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Polynomial Problem (Tomato subjective 72)

Problem: If \({\displaystyle{\alpha}, {\beta}, {\gamma}} \) are the roots of the equation \({\displaystyle{x^3 + px^2 + qx + r = 0}} \), find the equation whose roots are \({\displaystyle{\alpha} – {\frac{1}{{\beta}{\gamma}}}} \), \({\displaystyle{\beta} – {\frac{1}{{\alpha}{\gamma}}}} \), \({\displaystyle{\gamma} – {\frac{1}{{\alpha}{beta}}}} \) .

Solution: \({\displaystyle{\alpha}, {\beta}, {\gamma}} \) are roots of \({\displaystyle{x^3 + px^2 + qx + r = 0}} \)
\({\Rightarrow} \) \({\displaystyle{\alpha}, {\beta}, {\gamma}} \) = \({- r} \),
\({\alpha + \beta + \gamma} \) = \({- p} \)
\({{\alpha}{\beta} + {\gamma}{\alpha} + {\beta}{\gamma}} \) = \({q} \)
Now,
\({\displaystyle{\left({\alpha} – {\frac{1}{{\beta}{\gamma}}}\right)}} \) + \({\displaystyle{\left({\beta} – {\frac{1}{{\alpha}{\gamma}}}\right)}} \) + \({\displaystyle{\left({\gamma} – {\frac{1}{{\alpha}{\beta}}}\right)}} \)

= \({\displaystyle{\frac{{\alpha}{\beta}{\gamma} – 1}{{\alpha}{\beta}{\gamma}}}} \) \({\displaystyle{\left({\alpha} + {\beta} + {\gamma}\right)}} \)

= – \({\displaystyle{\frac{p(r+1)}{r}}} \)

 

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August 22, 2015

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