Question
Answer
$$(x^3-3)^4=x^{12}-12x^9+54x^6-108x^3+81$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x^3-3)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^{12}-12x^9+54x^6-108x^3+81\end{aligned} $$ | |
| ① | $$ (x^3-3)^4 = (x^3-3)^2 \cdot (x^3-3)^2 $$ |
| ② | Find $ \left(x^3-3\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}{ 3 }$. $$ \begin{aligned}\left(x^3-3\right)^2 = \color{blue}{\left( x^3 \right)^2} -2 \cdot x^3 \cdot 3 + \color{red}{3^2} = x^6-6x^3+9\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^6-6x^3+9}\right) $ by each term in $ \left( x^6-6x^3+9\right) $. $$ \left( \color{blue}{x^6-6x^3+9}\right) \cdot \left( x^6-6x^3+9\right) = x^{12}-6x^9+9x^6-6x^9+36x^6-54x^3+9x^6-54x^3+81 $$ |
| ④ | Combine like terms: $$ x^{12} \color{blue}{-6x^9} + \color{red}{9x^6} \color{blue}{-6x^9} + \color{green}{36x^6} \color{orange}{-54x^3} + \color{green}{9x^6} \color{orange}{-54x^3} +81 = \\ = x^{12} \color{blue}{-12x^9} + \color{green}{54x^6} \color{orange}{-108x^3} +81 $$ |
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