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Question

If the four roots of the equation z4+z3+2z2+z+1=0 form a quadrilateral on the Argand plane, then the area of the quadrilateral is

A
3+28
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B
3+24
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C
3+22
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D
3+23
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Solution

The correct option is B 3+24
z4+z3+2z2+z+1=0
On dividing the equation by z2,
z2+1z2+z+1z+2=0
(z+1z)2+(z+1z)=0
z+1z=0 or z+1z+1=0
z=±i or z=ω,ω2
where ω is cube root of unity.

Now, plotting solution on Argand plane,


The quadrilateral is a trapezium.
Thus, required area =12×(2+3)×12=3+24

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