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The strongest result in this direction that I've heard of is Sudbery's theorem (which was originally conjectured by Popoviciu and Erdös).

Theorem. Let $P(z)$ be a polynomial of degree $n\geq 2$ and let $\Pi(z)=\prod\limits_{k=0}^{n-1}P^{k}(z)$. \Pi(z)=\prod\limits_{k=0}^{n-1}P^{(k)}(z)$ where $P^{(k)}$ is the $k$th derivative of $P$. Then either $\Pi(z)$ has exactly one distinct zero root or $\Pi(z)$ has at least $n+1$ distinct zeroesroots.

See the original paper by Sudbery.
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The strongest result in this direction that I've heard of is Sudbery's theorem (which was originally conjectured by Popoviciu and Erdös).

Theorem. Let $P(z)$ be a polynomial of degree $n\geq 2$ and let $\Pi(z)=\prod\limits_{k=0}^{n-1}P^{k}(z)$. Then either $\Pi(z)$ has exactly one distinct zero or $\Pi(z)$ has at least $n+1$ distinct zeroes.

See the original paper by Sudbery.
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The strongest result in this direction that I've heard of is Sudbery's theorem.

Theorem. Let $P(z)$ be a polynomial of degree $n\geq 2$ and let $\Pi(z)=\prod\limits_{k=0}^{n-1}P^{k}(z)$. Then either $\Pi(z)$ has exactly one distinct zero or $\Pi(z)$ has at least $n+1$ distinct zeroes.

See the original paper by Sudbery.