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Grade: 12

                        

If 1, w, w2 are cube roots of unity, prove the following:(1 - w + w^2) (1+ w - w^2) = 4

If 1, w, w2 are cube roots of unity, prove the following:(1 - w + w^2) (1+ w - w^2) = 4
 

2 months ago

Answers : (3)

Anand Kumar Pandey
askIITians Faculty
4371 Points
							Dear Student

1+ w + w^2 = 0
so 1 + w = – w^2
1+ w^2 = – w
(1 - w + w^2)= –2w
(1+ w - w^2)= –2w^2
(1 - w + w^2) (1+ w - w^2)= (-2w)( –2w^2)
= 4w^3
and we know that w^3=1
Hence, proved

Thanks
2 months ago
Anshu Kumar
14 Points
							
As we know from cube root of unity is
the sum of cube root of unity is zero and product of cube root of unity is one, so
1+w+w^2 = 0 & w^3 = 1  which implies that
1+w = -w^2  …....(i) and
1+w^2 = -w ….....(ii)
Now form Left Hand side of given equation we have
(1-w+w^2)(1+w-w^2) we can write it as follows
(1+w^2-w)(1+w-w^2)
(-w-w)(-w^2-w^2) …........ from (i) & (ii)
(-2w)(-2w^2) = 4w^3 =4*1 = 4 .
Hence we get Right Hand Side of given Equation.
 
25 days ago
Ram Kushwah
106 Points
							
We know that 
If 1,ω,ω² are roots of unity then 
ω³=1..................................(1)
1+ω+ω²=0........................(2)
Now 
(1-ω+ω²)( 1+ω-ω²)
= ( 1+ω+ω²-2ω)(1+ω+ω²-2ω²)
 
putting 1+ω+ω²=0 we get
=(0-2ω)(0-2ω²)
=(-2 x -2) ω³
=4ω³
=4  (  As ω ³=1)
Hence proved 
Please aprove
Thanks
 
 
14 days ago
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