Найдите частное при делении многочлена \(\displaystyle 50y^{\,20}+25y^{\,18}-15y^{\,3}\) на одночлен \(\displaystyle 5y^{\,3}\) в столбик и восстановите процесс деления:
| \(\displaystyle -\) | \(\displaystyle 50y^{\,20}\) | \(\displaystyle +\) | \(\displaystyle 25y^{\,18}\) | \(\displaystyle -\) | \(\displaystyle 15y^{\,3}\) | \(\displaystyle 5y^{\,3}\) | |||||||
| \(\displaystyle -\) | |||||||||||||
| \(\displaystyle -\) | |||||||||||||
| \(\displaystyle 0\) | |||||||||||||
С учетом известного частного запишите разложение на множители:
Разделим многочлен \(\displaystyle 50y^{\,20}+25y^{\,18}-15y^{\,3}\) на одночлен \(\displaystyle 5y^{\,3}\) в столбик.
Шаг 1. Многочлен \(\displaystyle {\bf 50y^{\,20}+25y^{\,18}-15y^{\,3}}{\small .}\)
1. Выбираем одночлен старшей степени в записи многочлена \(\displaystyle \color{blue}{50y^{\,20}}+25y^{\,18}-15y^{\,3}{\small ,}\) это одночлен \(\displaystyle \color{blue}{50y^{\,20}}{\small .}\)
2. Делим одночлен \(\displaystyle \color{blue}{50y^{\,20}}\) на одночлен \(\displaystyle 5y^{\,3}{\small :}\)
\(\displaystyle \frac{\color{blue}{50y^{\,20}}}{5y^{\,3}}=\color{blue}{10y^{\, 17}}{\small .}\)
Записываем результат деления как первое слагаемое частного:
| \(\displaystyle \small \color{blue}{50y^{\,20}}+25y^{\,18}-15y^{\,3}\) | \(\displaystyle \small 5y^{\,3}\) |
| \(\displaystyle \small \color{blue}{10y^{\, 17}}\,?\) |
3. Вычитаем в столбик из многочлена \(\displaystyle \color{blue}{50y^{\,20}}+25y^{\,18}-15y^{\,3}\) одночлен \(\displaystyle \color{blue}{50y^{\,20}}=5y^{\,3}\cdot \color{blue}{10y^{\,17}} {\small :}\)
| \(\displaystyle -\) | \(\displaystyle \small \color{blue}{50y^{\,20}}\) | \(\displaystyle +\) | \(\displaystyle \small 25y^{\,18}\) | \(\displaystyle -\) | \(\displaystyle \small 15y^{\,3}\) | \(\displaystyle \small 5y^{\,3}\) |
| \(\displaystyle \small \color{blue}{50y^{\,20}}\) | \(\displaystyle \small\color{blue}{10y^{\,17}} \,?\) | |||||
| \(\displaystyle \small 25y^{\,18}\) | \(\displaystyle -\) | \(\displaystyle \small 15y^{\,3}\) |
Получаем многочлен \(\displaystyle 25y^{\,18}-15y^{\,3}{\small . }\)
Шаг 3. Многочлен \(\displaystyle {\bf -15y^{\,3}}{\small .}\)
1. Выбираем одночлен старшей степени в записи многочлена \(\displaystyle \color{orange}{-15y^{\,3}}{\small ,}\) это и есть сам одночлен \(\displaystyle \color{orange}{-15y^{\,3}}{\small .}\)
2. Делим одночлен \(\displaystyle \color{orange}{-15y^{\,3}}\) на одночлен \(\displaystyle 5y^{\,3}{\small :}\)
\(\displaystyle \frac{\color{orange}{-15y^{\,3}}}{5y^{\,3}}=\color{orange}{-3}{\small .}\)
Записываем результат деления как третье слагаемое частного со знаком \(\displaystyle "-"\):
| \(\displaystyle -\) | \(\displaystyle \small 50y^{\,20}\) | \(\displaystyle +\) | \(\displaystyle \small 25y^{\,18}\) | \(\displaystyle -\) | \(\displaystyle \small 15y^{\,3}\) | \(\displaystyle \small 5y^{\,3}\) | ||
| \(\displaystyle \small 50y^{\,20}\) | \(\displaystyle \small\color{blue}{10y^{\,17}}\color{green}{+5y^{\,15}}\,\color{orange}{-3}\) | |||||||
| \(\displaystyle -\) | \(\displaystyle \small 25y^{\,18}\) | \(\displaystyle -\) | \(\displaystyle \small 15y^{\,3}\) | |||||
| \(\displaystyle \small 25y^{\,18}\) | ||||||||
| \(\displaystyle \small \color{orange}{15y^{\,3}}\) | ||||||||
3. Вычитаем в столбик из одночлена \(\displaystyle \color{orange}{-15y^{\,3}}\) одночлен \(\displaystyle \color{orange}{-15y^{\,3}}=5y^{\,3}\cdot (\color{orange}{-3}) {\small :}\)
| \(\displaystyle -\) | \(\displaystyle \small 50y^{\,20}\) | \(\displaystyle +\) | \(\displaystyle \small 25y^{\,18}\) | \(\displaystyle -\) | \(\displaystyle \small 15y^{\,3}\) | \(\displaystyle \small 5y^{\,3}\) |
| \(\displaystyle \small 50y^{\,20}\) | \(\displaystyle \small\color{blue}{10y^{\,17}}\color{green}{+5y^{\,15}}\,\color{orange}{-3}\) | |||||
| \(\displaystyle -\) | \(\displaystyle \small 25y^{\,18}\) | \(\displaystyle -\) | \(\displaystyle \small 15y^{\,3}\) | |||
| \(\displaystyle \small 25y^{\,18}\) | ||||||
| \(\displaystyle -\) | \(\displaystyle \small \color{orange}{-15y^{\,3}}\) | |||||
| \(\displaystyle \small \color{orange}{-15y^{\,3}}\) | ||||||
| \(\displaystyle \small 0\) |
В итоге получаем \(\displaystyle 0{\small ,}\) процесс деления закончен.
Таким образом,
| \(\displaystyle -\) | \(\displaystyle \small\color{blue}{50y^{\,20}}\) | \(\displaystyle +\) | \(\displaystyle \small 25y^{\,18}\) | \(\displaystyle -\) | \(\displaystyle \small 15y^{\,3}\) | \(\displaystyle \small 5y^{\,3}\) |
| \(\displaystyle \small 50y^{\,20}\) | \(\displaystyle \small\color{blue}{10y^{\,17}}\color{green}{+5y^{\,15}}\,\color{orange}{-3}\) | |||||
| \(\displaystyle -\) | \(\displaystyle \small\color{green}{ 25y^{\,18}}\) | \(\displaystyle -\) | \(\displaystyle \small 15y^{\,3}\) | |||
| \(\displaystyle \small 25y^{\,18}\) | ||||||
| \(\displaystyle -\) | \(\displaystyle \small\color{orange}{ -15y^{\,3}}\) | |||||
| \(\displaystyle \small -15y^{\,3}\) | ||||||
| \(\displaystyle \small 0\) |
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