If $7^{\frac13} + 7^{\frac23}$ is a root of equation of minimum possible degree with rational coefficients, then what is the product of roots of this equation?
How do I solve it?
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Shaurya Gupta
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Are you familiar with Galois theory? – Alex Becker Apr 04 '14 at 05:41
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@AlexBecker No, it is a Tenth standard question... – Shaurya Gupta Apr 04 '14 at 05:43
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Ah. Well I have no idea how to do it without some relatively heavy math then, except by explicitly computing powers. – Alex Becker Apr 04 '14 at 05:44
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3$$y=7^{\frac13} + 7^{\frac23}\implies y^3=7+7^2+3\cdot 7^{\frac13}\cdot 7^{\frac23}y$$ – lab bhattacharjee Apr 04 '14 at 05:44
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@labbhattacharjee That was about to be my answer :( – evil999man Apr 04 '14 at 05:46
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@labbhattacharjee $y^3 = 56 + 21y$, So the equation of minimum degree will be $y^3 + 0y^2 - 21y - 56 = 0$, which means that our answer is 56...Thanks – Shaurya Gupta Apr 04 '14 at 05:49
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@shauryagupta, have you meant sum or product ? – lab bhattacharjee Apr 04 '14 at 05:52
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@shauryagupta, http://en.wikipedia.org/wiki/Vieta's_formulas – lab bhattacharjee Apr 04 '14 at 05:57
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@shauryagupta sum would be zero. See vieta's formulas – Guy Apr 04 '14 at 05:57
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@labbhattacharjee I'm sorry...I meant product! :) – Shaurya Gupta Apr 04 '14 at 05:58