Find all polynomial $P(x)$ with coefficient $\pm1$ and have all real roots.

My attempted work :

Let $P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$

$-P(x) = -a_nx^n - a_{n-1}x^{n-1} - ... - a_1x - a_0$

$P(x)$ have all real roots $\Leftrightarrow -P(x)$ have all real roots.

WLOG, $a_n = 1$

$P(x) = x^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$

Let $r_1, r_2, ..., r_n$ be roots of the equation.

$P(x) = (x-r_1)(x-r_2)...(x-r_n)$

Please suggest how to proceed.

  • 1
    $\begingroup$ This is far too general a question, but maybe Descartes' rule of signs can be of some help here. The Descartes' rule is an upper bound on the positive or negative real roots of a polynomial. $\endgroup$ – астон вілла олоф мэллбэрг Mar 18 '17 at 5:42
  • $\begingroup$ I still cannot figure out how to apply Descartes' rule in this problem. Can someone solve it ? $\endgroup$ – carat Mar 19 '17 at 14:49

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