Question
Answer
$$(x-i)(x^2-2i(i-2)x+1+i)=-4i^2x+3ix^2+x^3-i^2-ix+2x^2-i+x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-i)(x^2-2i(i-2)x+1+i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x-i)(x^2-(2i^2-4i)x+1+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x-i)(x^2-(-2-4i)x+1+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x-i)(x^2-(-2x-4ix)+1+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(x-i)(x^2+2x+4ix+1+i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(x-i)(4ix+x^2+i+2x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}-4i^2x+3ix^2+x^3-i^2-ix+2x^2-i+x\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{2i} $ by $ \left( i-2\right) $ $$ \color{blue}{2i} \cdot \left( i-2\right) = 2i^2-4i $$ |
| ② | $$ 2i^2 = 2 \cdot (-1) = -2 $$ |
| ③ | $$ \left( \color{blue}{-2-4i}\right) \cdot x = -2x-4ix $$ |
| ④ | Remove the parentheses by changing the sign of each term within them. $$ - \left( -2x-4ix \right) = 2x+4ix $$ |
| ⑤ | Combine like terms: $$ x^2+2x+4ix+1+i = 4ix+x^2+i+2x+1 $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{x-i}\right) $ by each term in $ \left( 4ix+x^2+i+2x+1\right) $. $$ \left( \color{blue}{x-i}\right) \cdot \left( 4ix+x^2+i+2x+1\right) = 4ix^2+x^3+ix+2x^2+x-4i^2x-ix^2-i^2-2ix-i $$ |
| ⑦ | Combine like terms: $$ \color{blue}{4ix^2} +x^3+ \color{red}{ix} +2x^2+x-4i^2x \color{blue}{-ix^2} -i^2 \color{red}{-2ix} -i = -4i^2x+ \color{blue}{3ix^2} +x^3-i^2 \color{red}{-ix} +2x^2-i+x $$ |
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