Question

# In the argand plane, the distinct roots of $$1+z+{ z }^{ 3 }+{ z }^{ 4 }=0$$ ($$z$$ is a complex number) represent vertices of

A
A square
B
An equilateral triangle
C
A rhombus
D
A rectangle

Solution

## The correct option is B An equilateral triangleGiven equation is$$1+z+{ z }^{ 3 }+{ z }^{ 4 }=0$$$$\Rightarrow \left( 1+z \right) +{ z }^{ 3 }\left( 1+z \right) =0$$$$\Rightarrow \left( 1+z \right) \left( 1+{ z }^{ 3 } \right) =1$$$$\Rightarrow z=-1$$$$\Rightarrow { z }^{ 3 }=-1$$$$\Rightarrow z=-1,z=-1,-\omega ,-{ \omega }^{ 2 }$$ Hence, roots are in cubic roots of unity.Hence, these roots are the vertices of an equilateral triangle.Maths

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