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$\textrm{Show that for any real numbers }$ $a_{3 } ,a_{4},........,a_{85 }, $ $\textrm{the roots of the equation }$ $a_{85}x^{85 }+ a_{84}x^{84}+ ..........+a_{3}+ 3x^2+ 2x+ 1=0$ $ \textrm{are not real}$

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2 Answers

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kaymant ·2010-06-25 05:37:17

That's rather doubtful,
If a₈₅ is not zero, then being an odd degree polynomial, the LHS will have at least one real root.
On the other hand for a₃ = 2, the equation is 2x³ + 3x² + 2x+1=0, which has x=-1 as a real root.

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Hari Shankar ·2010-06-25 06:25:08

In all probability he means that we have to prove that all roots cannot be real.

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