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OSP SMA 2005 Bagian Kedua No 3

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Jika $\alpha, \beta, \gamma$ adalah akar-akar dari persamaan $x^3-x-1=0$, tentukan nilai dari$$\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}.$$

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$\alpha, \beta, \gamma$ adalah akar-akar dari $x^3-x-1=0$. Maka $\alpha+\beta+\gamma=0$, $\alpha\beta+\beta\gamma+\gamma\alpha=-1$, dan $\alpha\beta\gamma=1$. \begin{align*}\sum_{\text{cyc}}\frac{1+\alpha}{1-\alpha}&=\frac{\displaystyle\sum_{\text{cyc}} \left\{(1+\alpha)(1-\beta)(1-\gamma)\right\}}{(1-\alpha)(1-\beta)(1-\gamma)}\\&=\frac{3-(\alpha+\beta+\gamma)-(\alpha\beta+\beta\gamma+\gamma\alpha)}{1-(\alpha+\beta+\gamma)+(\alpha\beta+\beta\gamma+\gamma\alpha)-\alpha\beta\gamma}\\&=\frac{3-0-(-1)}{1-0+(-1)-1}\\&=\boxed{-4}.\end{align*}

 

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