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Question

Find the value of $x^{3} + 7 x^{2} - x + 16$ , when $x = 1 + 2 i$ .

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Solution

$x = 1 + 2 i$
$x - 1 = 2 i$
$(x - 1)^{2} = 4 i^{2}$
$x^{2} - 2 x + 1 = - 4$
$x^{2} - 2 x + 5 = 0$
Therefore, $x^{3} + 7 x^{2} - x + 16 = x (x^{2} - 2 x + 5) + 9 (x^{2} - 2 x + 5) + (12 x - 29)$
$= x (0) + 9 (0) + 12 x - 29$
$= 12 (1 + 2 i) - 29 = - 17 + 24 i$

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