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

Answer 1

The correct option is B An equilateral triangle

Given equation is
$1+z+z^3+z^4=0$
$\Rightarrow (1+z)+z^3(1+z)=0$
$\Rightarrow (1+z)(1+z^3)=1$
$\Rightarrow z=-1$

$\Rightarrow z^3=-1$
$\Rightarrow z=-1,z=-1,-\omega ,-{\omega }^2$
Hence, roots are in cubic roots of unity.
Hence, these roots are the vertices of an equilateral triangle.