Answer
$$(1-2x)^3-7x(x^2-3)=-15x^3+12x^2+15x+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1-2x)^3-7x(x^2-3)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}1-6x+12x^2-8x^3-7x(x^2-3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}1-6x+12x^2-8x^3-(7x^3-21x) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}1-6x+12x^2-8x^3-7x^3+21x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-15x^3+12x^2+15x+1\end{aligned} $$ | |
| ① | Find $ \left(1-2x\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = 1 $ and $ B = 2x $. $$ \left(1-2x\right)^3 = 1^3-3 \cdot 1^2 \cdot 2x + 3 \cdot 1 \cdot \left( 2x \right)^2-\left( 2x \right)^3 = 1-6x+12x^2-8x^3 $$ |
| ② | Multiply $ \color{blue}{7x} $ by $ \left( x^2-3\right) $ $$ \color{blue}{7x} \cdot \left( x^2-3\right) = 7x^3-21x $$ |
| ③ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 7x^3-21x \right) = -7x^3+21x $$ |
| ④ | Combine like terms: $$ 1 \color{blue}{-6x} +12x^2 \color{red}{-8x^3} \color{red}{-7x^3} + \color{blue}{21x} = \color{red}{-15x^3} +12x^2+ \color{blue}{15x} +1 $$ |
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