Question
Answer
$$(-3x^3)(-2x+2)^4=-48x^7+192x^6-288x^5+192x^4-48x^3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(-3x^3)(-2x+2)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(-3x^3)(16x^4-64x^3+96x^2-64x+16) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-48x^7+192x^6-288x^5+192x^4-48x^3\end{aligned} $$ | |
| ① | $$ (-2x+2)^4 = (-2x+2)^2 \cdot (-2x+2)^2 $$ |
| ② | Find $ \left(-2x+2\right)^2 $ in two steps. S1: Change all signs inside bracket. S2: Apply formula $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 2x } $ and $ B = \color{red}{ 2 }$. $$ \begin{aligned}\left(-2x+2\right)^2& \xlongequal{ S1 } \left(2x-2\right)^2 \xlongequal{ S2 } \color{blue}{\left( 2x \right)^2} -2 \cdot 2x \cdot 2 + \color{red}{2^2} = \\[1 em] & = 4x^2-8x+4\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{4x^2-8x+4}\right) $ by each term in $ \left( 4x^2-8x+4\right) $. $$ \left( \color{blue}{4x^2-8x+4}\right) \cdot \left( 4x^2-8x+4\right) = 16x^4-32x^3+16x^2-32x^3+64x^2-32x+16x^2-32x+16 $$ |
| ④ | Combine like terms: $$ 16x^4 \color{blue}{-32x^3} + \color{red}{16x^2} \color{blue}{-32x^3} + \color{green}{64x^2} \color{orange}{-32x} + \color{green}{16x^2} \color{orange}{-32x} +16 = \\ = 16x^4 \color{blue}{-64x^3} + \color{green}{96x^2} \color{orange}{-64x} +16 $$ |
| ⑤ | Multiply $ \color{blue}{-3x^3} $ by $ \left( 16x^4-64x^3+96x^2-64x+16\right) $ $$ \color{blue}{-3x^3} \cdot \left( 16x^4-64x^3+96x^2-64x+16\right) = -48x^7+192x^6-288x^5+192x^4-48x^3 $$ |
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