Question
Answer
$$(1-x)^3+2(1-x)^2x+(1-x)x^2=-x+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1-x)^3+2(1-x)^2x+(1-x)x^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}1-3x+3x^2-x^3+2(1-2x+x^2)x+(1-x)x^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}1-3x+3x^2-x^3+(2-4x+2x^2)x+x^2-x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}1-3x+3x^2-x^3+2x-4x^2+2x^3+x^2-x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^3-x^2-x+1+x^2-x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-x+1\end{aligned} $$ | |
| ① | Find $ \left(1-x\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = 1 $ and $ B = x $. $$ \left(1-x\right)^3 = 1^3-3 \cdot 1^2 \cdot x + 3 \cdot 1 \cdot x^2-x^3 = 1-3x+3x^2-x^3 $$Find $ \left(1-x\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 1 } $ and $ B = \color{red}{ x }$. $$ \begin{aligned}\left(1-x\right)^2 = \color{blue}{1^2} -2 \cdot 1 \cdot x + \color{red}{x^2} = 1-2x+x^2\end{aligned} $$ |
| ② | Multiply $ \color{blue}{2} $ by $ \left( 1-2x+x^2\right) $ $$ \color{blue}{2} \cdot \left( 1-2x+x^2\right) = 2-4x+2x^2 $$$$ \left( \color{blue}{1-x}\right) \cdot x^2 = x^2-x^3 $$ |
| ③ | $$ \left( \color{blue}{2-4x+2x^2}\right) \cdot x = 2x-4x^2+2x^3 $$ |
| ④ | Combine like terms: $$ 1 \color{blue}{-3x} + \color{red}{3x^2} \color{green}{-x^3} + \color{blue}{2x} \color{red}{-4x^2} + \color{green}{2x^3} = \color{green}{x^3} \color{red}{-x^2} \color{blue}{-x} +1 $$ |
| ⑤ | Combine like terms: $$ \, \color{blue}{ \cancel{x^3}} \, \, \color{green}{ -\cancel{x^2}} \,-x+1+ \, \color{green}{ \cancel{x^2}} \, \, \color{blue}{ -\cancel{x^3}} \, = -x+1 $$ |
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