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Question
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Solution
\[ x = \frac{3 - 5i}{2}\]
\[ \Rightarrow x^2 = \left( \frac{3 - 5i}{2} \right)^2 \]
\[ = \frac{9 + 25 i^2 - 30i}{4}\]
\[ = \frac{- 16 - 30i}{4}\]
\[ \Rightarrow x^3 = \frac{- 16 - 30i}{4} \times \frac{3 - 5i}{2}\]
\[ = \frac{- 48 + 80i - 90i + 150 i^2}{8}\]
\[ = \frac{- 198 - 10i}{8}\]
\[ \therefore 2 x^3 + 2 x^2 - 7x + 72 = 2\left( \frac{- 198 - 10i}{8} \right) + 2\left( \frac{- 16 - 30i}{4} \right) - 7\left( \frac{3 - 5i}{2} \right) + 72\]
\[ = \frac{- 198 - 10i - 32 - 60i - 42 + 70i + 288}{4}\]
\[ = \frac{16}{4}\]
\[ = 4\]
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