I thought the error diminishes as we increase the terms in Taylor polynomial approximation, yet here I am trying to solve for the interest rate by defining the IRR in terms of net present value and solving for the rate in a 1st degree and a 2nd degree Taylor polynomial approximation only to notice that results from the 1st degree are more precise as compared to those that are found using the 2nd degree Taylor polynomial.

The root finding algorithm goes through the same number of iterations either with 1st degree or the 2nd degree polynomial and returns the roots that converge faster using the 1st degree approximation as compared to the 2nd degree approximation.

Any thoughts on why this is occurring?

As an example:

IRR [ -100,50,40,30,10 ]

Seed value = 0.10

Solving for interest rate in 1st degree Taylor polynomial approximation

IRR = 0.14488783107567296

Solving for interest rate in 2nd degree Taylor polynomial approximation

IRR 1 = null
IRR 2 = 0.14488785866378917