We know how to divide 1 by 8, Newton figured out we can always divide a complex expression such as (1-x) using exact same approach too.

Anything can be converted to power series, another example is

To take square root of a^2 + x^2, use root extraction.

Newton’s method to Approximate a root of a polynomial equation. let’s start from how he calculates the root of 2:

He converts the problem of finding the root of 2 to solving this

Generalized version

Algebraic Curves and Puiseux Series using Newton’s Polygon or Newton Diagram.