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Question
If 1,α1,α2,α3,α4 be the roots of z5−1=0 and ω be an imaginary cube root of unity,
then (ω−α1ω2−α1)(ω−α2ω2−α2)(ω−α3ω2−α3)(ω−α4ω2−α4) is ?
If 1,α1,α2,α3,α4 be the roots of z5−1=0 and ω be an imaginary cube root of unity,
then (ω−α1ω2−α1)(ω−α2ω2−α2)(ω−α3ω2−α3)(ω−α4ω2−α4) is ?
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Solution
The correct option is B ω
We have,
z5−1=(z−1)(z−α1)(z−α2)(z−α3)(z−α4)
putting z=w,
⇒w5−1=(w−1)(w−α1)(w−α2)(w−α3)(w−α4)..............(1)
Similarly putting z=w2
(w2)5−1=w10−1=(w2−1)(w2−α1)(w2−α2)(w2−α3)(w2−α4).........(2)
So the required expression reduces to
=(w5−1)/(w−1)(w10−1)/(w2−1)
=(w2−1)/(w−1)(w−1)/(w2−1)
[∵w3=1,w5=w3w2=w2]
=(w2−1)2(w−1)2
=(w−1)2(w+1)2(w−1)2
=1+2w+w2
=w[∵1+w2=−w]
We have,
z5−1=(z−1)(z−α1)(z−α2)(z−α3)(z−α4)
putting z=w,
⇒w5−1=(w−1)(w−α1)(w−α2)(w−α3)(w−α4)..............(1)
Similarly putting z=w2
(w2)5−1=w10−1=(w2−1)(w2−α1)(w2−α2)(w2−α3)(w2−α4).........(2)
So the required expression reduces to
=(w5−1)/(w−1)(w10−1)/(w2−1)
=(w2−1)/(w−1)(w−1)/(w2−1)
[∵w3=1,w5=w3w2=w2]
=(w2−1)2(w−1)2
=(w−1)2(w+1)2(w−1)2
=1+2w+w2
=w[∵1+w2=−w]
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