=\\[10px] \kern{10em} =-\frac{6}{1}x^{\,11-3}y^{\,18-2}z^{\,14-4}+\frac{4}{1}x^{\,9-3}y^{\,11-2}z^{\,7-4}+\frac{3}{1}x^{\,7-3}y^{\,5-2}z^{\,8-4}=\\[10px] \kern{21em} =-6x^{\,8}y^{\,16}z^{\,10}+4x^{\,6}y^{\,9}z^{\,3}+3x^{\,4}y^{\,3}z^{\,4} {\small .}\end{array}\)

Ответ: \(\displaystyle -6x^{\,8}y^{\,16}z^{\,10}+4x^{\,6}y^{\,9}z^{\,3}+3x^{\,4}y^{\,3}z^{\,4}{\small .}\)