Let $A$ be the ring $\Bbbk[\alpha_0, \alpha_1, \alpha_2, x_0, x_1, x_2]$ (where $\Bbbk$ is an infinite field, algebraically closed if it matters). Let $g \in \Bbbk[\alpha_0, \alpha_1, \alpha_2]$ be a homogeneous polynomial of degree at least one, such that $\alpha_0$ does not divide $g$. Let $$I = (g, \sum_i \alpha_i x_i).$$

Is $\alpha_0$ necessarily a nonzerodivisor in $A/I$?

A little bit of motivation: I've shown that $I$ has a certain nice property that I would like to carry over to its localization $I[\alpha_0^{-1}] \subset A[\alpha_0^{-1}]$. This will hold automatically if $I[\alpha_0^{-1}] \cap A = I$, which is true iff $\alpha_0$ is a nonzerodivisor in $A/I$. (Each of these is clearly equivalent to the statement: if $\alpha_0 f \in I$, then $f \in I$.)

I can show that at least $\alpha_0$ is not nilpotent, by showing that $(A/I)[\alpha_0^{-1}]$ is not the zero ring.