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.
Taylor series can be applied to polynomial function as a way of shifting toward a point where the higher degree terms converges to 0 faster than lower degree terms, so that the approximation can get by with fewer terms.
In fully expanded form like $f(x)=\sum_i c_i x^i$, if the center point $x_0$ of the polynomial is not $0$, then higher degree terms may actually increase faster than lower degree terms near $x_0$, which is not what we want. So we want to shift the center toward $x_0$ so that the function becomes $f(x)=\sum_i c'_i (x-x_0)^i$, in which case the higher degree terms will tend toward $0$ faster as $x$ approaches $x_0$, so we can get rid of the higher degree terms without too much error.
Example: We want to approximate near $x_0=3$, and we have $f(x)=x^2-5x+10$. If we put $x=3$, $x^2$ is still required for good result since it adds some value in the result. If we apply Taylor series on $x_0=3$, we get $f(x)=(x-3)^2+(x-3)+4$. If we put $x$ near $3$, we can see $(x-3)^2$ converges to 0 faster than other terms, so we can approximate the function as $f(x)\approx (x-3)+4$ or even $f(x)\approx 4$