Consider the family of polynomials
$$ p_n(z) = z^{n+1} + d \cdot z^n - v $$
over the complex numbers $\mathbb{C}$ with $n \in \mathbb{N}$, $d \in \mathbb{R}$, $d > 0$, and $v \in \mathbb{C}$ arbitrary.
I am looking for an explicit formula for the roots of this type of polynomial.
I am aware that in general there is no formula for polynomials of degree > 4, but I was hoping this very restricted family of polynomials might lend itself to a root formula.
So far I have been unable to come up with or find a formula. Things I have tried:
- Trying a variable substitution (inspired by bi-quadratic polynomials)
- Looking at Vieta's formulas
- Trying to figure out if that family of polynomials has a name
- Asking WolframAlpha and SymPy to find the roots
Next to the formula, I would also be very interested in a proof that no such formula can exist. I would also be interested in formulas for special cases. For example, restricting $v$ to the reals would be one such case, but I am very open to others.
Why am I interested in these roots? I was thinking about monomials and their root formulas, and I was fascinated by the idea of interpolating between $z^n$ and $z^{n+1}$ and seeing how the roots move around. In the process, I stumbled over the above polynomials. For a visual inspection, I could simply try to find the roots numerically, but a formula would feel more "satisfying" and "beautiful" to me :-)
Thank you for your help! Please, let me know if/how I can improve the post.