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
\(\displaystyle \begin{array}{l}(18x^{\,11}y^{\,18}z^{\,14}-12x^{\,9}y^{\,11}z^{\,7}-9x^{\,7}y^{\,5}z^{\,8}): (-3x^{\,3}y^{\,2}z^{\,4})=\\[10pt] =(18x^{\,11}y^{\,18}z^{\,14}): (-3x^{\,3}y^{\,2}z^{\,4})-(12x^{\,9}y^{\,11}z^{\,7}): (-3x^{\,3}y^{\,2}z^{\,4})-\\[10pt]\kern{20em}-(9x^{\,7}y^{\,5}z^{\,8}): (-3x^{\,3}y^{\,2}z^{\,4}){\small .}\end{array}\)
Заменим знак деления \(\displaystyle ":"\) на черту дроби:
\(\displaystyle \begin{array}{l}(18x^{\,11}y^{\,18}z^{\,14}): (-3x^{\,3}y^{\,2}z^{\,4})-(12x^{\,9}y^{\,11}z^{\,7}): (-3x^{\,3}y^{\,2}z^{\,4})-\\[10pt]-(9x^{\,7}y^{\,5}z^{\,8}): (-3x^{\,3}y^{\,2}z^{\,4})= \cfrac{18x^{\,11}y^{\,18}z^{\,14}}{-3x^{\,3}y^{\,2}z^{\,4}}-\cfrac{12x^{\,9}y^{\,11}z^{\,7}}{-3x^{\,3}y^{\,2}z^{\,4}}-\cfrac{9x^{\,7}y^{\,5}z^{\,8}}{-3x^{\,3}y^{\,2}z^{\,4}}{\small .}\end{array}\)
Сократим каждую дробь на \(\displaystyle -3\) и применим к степеням формулу частного степеней:
\(\displaystyle \begin{array}{l}\kern{2em}\cfrac{18x^{\,11}y^{\,18}z^{\,14}}{-3x^{\,3}y^{\,2}z^{\,4}}-\cfrac{12x^{\,9}y^{\,11}z^{\,7}}{-3x^{\,3}y^{\,2}z^{\,4}}-\cfrac{9x^{\,7}y^{\,5}z^{\,8}}{-3x^{\,3}y^{\,2}z^{\,4}}=\\[20px]=\cfrac{18}{-3}\cfrac{x^{\,11}y^{\,18}z^{\,14}}{x^{\,3}y^{\,2}z^{\,4}}-\cfrac{12}{-3}\cfrac{x^{\,9}y^{\,11}z^{\,7}}{x^{\,3}y^{\,2}z^{\,4}}-\cfrac{9}{-3}\cfrac{x^{\,7}y^{\,5}z^{\,8}}{x^{\,3}y^{\,2}z^{\,4}}=\\[20px] =-\cfrac{6}{1}\ \ x^{\,11-3}y^{\,18-2}z^{\,14-4}+\cfrac{4}{1}\ \ x^{\,9-3}y^{\,11-2}z^{\,7-4}+\cfrac{3}{1}\ \ x^{\,7-3}y^{\,5-2}z^{\,8-4}=\\[20px]\kern{3em} =-6x^{\,8}y^{\,16}z^{\,10}+4x^{\,6}y^{\,9}z^{\,3}+3x^{\,4}y^{\,3}z^{\,4}{\small .}\end{array}\)
Ответ: \(\displaystyle -6x^{\,8}y^{\,16}z^{\,10}+4x^{\,6}y^{\,9}z^{\,3}+3x^{\,4}y^{\,3}z^{\,4}{\small .}\)