If 1- - - 2 are the three cube roots of unity- then for - - - - - R- the expression - - - 2- 2- - 2- - is-

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Relations between Roots and Coefficients : Higher Order Equations

If 1, ω ,ω 2...

Question

If 1, $ω, ω^{2}$ are the three cube roots of unity, then for $α, β, γ, δ, ϵ R,$ , the expression $(\frac{α + β ω + γ ω^{2} + δ ω^{2}}{β + α ω^{2} + γ ω + δ ω})$ is:

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ω

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- ω

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ω^{- 1}

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Solution

The correct option is B $ω$
Let's first take the denominator,
Denominator $= β + α w^{2} + γ w + δ w$
$= \frac{1}{w} (β w + α w^{3} + γ w^{2} + δ w^{2})$ [ Multiplying and dividing by $w$ ]

$= \frac{1}{w} (β w + α + γ w^{2} + δ w^{2})$ [ Since, $w^{3} = 1$ ]

$= \frac{1}{w} (α + β w + γ w^{2} + δ w^{2})$ ----- ( 1 )

Now,
$(\frac{α + β w + γ w^{2} + δ w^{2}}{β + α w^{2} + γ w + δ w})$ $= \frac{α + β w + γ w^{2} + δ w^{2}}{\frac{1}{w} (α + β w + γ w^{2} + δ w^{2})}$ [ From ( 1 ) ]

$= w$

$∴$ $(\frac{α + β w + γ w^{2} + δ w^{2}}{β + α w^{2} + γ w + δ w}) = w$

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If 1,ω,ω2 are the three cube roots of unity, then for α,β,γ,δ,ϵR,, the expression (α+βω+γω2+δω2β+αω2+γω+δω) is:

If 1, $ω, ω^{2}$ are the three cube roots of unity, then for $α, β, γ, δ, ϵ R,$ , the expression $(\frac{α + β ω + γ ω^{2} + δ ω^{2}}{β + α ω^{2} + γ ω + δ ω})$ is: