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