> Horner's method or another representation of your polynomial might start to become important.

Horner's is the default way to evaluate a polynomial - and I think you're overstating the cost of evaluation.

> what the author is doing isn't much different than a Taylor approximation

Yes it is. The Taylor approximation is only valid around a point - and it's based on matching the nth order derivative. There are no error metrics. And it requires differentiation to find it.