Monic polynomials whose roots are their remaining coefficients

For the quadratic case $x^2+ax+b=0$, by Viète's formulas we have $$-a=a+b\qquad b=ab$$ The first formula implies $b=-2a$. Substituting this into the second equation gives $-2a=-2a^2$ or $a=a^2$, so $a=0,1$. These give corresponding $b$ values of 0 and $-2$ respectively, so the unique non-trivial quadratic "auto-solving" polynomial is $x^2+x-2=0$.

For the cubic case $x^3+ax^2+bx+c$, the same formulas give $$-a=a+b+c\qquad b=ab+bc+ca\qquad-c=abc$$ If $c=0$ the situation is just the quadratic case with another zero root. In general, if $p(x)$ is auto-solving, so is $p(x)x^k$ for any $k>0$, since only extra zeros are introduced to both the coefficient and root list. Thus we have for example $x^3+x^2-2x$, but these are trivial.

There is a nice non-trivial auto-solving cubic polynomial, $x^3+x^2-x-1$, but also a very ugly one given by $$a=0.56519771\dots$$ $$b=-1.76929235\dots$$ $$c=0.63889691\dots$$ (It turns out that these values are in the OEIS: A273065, $-$A273066, A273067.)

And this is just the all-real solution! There are two solutions with complex coefficients. One is given by $$a={-0.78259885\dots}-0.52171371\dots i$$ $$b={0.88464617\dots}-0.58974280\dots i$$ $$c={0.68055154\dots}+1.63317024\dots i$$ and the other is obtained by taking the complex conjugates of the values above.

For quartics, there is only one non-trivial real auto-solving polynomial, $x^4+x^3+ax^2+bx+c$ where $$a=-1.75487766\dots$$ $$b=-0.56984029\dots$$ $$c=0.32471795\dots$$ Even more startlingly, there are no auto-solving real polynomials of degree five or higher. This and the uniqueness of the quartic example were proved by Paul R. Stein in "On Polynomial Equations with Coefficients Equal to Their Roots" (March 1966), American Mathematical Monthly 73 (3), pp. 272-274.

Not a full solution to the general case, but at least two solutions to the general case and all solutions to the cubic.

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Well, we know $(x-p)(x-q)=x^2-(p+q)x+pq$, so if $a,b$ are zeroes of $x^2+ax+b$, then $a=-a-b$ and $b=ab$. From the last, we see $b=0$ or $a=1$, and so we get solutions $(a,b)=(0,0)$ and $(a,b)=(1,-2)$. Those are all.

For higher degree polynomials, we know $(a_n,\cdots,a_0)=(0,\cdots,0)$ is a solution (to the $n+1$-degree polynomial) so let's assume they're not all $0$. We have (again, our polynomial is degree $n+1$)

$$\prod_{k=0}^{n+1}(x-a_k)=x^{n+1}+\sum_{k=0}^{n}a_kx^k$$

This seems quite hard, so let's just try it for cubics first.

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Cubics.

Allright, so, let's jump right into it. We need:

$$x^3+ax^2+bx+c=(x-a)(x-b)(x-c)$$

and working out the parentheses on the right gives us the following equations:

\begin{align} a&=-a-b-c\\ b&=ab+bc+ca\\ c&=-abc \end{align}

So let's see what happens when $c=0$; then, $a=-a-b$ and $b=ab$. Hey! This is the solution to the second degree equation! Let's keep that in mind, maybe we can prove something about that for the general case. Now we proceed to assume $c\neq 0$. Then, $ab=-1$. Now let's see if we can express $a$ in terms of $b$ and $c$ to hopefully eliminate that variable:

\begin{align} a&=-\tfrac12(b+c)\\ a&=\frac{b-bc+1}{c}\\ a&=-\frac1b \end{align}

So now we express $c$ in terms of $b$ by pairing the last equation to the first two:

\begin{align} c&=\frac{2-b^2}{b}\\ c&=\frac{-b-b^2}{1-b^2} \end{align}

And so we get

$$b^4+b^3-2b^2+2=0$$

Ouch! That's ugly. But, it has a solution $b=-1$ (corresponding to $(a,b,c)=(1,-1,-1)$). The other three solutions are pretty ugly though; one real one and two complex ones (I won't work them out because they're too chaotic and seeing them won't accomplish anything. Feel free to use wolframalpha to get them though).

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As for our little trick with $c=0$ obtaining the solutions from the quadratic equation, that's actually quite simple to see. If $x^{n+1}+a_nx^n+\cdots+a_0$ has roots $a_n,\cdots,a_0$, then surely multiplying by $x$ doesn't change that: $x^{n+2}+a_nx^{n+1}+\cdots+a_0x+0$ now also has those roots, and since we multiplied by $x$, it has an extra root $0$. As a result, we may assume the constant term is nonzero, since if it is, we may divide the polynomial by $x$ to get to a lower degree case.

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}