Question
Answer
$$(r-(3+i))^2(r-(3-i))^2=i^4-2i^2r^2+r^4+12i^2r-12r^3-18i^2+54r^2-108r+81$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(r-(3+i))^2(r-(3-i))^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1r-3-i)^2(1r-3+i)^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(1i^2-2ir+r^2+6i-6r+9)(1i^2+2ir+r^2-6i-6r+9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}i^4-2i^2r^2+r^4+12i^2r-12r^3-18i^2+54r^2-108r+81\end{aligned} $$ | |
| ① | Remove the parentheses by changing the sign of each term within them. $$ - \left( 3+i \right) = -3-i $$Remove the parentheses by changing the sign of each term within them. $$ - \left( 3-i \right) = -3+i $$ |
| ② | Multiply each term of $ \left( \color{blue}{r-3-i}\right) $ by each term in $ \left( r-3-i\right) $. $$ \left( \color{blue}{r-3-i}\right) \cdot \left( r-3-i\right) = r^2-3r-ir-3r+9+3i-ir+3i+i^2 $$ |
| ③ | Combine like terms: $$ r^2 \color{blue}{-3r} \color{red}{-ir} \color{blue}{-3r} +9+ \color{green}{3i} \color{red}{-ir} + \color{green}{3i} +i^2 = i^2 \color{red}{-2ir} +r^2+ \color{green}{6i} \color{blue}{-6r} +9 $$Multiply each term of $ \left( \color{blue}{r-3+i}\right) $ by each term in $ \left( r-3+i\right) $. $$ \left( \color{blue}{r-3+i}\right) \cdot \left( r-3+i\right) = r^2-3r+ir-3r+9-3i+ir-3i+i^2 $$ |
| ④ | Combine like terms: $$ r^2 \color{blue}{-3r} + \color{red}{ir} \color{blue}{-3r} +9 \color{green}{-3i} + \color{red}{ir} \color{green}{-3i} +i^2 = i^2+ \color{red}{2ir} +r^2 \color{green}{-6i} \color{blue}{-6r} +9 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{i^2-2ir+r^2+6i-6r+9}\right) $ by each term in $ \left( i^2+2ir+r^2-6i-6r+9\right) $. $$ \left( \color{blue}{i^2-2ir+r^2+6i-6r+9}\right) \cdot \left( i^2+2ir+r^2-6i-6r+9\right) = \\ = i^4+ \cancel{2i^3r}+i^2r^2 -\cancel{6i^3}-6i^2r+9i^2 -\cancel{2i^3r}-4i^2r^2 -\cancel{2ir^3}+12i^2r+ \cancel{12ir^2} -\cancel{18ir}+i^2r^2+ \cancel{2ir^3}+r^4 -\cancel{6ir^2}-6r^3+9r^2+ \cancel{6i^3}+12i^2r+ \cancel{6ir^2}-36i^2 -\cancel{36ir}+ \cancel{54i}-6i^2r -\cancel{12ir^2}-6r^3+ \cancel{36ir}+36r^2-54r+9i^2+ \cancel{18ir}+9r^2 -\cancel{54i}-54r+81 $$ |
| ⑥ | Combine like terms: $$ \text{ \text{Text Is Too Long To Display} } = \\ = i^4 \color{orange}{-2i^2r^2} +r^4+ \color{orange}{12i^2r} \color{red}{-12r^3} \color{red}{-18i^2} + \color{orange}{54r^2} \color{blue}{-108r} +81 $$ |
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