Proved that with all n is positive integer, let equation:

$$\begin{pmatrix} n\\0\end{pmatrix} ^2x^n+\begin{pmatrix} n\\1\end{pmatrix} ^2x^{n-1}+\cdots+\begin{pmatrix} n\\n\end{pmatrix} ^2=0$$

Have n real-root diacritical and which are all negative real root!


I don not kow how! Please help!

  • 1
    $\begingroup$ The coefficients are all positive? $\endgroup$ – user99914 Nov 30 '13 at 5:03
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    $\begingroup$ Your constant term is 1 and the product of the roots must thus be 1. If they are all negative that won't work for n odd. Also I don't know what you mean by "diacritical". $\endgroup$ – Betty Mock Nov 30 '13 at 5:51
  • $\begingroup$ @BettyMock You got it wrong, a negative root $-a$ yields a factor $x+a$, in the product everything will be positive and allright. $\endgroup$ – Ewan Delanoy Nov 30 '13 at 6:35
  • $\begingroup$ @Ewan, so true. Thanks. $\endgroup$ – Betty Mock Dec 1 '13 at 0:14

This is a special case of the result shown here.

I don’t know if there is a simpler proof.


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