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Теория: Сложение и вычитание многочленов

Задание

Найдите сумму многочленов:
 

\(\displaystyle \left(3u^{\,3}s^{\,2}t-ust^{\,2}+ust+13u\right)+\left(-4ust^{\,2}+11u^{\,3}s^{\,2}t-3ust+7u\right)=\)

\(\displaystyle =\)\(\displaystyle u^{\,3}s^{\,2}t\)\(\displaystyle ust^{\,2}\)\(\displaystyle ust\)\(\displaystyle u\)

Решение

Сначала раскроем скобки:

\(\displaystyle \begin{array}{l} \left(3u^{\,3}s^{\,2}t-ust^{\,2}+ust+13u\right)+\left(-4ust^{\,2}+11u^{\,3}s^{\,2}t-3ust+7u\right)=\\ \kern{11em} =3u^{\,3}s^{\,2}t-ust^{\,2}+ust+13u-4ust^{\,2}+11u^{\,3}s^{\,2}t-3ust+7u {\small .}\end{array}\)

Теперь приведем подобные члены, сложив коэффициенты при одинаковых степенях:

\(\displaystyle \begin{array}{l} 3\color{blue}{u^{\,3}s^{\,2}t}-\color{green}{ust^{\,2}}+\color{red}{ust}+13u-4\color{green}{ust^{\,2}}+11\color{blue}{u^{\,3}s^{\,2}t}-3\color{red}{ust}+7u=\\ \kern{8em} =(3\color{blue}{u^{\,3}s^{\,2}t}+11\color{blue}{u^{\,3}s^{\,2}t}\,)+(-\color{green}{ust^{\,2}}-4\color{green}{ust^{\,2}})+(\color{red}{ust}-3\color{red}{ust}\,)+(13u+7u\,)=\\ \kern{14em} =(3+11)\color{blue}{u^{\,3}s^{\,2}t}+(-1-4)\color{green}{ust^{\,2}}+(1-3)\color{red}{ust}+(13+7)u=\\ \kern{24em} =14\color{blue}{u^{\,3}s^{\,2}t}-5\color{green}{ust^{\,2}}-2\color{red}{ust}+20u {\small .}\end{array}\)

 

Таким образом,

\(\displaystyle \begin{array}{l} \left(3u^{\,3}s^{\,2}t-ust^{\,2}+ust+13u\right)+\left(-4ust^{\,2}+11u^{\,3}s^{\,2}t-3ust+7u\right)=\\ \kern{21em} =14u^{\,3}s^{\,2}t-5ust^{\,2}-2ust+20u {\small .}\end{array}\)


Ответ: \(\displaystyle 14u^{\,3}s^{\,2}t-5ust^{\,2}-2ust+20u{\small .}\)