This is likely incredibly naïve and basic, but ignorance is worse... so here it goes...
I understand that Taylor series are great to approximate a difficult function (trigonometric, for example) with a polynomial built up via derivatives of the original function at a nearby point we know well in the original function. The derivatives become the coefficients of the newly-created polynomial (modified by the factorials in the denominators).
But I don't see what is the point or how to use Taylor expansions when we already have a polynomial, and we are talking about the roots or zeros.
The context of this question is this other post. I am trying to build up my background knowledge to actually understand the answer on the prior post.