Question
Answer
$$(x-2+i)(x-4)(x-2-i)=-i^2x+x^3+4i^2-8x^2+20x-16$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-2+i)(x-4)(x-2-i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-4x-2x+8+ix-4i)(x-2-i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1ix+x^2-4i-6x+8)(x-2-i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-i^2x+x^3+4i^2-8x^2+20x-16\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x-2+i}\right) $ by each term in $ \left( x-4\right) $. $$ \left( \color{blue}{x-2+i}\right) \cdot \left( x-4\right) = x^2-4x-2x+8+ix-4i $$ |
| ② | Combine like terms: $$ x^2 \color{blue}{-4x} \color{blue}{-2x} +8+ix-4i = ix+x^2-4i \color{blue}{-6x} +8 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{ix+x^2-4i-6x+8}\right) $ by each term in $ \left( x-2-i\right) $. $$ \left( \color{blue}{ix+x^2-4i-6x+8}\right) \cdot \left( x-2-i\right) = \\ = \cancel{ix^2}-2ix-i^2x+x^3-2x^2 -\cancel{ix^2}-4ix+ \cancel{8i}+4i^2-6x^2+12x+6ix+8x-16 -\cancel{8i} $$ |
| ④ | Combine like terms: $$ \, \color{blue}{ \cancel{ix^2}} \, \color{green}{-2ix} -i^2x+x^3 \color{orange}{-2x^2} \, \color{blue}{ -\cancel{ix^2}} \, \color{blue}{-4ix} + \, \color{red}{ \cancel{8i}} \,+4i^2 \color{orange}{-6x^2} + \color{orange}{12x} + \color{blue}{6ix} + \color{orange}{8x} -16 \, \color{red}{ -\cancel{8i}} \, = -i^2x+x^3+4i^2 \color{orange}{-8x^2} + \color{orange}{20x} -16 $$ |
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