The following result about polynomials is known:
Proposition: Let $K$ be a subfield of $\mathbb{C}$, $f(x) \in K[x]$ a polynomial with degree $n \geq 1$ and $\alpha \in \mathbb{C}$ a root of $f(x)$. Then
a) $\alpha$ is a simple root of $f(x)$ $\iff$ $f(\alpha) = 0$ and $f'(\alpha) = 0$;
b) if $f(x)$ is irredutible over $K$ then all the roots of $f(x)$ are simple.
Is this result true if we exchange $\mathbb{C}$ for any algebraically closed field?
I believe you want to say that $\alpha$ is a simple root implies that $f(\alpha)=0, f'(\alpha)\neq 0$. This is not true is the characteristic is $p$, $X^p-a=(X-a)^p$ and $a$ is simple.