Does the Quantifier apply to all?

Question

I have the following question:

Let $f(x, y, z) = x^2y+z^3$, where $x, y, z \in \mathbb{Z}$. For each of the following determine its truth value. Justify your answers.

(a)$\exists x, y, z: f(x, y, z) = 0$

My question: Does the $\exists$ affect the "$y$" and "$z$" as well? or is it only the $x$?

Also, if it does not affect the "$y$" and "$z$" variable, would those variables be some constant or something? Not too sure about that part.

Answer 1

The $\exists$ affects all the variables. Writing $\exists x,y,z$ is short for $\exists x\exists y\exists z$.

If $\exists$ only affected one variable, then the formula wouldn't even be a statement exactly because there wouldn't be any quantification on the missing variables.

Answer 2

$\exists x,y,z$ is shorthand for $\exists x \exists y \exists z$ so your statement is true if you can find any triple $(x,y,z)$ that satisfies the equation.