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Теория: Деление многочлена на одночлен в столбик (* доп. раздел)

Задание

Найдите частное при делении многочлена \(\displaystyle -77z^{\,11}+49z^{\,7}-7z^{\,5}\) на одночлен \(\displaystyle 7z^{\,3}\) в столбик, при известном процессе деления:
 

\(\displaystyle -\) \(\displaystyle \small\color{blue}{\bf -77z^{\,11}}\) \(\displaystyle +\) \(\displaystyle \small 49z^{\,7}\) \(\displaystyle -\) \(\displaystyle \small 7z^{\,5}\) \(\displaystyle \small 7z^{\,3}\)

\(\displaystyle \small -77z^{\,11}\)

       
-11z^8
+7z^4
-z^2
    \(\displaystyle -\) \(\displaystyle \small\color{yellowgreen}{\bf 49z^{\,7}}\) \(\displaystyle -\) \(\displaystyle \small 7z^{\,5}\)      
   

\(\displaystyle \small 49z^{\,7}\)

         
        \(\displaystyle -\) \(\displaystyle \small\color{orange}{\bf -7z^{\,5}}\)      
       

\(\displaystyle \small -7z^{\,5}\)

     
          \(\displaystyle \small0\)      


С учетом найденного частного запишите разложение на множители:

\(\displaystyle -77z^{\,11}+49z^{\,7}-7z^{\,5}=7z^{\, 3}\cdot\big(\)
-11z^8+7z^4-z^2
\(\displaystyle \big){\small . }\)
Решение

Разделим многочлен \(\displaystyle -77z^{\,11}+49z^{\,7}-7z^{\,5}\) на одночлен \(\displaystyle 7z^{\,3}\) в столбик.

Шаг 1

Шаг 2

Шаг 3

Таким образом,

\(\displaystyle -\) \(\displaystyle \small\color{blue}{-77z^{\,11}}\) \(\displaystyle +\) \(\displaystyle \small 49z^{\,7}\) \(\displaystyle -\) \(\displaystyle \small 7z^{\,5}\) \(\displaystyle \small 7z^{\,3}\)
\(\displaystyle \small -77z^{\,11}\)         \(\displaystyle \small\color{blue}{-11z^{\,8}} \color{green}{+7z^{\,4}}\color{orange}{-z^{\,2}}\)
    \(\displaystyle -\) \(\displaystyle \small\color{green}{ 49z^{\,7}}\) \(\displaystyle -\) \(\displaystyle \small 7z^{\,5}\)  
    \(\displaystyle \small 49z^{\,7}\)      
        \(\displaystyle -\) \(\displaystyle \small\color{orange}{ -7z^{\,5}}\)  
        \(\displaystyle \small -7z^{\,5}\)  
          \(\displaystyle \small 0\)  

и

\(\displaystyle -77z^{\,11}+49z^{\,7}-7z^{\,5}=7z^{\, 3} \cdot (-11z^{\,8}+7z^{\,4}-z^{\,2}){\small .}\)