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Question
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
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Solution
\[ x = - 1 + \sqrt{2}i\]
\[ \Rightarrow x^2 = \left( - 1 + \sqrt{2}i \right)^2 \]
\[ = 1 + 2 i^2 - 2\sqrt{2}i\]
\[ = - 1 - 2\sqrt{2}i\]
\[ \Rightarrow x^3 = \left( - 1 - 2\sqrt{2}i \right) \times \left( - 1 + \sqrt{2}i \right)\]
\[ = 1 - \sqrt{2}i + 2\sqrt{2}i - 4 i^2 \]
\[ = 5 + \sqrt{2}i\]
\[ \Rightarrow x^4 = \left( - 1 - 2\sqrt{2}i \right)^2 \]
\[ = 1 + 8 i^2 + 4\sqrt{2}i\]
\[ = - 7 + 4\sqrt{2}i\]
\[ \Rightarrow x^4 + 4 x^3 + 6 x^2 + 4x + 9 = - 7 + 4\sqrt{2}i + 4\left( 5 + \sqrt{2}i \right) + 6\left( - 1 - 2\sqrt{2}i \right) + 4\left( - 1 + \sqrt{2}i \right) + 9\]
\[ = - 7 + 4\sqrt{2}i + 20 + 4\sqrt{2}i - 6 - 12\sqrt{2}i - 4 + 4\sqrt{2}i + 9\]
\[ = 12\]
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