Roots of a Polynomial Minus It's Constant Term

Question

Suppose we have a sequence of integers $a_1,\dots,a_n$. Is there any way to determine the roots of the polynomial

$$P(x) = (x+a_1)\dots(x+a_n) - a_1\dots a_n$$

Clearly $P(0) = 0$, but can anything be said about the other roots? Can they be expressed in some way related to the original integers $a_1,\dots,a_n$?

Any answer or reference would be appreciated.

Edit: If need be, you may assume $a_i|a_{i+1}$ for all $i$.

Comment: The new roots need not be integers. I would be satisfied with finding complex roots.

Answer 1

It doesn't appear to be easier than solving a polynomial of degree $n-1$ "from scratch". For example, if $Q(x) = (x+1)(x+2)(x+4)(x+8)(x+16)(x+32)$, the Galois group of $$P(z)/z = (Q(z)-Q(0))/z = z^5 + 63 z^4 + 1302 z^3 + 11160 z^2 + 41664 z + 64512$$ is $S_5$, so this is not solvable by radicals.

On the other hand, it may be interesting to look at the roots of $Q(x) - t$ as functions of $t$: these are analytic except at the points where they collide (the roots of the discriminant of $Q(x)-t$), which can be branch points. In the above example, the root that is $-32$ at $t=0$ has the Maclaurin series $$ -32-{\frac {1}{9999360}}t+{\frac {1189}{578592599703552000}}{t}^{2}-{ \frac {5752091}{84743927083111118222131200000}}{t}^{3}+{\frac { 28255922633}{10460874099698222880422457709166592000000}}{t}^{4}-{ \frac {8129746966487}{ 68399931666747186847737020565295881781248000000000}}{t}^{5} +\ldots$$ I believe this has radius of convergence approximately $1.61741 \times 10^7$ (one of the roots of that discriminant), so it should certainly converge quite nicely at $t=Q(0)$.

Answer 2

Can they be expressed in some way related to the original integers $a_1,\dots,a_n$ ?

No. Take $x^5-x\pm1=0$. Its roots are inexpressible in radicals. But $x^5-x=x(x^4-1)=$

$=x(x^2-1)(x^2+1)=(x-1)x(x+1)(x^2+1)$ has all roots expressible in radicals.