Question
Answer
$$(z-2i)(z+2i)(z-1-4i)(z-1+4i)=64i^4-20i^2z^2+z^4+8i^2z-2z^3-4i^2+z^2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(z-2i)(z+2i)(z-1-4i)(z-1+4i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1z^2+2iz-2iz-4i^2)(z-1-4i)(z-1+4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(-4i^2+z^2)(z-1-4i)(z-1+4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(-4i^2z+4i^2+16i^3+z^3-z^2-4iz^2)(z-1+4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}64i^4-20i^2z^2+z^4+8i^2z-2z^3-4i^2+z^2\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{z-2i}\right) $ by each term in $ \left( z+2i\right) $. $$ \left( \color{blue}{z-2i}\right) \cdot \left( z+2i\right) = z^2+ \cancel{2iz} -\cancel{2iz}-4i^2 $$ |
| ② | Combine like terms: $$ z^2+ \, \color{blue}{ \cancel{2iz}} \, \, \color{blue}{ -\cancel{2iz}} \,-4i^2 = -4i^2+z^2 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{-4i^2+z^2}\right) $ by each term in $ \left( z-1-4i\right) $. $$ \left( \color{blue}{-4i^2+z^2}\right) \cdot \left( z-1-4i\right) = -4i^2z+4i^2+16i^3+z^3-z^2-4iz^2 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{-4i^2z+4i^2+16i^3+z^3-z^2-4iz^2}\right) $ by each term in $ \left( z-1+4i\right) $. $$ \left( \color{blue}{-4i^2z+4i^2+16i^3+z^3-z^2-4iz^2}\right) \cdot \left( z-1+4i\right) = \\ = -4i^2z^2+4i^2z -\cancel{16i^3z}+4i^2z-4i^2+ \cancel{16i^3}+ \cancel{16i^3z} -\cancel{16i^3}+64i^4+z^4-z^3+ \cancel{4iz^3}-z^3+z^2 -\cancel{4iz^2} -\cancel{4iz^3}+ \cancel{4iz^2}-16i^2z^2 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-4i^2z^2} + \color{red}{4i^2z} \, \color{green}{ -\cancel{16i^3z}} \,+ \color{red}{4i^2z} -4i^2+ \, \color{blue}{ \cancel{16i^3}} \,+ \, \color{green}{ \cancel{16i^3z}} \, \, \color{blue}{ -\cancel{16i^3}} \,+64i^4+z^4 \color{green}{-z^3} + \, \color{orange}{ \cancel{4iz^3}} \, \color{green}{-z^3} +z^2 \, \color{red}{ -\cancel{4iz^2}} \, \, \color{orange}{ -\cancel{4iz^3}} \,+ \, \color{red}{ \cancel{4iz^2}} \, \color{blue}{-16i^2z^2} = \\ = 64i^4 \color{blue}{-20i^2z^2} +z^4+ \color{red}{8i^2z} \color{green}{-2z^3} -4i^2+z^2 $$ |
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