Question

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?

Answer 1

Yes you can use the discriminant of a quadratic/polynomial. For instance, if
$b^2 - 4ac = 0$ there is one real root. If $b^2 - 4ac > 0$ there are two real roots and i$b^2 - 4ac < 0$  there are no real roots

The discriminant comes from the quadratic equation, which is the following.
$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$