If 1, α1, α2, α3, α4 be the roots of x5-1=0, then the value of (ω-α1/ω2-α1) (ω-α2/ω2-α2) (ω-α3/ω2-α3) (ω-α4/ω2-α4) ldots is . (where ω is imaginary cube root of unity.)

Question

If 1, α1, α2, α3, α4 be the roots of x5-1=0, then the value of (ω-α1/ω2-α1) (ω-α2/ω2-α2) (ω-α3/ω2-α3) (ω-α4/ω2-α4) ldots is . (where ω is imaginary cube root of unity.) (A)  ω (B) ω2 (C) 2 ω (D) 2 ω2

Tardigrade Admin · Accepted Answer

(z-1)(z-α1) ldots ldots . .(z-α4)=z5-1 Put z=ω, z=ω2 and divide ((ω-1)(ω-α1)(ω-α2)(ω-α3)(ω-α4)/(ω2-1)(ω2-α1)(ω2-α2)(ω2-α3)(ω2-α4))=(ω5-1/ω10-1) ((ω-α1)(ω-α2)(ω-α3)(ω-α4)/(ω2-α1)(ω2-α2)(ω2-α3)(ω2-α4)) =((ω2-1)2/(ω-1)2)=(ω+1)2=ω4=ω

$ω$

$ω^{2}$

$2 ω$

$2 ω^{2}$

Q. If 1,α1​,α2​,α3​,α4​ be the roots of x5−1=0, then the value of ω2−α1​ω−α1​​ω2−α2​ω−α2​​ω2−α3​ω−α3​​ω2−α4​ω−α4​​… is . (where ω is imaginary cube root of unity.)

ω

ω2

2ω

2ω2

$ω$

$ω^{2}$

$2 ω$

$2 ω^{2}$