Задание
Разделите многочлен \(\displaystyle 6s^{\,3}-9s^{\,2}-23s+25\) на многочлен \(\displaystyle 2s-5\) в столбик:
| \(\displaystyle -\) | \(\displaystyle \phantom{\,\,\,}6s^{\,3}-9s^{\,2}-23s+25\) | \(\displaystyle 2s-5\) | |||||
| \(\displaystyle \phantom{ 9xx} -\) | |||||||
| \(\displaystyle -\) | |||||||
| \(\displaystyle \phantom{xxx}\) | \(\displaystyle \phantom{xxxx}\) | ||||||
и запишите разложение:
\(\displaystyle 6s^{\,3}-9s^{\,2}-23s+25=(2s-5)\cdot \big(\)
\(\displaystyle \big)+\)
\(\displaystyle \overset{неполное}{\footnotesize{частное}}\) \(\displaystyle \overset{остаток}{\footnotesize{\phantom{x}}}\)
Решение
Разделим многочлен \(\displaystyle 6s^{\,3}-9s^{\,2}-23s+25\) на многочлен \(\displaystyle 2s-5\) в столбик.
Одночлен старшей степени у делителя \(\displaystyle 2s-5\) – это одночлен \(\displaystyle \color{red}{2s}{\small .}\)
Шаг 1. Деление многочлена \(\displaystyle {\small \color{blue}{6s^{\,3}-9s^{\,2}-23s+25}}\)
Шаг 2. Деление многочлена \(\displaystyle {\small \color{green}{6s^{\,2}-23s+25}}\)
Шаг 3. Деление многочлена \(\displaystyle {\small \color{orange}{-8s+25}}\)
Таким образом,
| \(\displaystyle -\) | \(\displaystyle \color{blue}{ 6s^{\,3}-\phantom{1\,}9s^{\,2}-23s+25}\) | \(\displaystyle 2s-5\) | ||||||||||
| \(\displaystyle 6s^{\,3}-15s^{\,2}\) | \(\displaystyle 3s^{\,2}+3s\color{orange}{-4}\) | |||||||||||
| \(\displaystyle \phantom{s^{\,3}+}-\) | \(\displaystyle \color{green}{ 6s^{\,2}-23s+25}\) | |||||||||||
| \(\displaystyle 6s^{\,2}-15s\) | ||||||||||||
| \(\displaystyle \phantom{ 9s^{\,2}+\,} -\) | \(\displaystyle \color{orange}{-8s+25}\) | |||||||||||
| \(\displaystyle -8s+20\) | ||||||||||||
| \(\displaystyle 5\) | ||||||||||||
и
\(\displaystyle 6s^{\,3}-9s^{\,2}-23s+25=(2s-5)\cdot ({\bf 3s^{\,2}+3s-4})+5{\small .}\)