Question
Answer
$$3(x^2-4)(x-2)^4=3x^6-24x^5+60x^4-240x^2+384x-192$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}3(x^2-4)(x-2)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}3(x^2-4)(x^4-8x^3+24x^2-32x+16) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(3x^2-12)(x^4-8x^3+24x^2-32x+16) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}3x^6-24x^5+60x^4-240x^2+384x-192\end{aligned} $$ | |
| ① | $$ (x-2)^4 = (x-2)^2 \cdot (x-2)^2 $$ |
| ② | Find $ \left(x-2\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 } $ and $ B = \color{red}{ 2 }$. $$ \begin{aligned}\left(x-2\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 2 + \color{red}{2^2} = x^2-4x+4\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-4x+4}\right) $ by each term in $ \left( x^2-4x+4\right) $. $$ \left( \color{blue}{x^2-4x+4}\right) \cdot \left( x^2-4x+4\right) = x^4-4x^3+4x^2-4x^3+16x^2-16x+4x^2-16x+16 $$ |
| ④ | Combine like terms: $$ x^4 \color{blue}{-4x^3} + \color{red}{4x^2} \color{blue}{-4x^3} + \color{green}{16x^2} \color{orange}{-16x} + \color{green}{4x^2} \color{orange}{-16x} +16 = \\ = x^4 \color{blue}{-8x^3} + \color{green}{24x^2} \color{orange}{-32x} +16 $$ |
| ⑤ | Multiply $ \color{blue}{3} $ by $ \left( x^2-4\right) $ $$ \color{blue}{3} \cdot \left( x^2-4\right) = 3x^2-12 $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{3x^2-12}\right) $ by each term in $ \left( x^4-8x^3+24x^2-32x+16\right) $. $$ \left( \color{blue}{3x^2-12}\right) \cdot \left( x^4-8x^3+24x^2-32x+16\right) = \\ = 3x^6-24x^5+72x^4 -\cancel{96x^3}+48x^2-12x^4+ \cancel{96x^3}-288x^2+384x-192 $$ |
| ⑦ | Combine like terms: $$ 3x^6-24x^5+ \color{blue}{72x^4} \, \color{red}{ -\cancel{96x^3}} \,+ \color{orange}{48x^2} \color{blue}{-12x^4} + \, \color{red}{ \cancel{96x^3}} \, \color{orange}{-288x^2} +384x-192 = 3x^6-24x^5+ \color{blue}{60x^4} \color{orange}{-240x^2} +384x-192 $$ |
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