Question
Answer
$$(z-1)(z^6-6z^5+31z^4+12z^3+31z^2-6z+1)=z^7-7z^6+37z^5-19z^4+19z^3-37z^2+7z-1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(z-1)(z^6-6z^5+31z^4+12z^3+31z^2-6z+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}z^7-7z^6+37z^5-19z^4+19z^3-37z^2+7z-1\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{z-1}\right) $ by each term in $ \left( z^6-6z^5+31z^4+12z^3+31z^2-6z+1\right) $. $$ \left( \color{blue}{z-1}\right) \cdot \left( z^6-6z^5+31z^4+12z^3+31z^2-6z+1\right) = \\ = z^7-6z^6+31z^5+12z^4+31z^3-6z^2+z-z^6+6z^5-31z^4-12z^3-31z^2+6z-1 $$ |
| ② | Combine like terms: $$ z^7 \color{blue}{-6z^6} + \color{red}{31z^5} + \color{green}{12z^4} + \color{orange}{31z^3} \color{blue}{-6z^2} + \color{red}{z} \color{blue}{-z^6} + \color{red}{6z^5} \color{green}{-31z^4} \color{orange}{-12z^3} \color{blue}{-31z^2} + \color{red}{6z} -1 = \\ = z^7 \color{blue}{-7z^6} + \color{red}{37z^5} \color{green}{-19z^4} + \color{orange}{19z^3} \color{blue}{-37z^2} + \color{red}{7z} -1 $$ |
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