Question
Answer
$$2(x-1)(2x+2)^3=16x^4+32x^3-32x-16$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}2(x-1)(2x+2)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}2(x-1)(8x^3+24x^2+24x+8) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(2x-2)(8x^3+24x^2+24x+8) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}16x^4+32x^3-32x-16\end{aligned} $$ | |
| ① | Find $ \left(2x+2\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 2x $ and $ B = 2 $. $$ \left(2x+2\right)^3 = \left( 2x \right)^3+3 \cdot \left( 2x \right)^2 \cdot 2 + 3 \cdot 2x \cdot 2^2+2^3 = 8x^3+24x^2+24x+8 $$ |
| ② | Multiply $ \color{blue}{2} $ by $ \left( x-1\right) $ $$ \color{blue}{2} \cdot \left( x-1\right) = 2x-2 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{2x-2}\right) $ by each term in $ \left( 8x^3+24x^2+24x+8\right) $. $$ \left( \color{blue}{2x-2}\right) \cdot \left( 8x^3+24x^2+24x+8\right) = \\ = 16x^4+48x^3+ \cancel{48x^2}+16x-16x^3 -\cancel{48x^2}-48x-16 $$ |
| ④ | Combine like terms: $$ 16x^4+ \color{blue}{48x^3} + \, \color{red}{ \cancel{48x^2}} \,+ \color{orange}{16x} \color{blue}{-16x^3} \, \color{red}{ -\cancel{48x^2}} \, \color{orange}{-48x} -16 = 16x^4+ \color{blue}{32x^3} \color{orange}{-32x} -16 $$ |
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