A root of $4 x^{3} + 12 x^{2} + 9 x + 27 = 0$ is $(2 x - 3 i)$ . Which of the following is another root?

x + 3

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x - 3

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2 x - 2 i

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2 x + 2 i

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2 x - 3

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Solution

The correct option is A $x + 3$
Given that one root of the equation is $2 x - 3 i$
Since it is complex number , the conjugate is also a root of it
So $2 x - 3 i$ and $2 x + 3 i$
Let the another root be $a x + b$
So we have $(2 x + 3 i) (2 x - 3 i) (a x + b) = 4 x^{3} + 12 x^{2} + 9 x + 27$
$\Rightarrow 4 a x^{3} + 4 b x^{2} + 9 a x + 9 b = 4 x^{3} + 12 x^{2} + 9 x + 27$
By comparing we get $a = 1$ and $b = 3$
Therefore $x + 3$ is another root

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