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

Question

Harshit Singh · Answer

Dear Student 1+ w + w^2 = 0 so 1 + w = &ndash; w^2 1+ w^2 = &ndash; w (1 - w + w^2)= &ndash;2w (1+ w - w^2)= &ndash;2w^2 (1 - w + w^2) (1+ w - w^2)= (-2w)( &ndash;2w^2) = 4w^3 and we know that w^3=1 Hence, proved Thanks

Anshu Kumar · Answer

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.

Ram Kushwah · Answer

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

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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

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