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