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Question
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
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Solution
\[x = 3 + 2i\]
\[ \Rightarrow x^2 = \left( 3 + 2i \right)^2 \]
\[ = 9 + 4 i^2 + 12i\]
\[ = 5 + 12i\]
\[ \Rightarrow x^3 = x^2 \times x\]
\[ = \left( 5 + 12i \right) \times \left( 3 + 2i \right)\]
\[ = 15 + 10i + 36i - 24\]
\[ = - 9 + 46i\]
\[ \Rightarrow x^4 = \left( x^2 \right)^2 \]
\[ = \left( 5 + 12i \right)^2 \]
\[ = 25 + 144 i^2 + 120i\]
\[ = - 119 + 120i\]
\[ \Rightarrow x^4 - 4 x^3 + 4 x^2 + 8x + 44 = - 119 + 120i - 4\left( - 9 + 46i \right) + 4\left( 5 + 12i \right) + 8\left( 3 + 2i \right) + 44\]
\[ = - 119 + 120i + 36 - 184i + 20 + 48i + 24 + 16i + 44\]
\[ = 5\]
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