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Sum of n Terms

If z = 2 + ...

Question

If $z = 2 + 3 i$ , then the expression $z^{4} - z^{3} + 10 z^{2} + 3 z - 5$ reduces to a real number equal to

122

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- 122

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160

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- 160

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Solution

The correct option is B $- 122$
We know that $z = 2 + 3 i$
$∴ z^{4} - z^{3} + 10 z^{2} + 3 z - 5$
$\Rightarrow z^{2} = {(2 + 3 i)}^{2} = 4 + 12 i - 9 = 12 i - 5$
$z^{3} = {(2 + 3 i)}^{3} = 8 - 27 i + 36 i - 54 = 9 i - 46$
$z^{4} = {(2 + 3 i)}^{4} = - 119 - 120 i$
$∴ z^{4} - z^{3} + 10 z^{2} + 3 z - 5$
$= - 119 - 120 i - 9 i + 46 + 120 i - 50 - 5 + 6 + 9 i$
$= - 122.$
Hence, the answer is $- 122.$

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