If $z = 2 + i$ and $z^{3} + 3 z^{2} - 9 z + 8 = a + i b$ , then the value of $a + b$ is

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Solution

Given : $z^{3} + 3 z^{2} - 9 z + 8 = a + i b$
Now,
$z = 2 + i \Rightarrow (z - 2)^{2} = - 1 \Rightarrow z^{2} - 4 z + 5 = 0 \dots (1)$

Using equation $(1)$ , we get
$z^{3} + 3 z^{2} - 9 z + 8 = z (z^{2} - 4 z + 5) + 7 z^{2} - 14 z + 8 = 0 + 7 (z^{2} - 4 z + 5) + 14 z - 27 = 14 z - 27 = 28 + 14 i - 27 = 1 + 14 i \Rightarrow a = 1, b = 14 ∴ a = 15$

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