Is there a way to simplify the process of plugging in factors through synthetic division to find the zeros of a polynomial? i am aware of descarte's rule of signs, but is there any other way to make it faster than guess-and-check?

Yes you can use the discriminant of a quadratic/polynomial. For instance, if
there is one real root. If there are two real roots and i  there are no real roots

The discriminant comes from the quadratic equation, which is the following.

Two because two of one thing is two.