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if 1,a1,a2......an-1are n roots of unity , then the value of (1-a1)(1-a2).....(1-an-1) is?

if 1,a1,a2......an-1are n roots of unity , then the value of (1-a1)(1-a2).....(1-an-1) is?

Grade:12th pass

3 Answers

Nishant Vora IIT Patna
askIITians Faculty 2467 Points
6 years ago
Basically 1,a1,a2......an-1 all these are the roots of this equation x3 – 1 =0

So I can write

x^3 -1= (x-1)(x-a_1)(x-a_2).....(x-a_{n-1})
\frac{x^3 -1}{(x-1)}= (x-a_1)(x-a_2).....(x-a_{n-1})
x^2 + x +1= (x-a_1)(x-a_2).....(x-a_{n-1})
Put x=1 on both the sides now
Ans =3


Jayakeerti
17 Points
3 years ago
x^n - 1/(x-1) = (x-@1)(x-@2)......(x-@n-1)
 
(x^(n-1) +x^(n-2)1 + ...... + 1^(n-1)) 
Put x = 1
Thus, answer is n
Kushagra Madhukar
askIITians Faculty 629 Points
one year ago
Dear student,
Please find the solution to your problem below.
 
x3 – 1 = (x – 1)(x – a1)(x – a2)......(x – an-1)
Hence, (x3 – 1) / (x – 1) = (x – a1)(x – a2)......(x – an-1)
or, (x2 + x + 1) = (x – a1)(x – a2)......(x – an-1)
Put x = 1
Hence, (1 – a1)(1 – a2)........(1 – an-1)  = (12 + 1 + 1) = 3
Hence the answer is 3
 
Hope it helps.
Thanks and regards,
Kushagra

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