Is there a symbol used for an element which a function sends to zero?
Like this: $f(?) = 0$, where "$?$" would be the symbol for which I am wondering if exists.
I think this could be useful for instance in factoring polynomials: $ax^2+bx+c = a(x-?_1)(x-?_2)$ and the zeroes of functions are always interesting.
On
In almost twenty years of mathematics I've never seen such a symbol. I believe that it would be pretty useless, since in many cases zero is nothing really special. It is mainly for convenience that we speak of zeroes of a function: everybody learns that $f(x)=y$ is equivalent to $f(x)-y=0$ (I am thinking of real- or -complex-valued functions here).
Everybody can introduce his own notation, but I guess you can hardly avoid a sentence like "Let $x_1$ be a zero of $f$". How can you distinguish different zeroes with a unique symbol?
You could write $f^{-1}(0)$. But note that this is a set, not a single specific value, because there might be several different values that the function $f$ sends to zero.
Another notation you might see is $Z(f)$, standing for the "zero set" of $f$: $$ Z(f) = \{ x : f(x)=0\} $$
In some fields, like linear algebra, $Z(f)$ is called the "kernel" of $f$. See here.
Personally, I don't like using special symbols when words will serve instead. So, in my view, "$x$ is a zero of $f$" is much better than "$x \in Z(f)$". Of course, "$f(x)=0$" is pleasantly clear and concise, too.