Question
Answer
$$(x-3i)(x+3i)(x-2-i)(x-2+i)=9i^4-10i^2x^2+x^4+36i^2x-4x^3-36i^2+4x^2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-3i)(x+3i)(x-2-i)(x-2+i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+3ix-3ix-9i^2)(x-2-i)(x-2+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(-9i^2+x^2)(x-2-i)(x-2+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(-9i^2x+18i^2+9i^3+x^3-2x^2-ix^2)(x-2+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}9i^4-10i^2x^2+x^4+36i^2x-4x^3-36i^2+4x^2\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x-3i}\right) $ by each term in $ \left( x+3i\right) $. $$ \left( \color{blue}{x-3i}\right) \cdot \left( x+3i\right) = x^2+ \cancel{3ix} -\cancel{3ix}-9i^2 $$ |
| ② | Combine like terms: $$ x^2+ \, \color{blue}{ \cancel{3ix}} \, \, \color{blue}{ -\cancel{3ix}} \,-9i^2 = -9i^2+x^2 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{-9i^2+x^2}\right) $ by each term in $ \left( x-2-i\right) $. $$ \left( \color{blue}{-9i^2+x^2}\right) \cdot \left( x-2-i\right) = -9i^2x+18i^2+9i^3+x^3-2x^2-ix^2 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{-9i^2x+18i^2+9i^3+x^3-2x^2-ix^2}\right) $ by each term in $ \left( x-2+i\right) $. $$ \left( \color{blue}{-9i^2x+18i^2+9i^3+x^3-2x^2-ix^2}\right) \cdot \left( x-2+i\right) = \\ = -9i^2x^2+18i^2x -\cancel{9i^3x}+18i^2x-36i^2+ \cancel{18i^3}+ \cancel{9i^3x} -\cancel{18i^3}+9i^4+x^4-2x^3+ \cancel{ix^3}-2x^3+4x^2 -\cancel{2ix^2} -\cancel{ix^3}+ \cancel{2ix^2}-i^2x^2 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-9i^2x^2} + \color{red}{18i^2x} \, \color{green}{ -\cancel{9i^3x}} \,+ \color{red}{18i^2x} -36i^2+ \, \color{blue}{ \cancel{18i^3}} \,+ \, \color{green}{ \cancel{9i^3x}} \, \, \color{blue}{ -\cancel{18i^3}} \,+9i^4+x^4 \color{green}{-2x^3} + \, \color{orange}{ \cancel{ix^3}} \, \color{green}{-2x^3} +4x^2 \, \color{red}{ -\cancel{2ix^2}} \, \, \color{orange}{ -\cancel{ix^3}} \,+ \, \color{red}{ \cancel{2ix^2}} \, \color{blue}{-i^2x^2} = \\ = 9i^4 \color{blue}{-10i^2x^2} +x^4+ \color{red}{36i^2x} \color{green}{-4x^3} -36i^2+4x^2 $$ |
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