The common roots of the equations z³+2z²+2z+1=0 and z¹985}+z¹00}+1=0 are

Correct Answer is : ω,ω2ω, ω²ω,ω2

Correct Answer is : ω,ω2ω, ω²ω,ω2

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CGPET 2021 Solved Paper

Section: Mathematics

Question : 118 of 150

Marks: +1, -0

z^3+2z^2+2z+1=0

z^{1985}+z^{100}+1=0

Solution:

z^3+2z^2+2z+1=0

⇒

(z+1)(z^2+z+1)=0

z+1=0 \text{ or } z^2+z+1=0

when

z=-1, z^{1985}+z^{100}+1=(-1)^{1985}+(-1)^{100}+1

=-1+1+1=1\neq 0

∵

z=-1

do not satisfy the equation

z^{1985}+z^{100}+1=0

Then, it is not the common roots.

∴ The common roots are the roots of

z^2+z+1=0

We know that

\omega

and

\omega^2

are the roots of

z^2+z+1=0

(where

\omega

is cube root of unity)

When

z=\omega

(\omega)^{1985}+(\omega)^{100}+1=(\omega^3)^{661}\omega^2+(\omega^3)^{33}\omega+1

=\omega^2+\omega+1=0

(true)

When

z=\omega

(\omega^2)^{1985}+(\omega^2)^{100}+1=(\omega)^{3970}+\omega^{200}+1

=(\omega^2)^{1323}\omega+(\omega^3)^{66}\omega^2+1

=(1)^{1323}\omega+(1)^{66}\omega^2+1

=\omega+\omega^2+1=0

(true)

∴

\omega, \omega^2

are the common roots of

z^{1985}+z^{100}+1=0

and

z^3+2z^2+2z+1=0

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