If α,β,γ are the roots of x3−3x2+3x+7=0 (ω is the cube root of unity), then α−1β−1+β−1γ−1+γ−1α−1 is

x3−3x2+3x+7=0 (x−1)3=−8 (x−1)3−(−2)3=0 (∵a3−b3=(a−b)(a2+ab+b2)) (x+1)(x2−4x+7)=0 α=−1 , β=2+i3 , γ=2−i3 α−1=−2 , β−1=1+i3 , γ−1=1−i3 α−1=−2 , β−1=−2ω , γ−1=−2ω2 ⇒α−1β−1+β−1γ−1+γ−1α−1=(−2−2ω)+(−2ω−2ω2)+(−2ω2−2)=1ω+1ω+ω2 ⇒2ω+ω2=3ω (∵ω3=1)

If α,β,γ are the roots of x3−3x2+3x+7=0 (ω is the cube root of unity), then α−1β−1+β−1γ−1+γ−1α−1 is

x3−3x2+3x+7=0 (x−1)3=−8 (x−1)3−(−2)3=0 (∵a3−b3=(a−b)(a2+ab+b2)) (x+1)(x2−4x+7)=0 α=−1 , β=2+i3 , γ=2−i3 α−1=−2 , β−1=1+i3 , γ−1=1−i3 α−1=−2 , β−1=−2ω , γ−1=−2ω2 ⇒α−1β−1+β−1γ−1+γ−1α−1=(−2−2ω)+(−2ω−2ω2)+(−2ω2−2)=1ω+1ω+ω2 ⇒2ω+ω2=3ω (∵ω3=1)

AI Mentor

Check Your IQ

Free Expert Demo

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

If $α, β, γ$ are the roots of $x^{3} - 3 x^{2} + 3 x + 7 = 0$ ( $ω$ is the cube root of unity), then $\frac{α - 1}{β - 1} + \frac{β - 1}{γ - 1} + \frac{γ - 1}{α - 1}$ is

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

ω^{2}

2 ω^{2}

\frac{3}{ω}

3 ω

answer is A.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

x^{3} - 3 x^{2} + 3 x + 7 = 0

${(x - 1)}^{3} = - 8$

${(x - 1)}^{3} - {(- 2)}^{3} = 0$ $(∵ a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2}))$

$(x + 1) (x^{2} - 4 x + 7) = 0$

$α = - 1, β = 2 + i \sqrt{3}, γ = 2 - i \sqrt{3}$

$α - 1 = - 2, β - 1 = 1 + i \sqrt{3}, γ - 1 = 1 - i \sqrt{3}$

$α - 1 = - 2, β - 1 = - 2 ω, γ - 1 = - 2 ω^{2}$

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