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A tough question- Brain Teaser

Posted May 31st, 2010 by alpha
in

The eqn. $ x^2-x+1=0 $ has root $ \alpha and \beta $ . Find $ \alpha^{2009 }+\beta^{2009 } $

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answer

On June 1st, 2010 Structure says:

$ x^3 + 1 = (x + 1)(x^2 - x + 1) $

So $ \alpha $ and $ \beta $ are roots of $ x^3 + 1 = 0 $

$ \alpha^{2009 }+ \beta^{2009 }= \alpha^2\alpha^{2007 }+ \beta^2\beta^{2007} = $
$ = \alpha^2\alpha^{3 \times 669}+ \beta^2\beta^{3 \times 669}=-\alpha^2-\beta^2=-\alpha+1 -\beta+1=1 $

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