Найдите сумму многочленов:
\(\displaystyle \left(3u^{\,3}s^{\,2}t-ust^{\,2}+ust+13u\right)+\left(-4ust^{\,2}+11u^{\,3}s^{\,2}t-3ust+7u\right)=\)
\(\displaystyle =\)\(\displaystyle u^{\,3}s^{\,2}t\)\(\displaystyle ust^{\,2}\)\(\displaystyle ust\)\(\displaystyle u\)
Сначала раскроем скобки:
\(\displaystyle \begin{array}{l}\left(3u^{\,3}s^{\,2}t-ust^{\,2}+ust+13u\right)+\left(-4ust^{\,2}+11u^{\,3}s^{\,2}t-3ust+7u\right)=\\[10pt]\kern{5em} =3u^{\,3}s^{\,2}t-ust^{\,2}+ust+13u-4ust^{\,2}+11u^{\,3}s^{\,2}t-3ust+7u{\small .}\end{array}\)
Теперь приведем подобные члены, сложив коэффициенты при одинаковых степенях:
\(\displaystyle \begin{array}{l}3\color{blue}{u^{\,3}s^{\,2}t}-\color{green}{ust^{\,2}}+\color{red}{ust}+13u-4\color{green}{ust^{\,2}}+11\color{blue}{u^{\,3}s^{\,2}t}-3\color{red}{ust}+7u=\\[10pt]\kern{3em} =(3\color{blue}{u^{\,3}s^{\,2}t}+11\color{blue}{u^{\,3}s^{\,2}t}\,)+(-\color{green}{ust^{\,2}}-4\color{green}{ust^{\,2}})+(\color{red}{ust}-3\color{red}{ust}\,)+(13u+7u\,)=\\[10pt]\kern{6em} =(3+11)\color{blue}{u^{\,3}s^{\,2}t}+(-1-4)\color{green}{ust^{\,2}}+(1-3)\color{red}{ust}+(13+7)u=\\[10pt]\kern{9em} =14\color{blue}{u^{\,3}s^{\,2}t}-5\color{green}{ust^{\,2}}-2\color{red}{ust}+20u{\small .}\end{array}\)
Таким образом,
\(\displaystyle \begin{array}{l}\left(3u^{\,3}s^{\,2}t-ust^{\,2}+ust+13u\right)+\left(-4ust^{\,2}+11u^{\,3}s^{\,2}t-3ust+7u\right)=\\[10pt]\kern{6em} =14u^{\,3}s^{\,2}t-5ust^{\,2}-2ust+20u{\small .}\end{array}\)
Ответ: \(\displaystyle 14u^{\,3}s^{\,2}t-5ust^{\,2}-2ust+20u{\small .}\)