I realise this is a general question. I am self-teaching mathematics and I have observed that the zeros of real and complex functions are of much interest.
Question: Why are the zeros of real or complex so important?
I was going to say that the zeros completely determine a function because the function can be factorised into factors, with each factor corresponding to one of the zeros. For example, $f(x) = (x-1)(x-3)$ is completely determined by its zeros at $x=1$ and $x=3$.
However this doesn't work for functions like $g(x) = (x-1)(x-3) + 2$.
I appreciate your patience with a question that might be naive.
Best Answer
Often the zeros carry special significance.
For example the location of non-trivial zeros of the Riemann-Zeta function has far reaching consequences in mathematics, especially for theories of prime numbers.
In ordinary linear differential equations, the zeros of these “functions of functions” are called the “homogenous solutions” and form a vector space of solutions (since every solution evaluates to zero you can take linear combinations and still end up with zero).
The study of the roots of polynomials led to Galois Theory which connected fields and groups and helped resolve longstanding issues of which polynomials are solvable and also issues about provability in geometry.
Finally, a trivial interest in zeros is that $f(x)=c \iff g(x):=f(x)-c=0$.
Just a smattering of examples to illustrate.