Question
Answer
$$6(x^2-3)^4=6x^8-72x^6+324x^4-648x^2+486$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}6(x^2-3)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}6(x^8-12x^6+54x^4-108x^2+81) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}6x^8-72x^6+324x^4-648x^2+486\end{aligned} $$ | |
| ① | $$ (x^2-3)^4 = (x^2-3)^2 \cdot (x^2-3)^2 $$ |
| ② | Find $ \left(x^2-3\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^2 } $ and $ B = \color{red}{ 3 }$. $$ \begin{aligned}\left(x^2-3\right)^2 = \color{blue}{\left( x^2 \right)^2} -2 \cdot x^2 \cdot 3 + \color{red}{3^2} = x^4-6x^2+9\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^4-6x^2+9}\right) $ by each term in $ \left( x^4-6x^2+9\right) $. $$ \left( \color{blue}{x^4-6x^2+9}\right) \cdot \left( x^4-6x^2+9\right) = x^8-6x^6+9x^4-6x^6+36x^4-54x^2+9x^4-54x^2+81 $$ |
| ④ | Combine like terms: $$ x^8 \color{blue}{-6x^6} + \color{red}{9x^4} \color{blue}{-6x^6} + \color{green}{36x^4} \color{orange}{-54x^2} + \color{green}{9x^4} \color{orange}{-54x^2} +81 = \\ = x^8 \color{blue}{-12x^6} + \color{green}{54x^4} \color{orange}{-108x^2} +81 $$ |
| ⑤ | Multiply $ \color{blue}{6} $ by $ \left( x^8-12x^6+54x^4-108x^2+81\right) $ $$ \color{blue}{6} \cdot \left( x^8-12x^6+54x^4-108x^2+81\right) = 6x^8-72x^6+324x^4-648x^2+486 $$ |
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