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