Question

if alpha and beta are complex cube root of unity prove that (1-α)×(1-beta)×(1-alpha)2(1-beta)2=9

Answer 1

let alpha = w
beta = w^2

Answer 2

/* There is a mistake in the question .It should be like this this */

$If \:\alpha \: and \:\beta \: are\: complex \\cube \: root \: of \:unity$

$then \: \alpha = \omega \: and \: \beta = \omega^{2}$

$\underline { \red { To \:prove :} }$

$(1-\alpha)(1-\beta)(1-\alpha)^{2}(1-\beta)^{2} = 27$

$\underline { \pink { Proof :}}$

$LHS = (1-\alpha)(1-\beta)(1-\alpha)^{2}(1-\beta)^{2}\\= (1-\alpha)^{3}(1-\beta)^{3}\\= [(1-\alpha)(1-\beta)]^{3} \\= ( 1 - \beta - \alpha - \alpha \beta )^{3} \\= (1 - \omega^{2} - \omega - \omega \times \omega^{2})^{3} \\= [ 1 - (\omega + \omega^{2} ) + \omega^{3} ]^{3} \\= [1-(-1) + 1 ]^{3}$

$\boxed { \pink { i) 1+\omega + \omega^{2} = 0 \:and \: ii )\omega^{3} = 1 }}$

$= ( 1 + 1 + 1 )^{3} \\= 3^{3} \\= 27 \\= RHS$

$Hence \: proved$

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