Monic polynomials whose roots are their remaining coefficients I was bored in class one day and wondered to myself if there were any quadratics $x^2+ax+b$ such that $a$ and $b$ are the zeros. I found two:  $x^2+x-2,$ and $x^2 -{1\over2}x -{1\over2}$. The comments suggested $x^2+0x+0$, though this seems trivial.  I wonder if this applies to other degree polynomials. Clearly it never works for a linear, except for $x+0=0$, as if $x+a=0$, $x=-a$, not $a$. What about cubics, quadratics, or even higher powers? In general, $x^n$ works. What about non-trivial solutions?

Cubics:
\begin{align} 
x^3+x^2-2\,x=0\quad &\Rightarrow 1,-2,\phantom{-}0
,\\
x^3+x^2-x-1=0\quad &\Rightarrow 1,-1,-1
.
\end{align}  
Similarly,
\begin{align} 
x^4+x^3-2x^2=0\quad &\Rightarrow 1,-2,0,0
,\\
x^4+x^3-x^2-x=0\quad &\Rightarrow 1,-1,-1,0
.
\end{align}  
