Question
Answer
$$(1+t+t^2)(t^1+t^2+t^3)(t^0+t^1+t^2+t^3+t^4)=t^9+3t^8+6t^7+8t^6+9t^5+8t^4+6t^3+3t^2+t$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+t+t^2)(t^1+t^2+t^3)(t^0+t^1+t^2+t^3+t^4)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1t^5+2t^4+3t^3+2t^2+t)(t^0+t^1+t^2+t^3+t^4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}t^9+3t^8+6t^7+8t^6+9t^5+8t^4+6t^3+3t^2+t\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{1+t+t^2}\right) $ by each term in $ \left( t+t^2+t^3\right) $. $$ \left( \color{blue}{1+t+t^2}\right) \cdot \left( t+t^2+t^3\right) = t+t^2+t^3+t^2+t^3+t^4+t^3+t^4+t^5 $$ |
| ② | Combine like terms: $$ t+ \color{blue}{t^2} + \color{red}{t^3} + \color{blue}{t^2} + \color{green}{t^3} + \color{orange}{t^4} + \color{green}{t^3} + \color{orange}{t^4} +t^5 = \\ = t^5+ \color{orange}{2t^4} + \color{green}{3t^3} + \color{blue}{2t^2} +t $$ |
| ③ | Multiply each term of $ \left( \color{blue}{t^5+2t^4+3t^3+2t^2+t}\right) $ by each term in $ \left( 1+t+t^2+t^3+t^4\right) $. $$ \left( \color{blue}{t^5+2t^4+3t^3+2t^2+t}\right) \cdot \left( 1+t+t^2+t^3+t^4\right) = \\ = t^5+t^6+t^7+t^8+t^9+2t^4+2t^5+2t^6+2t^7+2t^8+3t^3+3t^4+3t^5+3t^6+3t^7+2t^2+2t^3+2t^4+2t^5+2t^6+t+t^2+t^3+t^4+t^5 $$ |
| ④ | Combine like terms: $$ \color{blue}{t^5} + \color{red}{t^6} + \color{green}{t^7} + \color{orange}{t^8} +t^9+ \color{blue}{2t^4} + \color{red}{2t^5} + \color{green}{2t^6} + \color{orange}{2t^7} + \color{orange}{2t^8} + \color{blue}{3t^3} + \color{red}{3t^4} + \color{green}{3t^5} + \color{orange}{3t^6} + \color{orange}{3t^7} + \color{blue}{2t^2} + \color{red}{2t^3} + \color{green}{2t^4} + \color{orange}{2t^5} + \color{orange}{2t^6} +t+ \color{blue}{t^2} + \color{red}{t^3} + \color{green}{t^4} + \color{orange}{t^5} = \\ = t^9+ \color{orange}{3t^8} + \color{orange}{6t^7} + \color{orange}{8t^6} + \color{orange}{9t^5} + \color{green}{8t^4} + \color{red}{6t^3} + \color{blue}{3t^2} +t $$ |
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