Question
Answer
$$(1-x)^2x+(1-x)x^2\cdot2+x^3=x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1-x)^2x+(1-x)x^2\cdot2+x^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1-2x+x^2)x+(1-x)x^2\cdot2+x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x-2x^2+x^3+(x^2-x^3)\cdot2+x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x-2x^2+x^3+2x^2-2x^3+x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-x^3+x+x^3 \xlongequal{ } \\[1 em] & \xlongequal{ } -\cancel{x^3}+x+ \cancel{x^3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x\end{aligned} $$ | |
| ① | 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 | $$ \left( \color{blue}{1-2x+x^2}\right) \cdot x = x-2x^2+x^3 $$$$ \left( \color{blue}{1-x}\right) \cdot x^2 = x^2-x^3 $$ |
| ③ | $$ \left( \color{blue}{x^2-x^3}\right) \cdot 2 = 2x^2-2x^3 $$ |
| ④ | Combine like terms: $$ x \, \color{blue}{ -\cancel{2x^2}} \,+ \color{green}{x^3} + \, \color{blue}{ \cancel{2x^2}} \, \color{green}{-2x^3} = \color{green}{-x^3} +x $$ |
| ⑤ | Combine like terms: $$ \, \color{blue}{ -\cancel{x^3}} \,+x+ \, \color{blue}{ \cancel{x^3}} \, = x $$ |
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