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