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In the argand plane, the distinct roots of $1+z+z^{3}+z^{4}=0(z$ is a complex number) represent vertices of
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Verified Answer
The correct answer is:
an equilateral triangle
Given equation is
$1+z+z^{3}+z^{4}=0$
$\Rightarrow \quad(1+z)+z^{3}(1+z)=0$
$\Rightarrow \quad(1+z)\left(1+z^{3}\right)=1$
$\Rightarrow \quad z=-1, z^{3}=-1$
$\Rightarrow \quad z=-1, z=-1,-\omega-\omega^{2}$
Hence, these roots are the vertices of an equilateral triangle.
$1+z+z^{3}+z^{4}=0$
$\Rightarrow \quad(1+z)+z^{3}(1+z)=0$
$\Rightarrow \quad(1+z)\left(1+z^{3}\right)=1$
$\Rightarrow \quad z=-1, z^{3}=-1$
$\Rightarrow \quad z=-1, z=-1,-\omega-\omega^{2}$
Hence, these roots are the vertices of an equilateral triangle.
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