Question
Answer
$$(s^2+s+1)((s^2+s+1)^2-1)+s(-s(s^2+s+1))=s^6+3s^5+5s^4+6s^3+4s^2+2s$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(s^2+s+1)((s^2+s+1)^2-1)+s(-s(s^2+s+1))& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(s^2+s+1)(1s^4+2s^3+3s^2+2s+1-1)+s(-s(s^2+s+1)) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(s^2+s+1)(1s^4+2s^3+3s^2+2s)+s(-s(s^2+s+1)) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}s^6+3s^5+6s^4+7s^3+5s^2+2s+s(-(1s^3+s^2+s)) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}s^6+3s^5+6s^4+7s^3+5s^2+2s+s(-s^3-s^2-s) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}s^6+3s^5+6s^4+7s^3+5s^2+2s-s^4-s^3-s^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}s^6+3s^5+5s^4+6s^3+4s^2+2s\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{s^2+s+1}\right) $ by each term in $ \left( s^2+s+1\right) $. $$ \left( \color{blue}{s^2+s+1}\right) \cdot \left( s^2+s+1\right) = s^4+s^3+s^2+s^3+s^2+s+s^2+s+1 $$ |
| ② | Combine like terms: $$ s^4+ \color{blue}{s^3} + \color{red}{s^2} + \color{blue}{s^3} + \color{green}{s^2} + \color{orange}{s} + \color{green}{s^2} + \color{orange}{s} +1 = s^4+ \color{blue}{2s^3} + \color{green}{3s^2} + \color{orange}{2s} +1 $$ |
| ③ | Combine like terms: $$ s^4+2s^3+3s^2+2s+ \, \color{blue}{ \cancel{1}} \, \, \color{blue}{ -\cancel{1}} \, = s^4+2s^3+3s^2+2s $$ |
| ④ | Multiply each term of $ \left( \color{blue}{s^2+s+1}\right) $ by each term in $ \left( s^4+2s^3+3s^2+2s\right) $. $$ \left( \color{blue}{s^2+s+1}\right) \cdot \left( s^4+2s^3+3s^2+2s\right) = \\ = s^6+2s^5+3s^4+2s^3+s^5+2s^4+3s^3+2s^2+s^4+2s^3+3s^2+2s $$ |
| ⑤ | Combine like terms: $$ s^6+ \color{blue}{2s^5} + \color{red}{3s^4} + \color{green}{2s^3} + \color{blue}{s^5} + \color{orange}{2s^4} + \color{blue}{3s^3} + \color{red}{2s^2} + \color{orange}{s^4} + \color{blue}{2s^3} + \color{red}{3s^2} +2s = \\ = s^6+ \color{blue}{3s^5} + \color{orange}{6s^4} + \color{blue}{7s^3} + \color{red}{5s^2} +2s $$Multiply $ \color{blue}{s} $ by $ \left( s^2+s+1\right) $ $$ \color{blue}{s} \cdot \left( s^2+s+1\right) = s^3+s^2+s $$ |
| ⑥ | Remove the parentheses by changing the sign of each term within them. $$ - \left(s^3+s^2+s \right) = -s^3-s^2-s $$ |
| ⑦ | Multiply $ \color{blue}{s} $ by $ \left( -s^3-s^2-s\right) $ $$ \color{blue}{s} \cdot \left( -s^3-s^2-s\right) = -s^4-s^3-s^2 $$ |
| ⑧ | Combine like terms: $$ s^6+3s^5+ \color{blue}{6s^4} + \color{red}{7s^3} + \color{green}{5s^2} +2s \color{blue}{-s^4} \color{red}{-s^3} \color{green}{-s^2} = \\ = s^6+3s^5+ \color{blue}{5s^4} + \color{red}{6s^3} + \color{green}{4s^2} +2s $$ |
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