Задание
Найдите частное при делении многочлена \(\displaystyle -77z^{\,11}+49z^{\,7}-7z^{\,5}\) на одночлен \(\displaystyle 7z^{\,3}\) в столбик, при известном процессе деления:
| \(\displaystyle -\) | \(\displaystyle \small\color{blue}{\bf -77z^{\,11}}\) | \(\displaystyle +\) | \(\displaystyle \small 49z^{\,7}\) | \(\displaystyle -\) | \(\displaystyle \small 7z^{\,5}\) | \(\displaystyle \small 7z^{\,3}\) | ||
\(\displaystyle \small -77z^{\,11}\) |
|
|
| |||||
| \(\displaystyle -\) | \(\displaystyle \small\color{yellowgreen}{\bf 49z^{\,7}}\) | \(\displaystyle -\) | \(\displaystyle \small 7z^{\,5}\) | |||||
\(\displaystyle \small 49z^{\,7}\) | ||||||||
| \(\displaystyle -\) | \(\displaystyle \small\color{orange}{\bf -7z^{\,5}}\) | |||||||
\(\displaystyle \small -7z^{\,5}\) | ||||||||
| \(\displaystyle \small0\) | ||||||||
С учетом найденного частного запишите разложение на множители:
\(\displaystyle -77z^{\,11}+49z^{\,7}-7z^{\,5}=7z^{\, 3}\cdot\big(\)
\(\displaystyle \big){\small . }\)
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
\(\displaystyle -77z^{\,11}+49z^{\,7}-7z^{\,5}=7z^{\, 3} \cdot (-11z^{\,8}+7z^{\,4}-z^{\,2}){\small .}\)