parmenides51 wrote: Find all polynomials $P(x)$ which have the properties: 1) $P(x)$ is not a constant polynomial and is a mononic polynomial. 2) $P(x)$ has all real roots and no duplicate roots. 3) If $P(a)=0$ then $P(a|a|)=0$ (nooonui) Obviously, since finitelay many roots, we need $|a|\in\{0,1\}$ and so $x$ $x-1$ $x+1$ $x^2-x$ $x^2+x$ $x^2-1$ $x^3-x$