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Question
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
Solution
\[x = - 2 - \sqrt{3}i\]
\[ \Rightarrow x^2 = \left( - 2 - \sqrt{3}i \right)^2 \]
\[ = ( - 2 )^2 + ( - \sqrt{3}i )^2 + 2( - 2)( - \sqrt{3}i)\]
\[ = 4 + 3 i^2 + 4\sqrt{3}i\]
\[ = 4 - 3 + 4\sqrt{3}i [ \because i^2 = - 1]\]
\[ = 1 + 4\sqrt{3}i\]
\[ \Rightarrow x^3 = \left( 1 + 4\sqrt{3}i \right) \times \left( - 2 - \sqrt{3}i \right)\]
\[ = - 2 - \sqrt{3}i - 8\sqrt{3}i - 12 i^2 \]
\[ = 10 - 9\sqrt{3}i [ \because i^2 = - 1]\]
\[ \Rightarrow x^4 = \left( 1 + 4\sqrt{3}i \right)^2 \]
\[ = 1 + 48 i^2 + 8\sqrt{3}i\]
\[ = - 47 + 8\sqrt{3}i [ \because i^2 = - 1]\]
\[ \Rightarrow 2 x^4 + 5 x^3 + 7 x^2 - x + 41 = 2( - 47 + 8\sqrt{3}i ) + 5\left( 10 - 9\sqrt{3}i \right) + 7\left( 1 + 4\sqrt{3}i \right) - \left( - 2 - \sqrt{3}i \right) + 41\]
\[ = - 94 + 16\sqrt{3}i + 50 - 45\sqrt{3}i + 7 + 28\sqrt{3}i + 2 + \sqrt{3}i + 41\]
\[ = 6\]
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