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

Explanation for the correct option:

Finding the distinct root of

$$\begin{gathered} \Rightarrow 1+z+z^3+z^4=0 \\ \Rightarrow (1+z)+z^3(1+z)=0 \\ \Rightarrow (1+z)(1+z^3)=0 \\ \Rightarrow z=-1or{z}^3=-1 \\ \Rightarrow z=-1,-\omega ,-{\omega }^2 \end{gathered}$$

From the above values, we know that the roots are the cube root of unity.

So, these roots are the vertices of an equilateral triangle

Hence, option (B) is the correct answer.