Question
Answer
$$(x^3-1)^4-x^{12}+x^6=-4x^9+7x^6-4x^3+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x^3-1)^4-x^{12}+x^6& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^{12}-4x^9+6x^6-4x^3+1-x^{12}+x^6 \xlongequal{ } \\[1 em] & \xlongequal{ } \cancel{x^{12}}-4x^9+6x^6-4x^3+1 -\cancel{x^{12}}+x^6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-4x^9+7x^6-4x^3+1\end{aligned} $$ | |
| ① | $$ (x^3-1)^4 = (x^3-1)^2 \cdot (x^3-1)^2 $$ |
| ② | Find $ \left(x^3-1\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x^3 } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(x^3-1\right)^2 = \color{blue}{\left( x^3 \right)^2} -2 \cdot x^3 \cdot 1 + \color{red}{1^2} = x^6-2x^3+1\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^6-2x^3+1}\right) $ by each term in $ \left( x^6-2x^3+1\right) $. $$ \left( \color{blue}{x^6-2x^3+1}\right) \cdot \left( x^6-2x^3+1\right) = x^{12}-2x^9+x^6-2x^9+4x^6-2x^3+x^6-2x^3+1 $$ |
| ④ | Combine like terms: $$ x^{12} \color{blue}{-2x^9} + \color{red}{x^6} \color{blue}{-2x^9} + \color{green}{4x^6} \color{orange}{-2x^3} + \color{green}{x^6} \color{orange}{-2x^3} +1 = \\ = x^{12} \color{blue}{-4x^9} + \color{green}{6x^6} \color{orange}{-4x^3} +1 $$ |
| ⑤ | Combine like terms: $$ \, \color{blue}{ \cancel{x^{12}}} \,-4x^9+ \color{green}{6x^6} -4x^3+1 \, \color{blue}{ -\cancel{x^{12}}} \,+ \color{green}{x^6} = -4x^9+ \color{green}{7x^6} -4x^3+1 $$ |
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