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Properties of nth Root of a Complex Number

If z1 and ...

Question

If $z_{1}$ and $z_{2}$ are the complex roots of the equation $(x - 3)^{3} + 1 = 0$ , then $z_{1} + z_{2}$ equals to

1

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3

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5

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7

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Solution

The correct option is D $7$
$(x - 3)^{3} + 1 = 0$
$\Rightarrow {(\frac{x - 3}{- 1})}^{3} = 1$
$\Rightarrow \frac{x - 3}{- 1} = 1, ω, ω^{2}$
$\Rightarrow x = 2, 3 - ω, 3 ω^{2}$
Hence, the sum of complex roots is $6 - (ω + ω)^{2} = 6 + 1 = 7$ .

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