Polynomial Power Sum Reduction

This manuscript presents an alternative pedagogical framework using a polynomial reduction method structured to evaluate high-order root power sums (S_N) by bypassing intermediate values. While standard curricula approach power sums via the step-by-step arithmetic ladder of the Newton-Girard identities, this expository note applies modular polynomial reduction directly within the variable space. This collapses the targeted power expression into a lower-degree remainder, illustrating why a student can dynamically leapfrog over intermediate sequence dependencies (such as moving directly from S_1 to S_3 in a quadratic case, or bypassing S_4 when determining S_5 in a quartic system). The note provides a side-by-side verification against traditional Newton recurrences using numerical examples and discusses how the framework scales across quadratic-like trinomial forms of arbitrary degrees. This educational approach was structured by the author during problem design for the textbook Pre-University Further Pure Mathematics: Contains Non-Trivial Problems to highlight structural algebraic pathways over rote algorithmic bookkeeping.