This is a generalization of an MSE question, Polynomials that share at least one root.

Let $P(x)$ be a specific polynomial of degree $d$, with given real coefficients $A_i$ ($A_d=1$), and real roots: $$P(x) = x^d + A_{d-1}x^{d-1} + A_{d-2}x^{d-2} + \cdots + A_{0}\;. $$

. What does the set of polynomials, with real coefficients $a_i$, $$p(x) = x^d + a_{d-1}x^{d-1} + a_{d-2}x^{d-2} + \cdots + a_{0} \;,$$ look like (geometrically) in $\mathbb{R}^d$, when each $p(x)$ shares at least one root with $P(x)$?Q

I am seeking a description of this set in the space of the $d$ coefficients, $(a_0,\ldots,a_{d-1})$. The reason there is hope for a nice description, is that it makes a pretty picture for $d=2$.

Let $P(x) = x^2 + A_1x + A_0$, with $A_1=3$ and $A_0=-1$. The plot in the $a_0 a_1$-plane of all other $p(x)= x^2 + a_1x + a_0$ that share a root with $P(x)$, looks like this:

^{ Lines intersect at $(a_0,a_1)=(-1,3)$. The discriminant is $a_1^2 = 4 a_0$. }

All those $(a_0,a_1)$ on the two lines share a root with $x^2 + 3x -1$. The lines are tangent to the discriminant parabola.