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