On the Average-Offset Pair-Partitioning of Polynomial Roots

We develop a unified framework for parameterizing the roots of depressed polynomial equations of any degree, extending Po-Shen Loh's elegant "average ± offset" approach to the quadratic formula. For a depressed polynomial of degree n, the n roots (which sum to zero by Vieta's relations) are partitioned into ⌊n/2⌋ pairs, each of the form Ai±Bi, together with a lone root −2∑Ai when n is odd. This parameterization automatically satisfies the sum constraint and reduces the root-finding problem to a system of equations in the pair-center and pair-spread parameters. We carry out this program explicitly for degrees 2 through 5, recovering and unifying classical results (the quadratic formula, Vieta's trigonometric solution for cubics, Ferrari's method for quartics) within a single conceptual framework. For each degree, we derive novel structural results: a canonical factorization of the discriminant into intra-pair, cross-pair, and lone-root factors; a Sign Purity Theorem relating the discriminant's sign directly to the pair-spread parameters; necessary conditions on coefficients for all-real roots via a positive-definite energy form; and a complete classification of codimension-1 bifurcations governing root-type transitions. At degree 5, we give a precise accounting of why the resolvent system exceeds degree 4, providing a concrete geometric interpretation of the Abel-Ruffini obstruction through the lens of pair-partition combinatorics and the simplicity of A5. The framework extends naturally to arbitrary degree n, where the number of pair-partitions grows as (n−1)!! for even n, yielding resolvents whose degrees exceed 4 for all n≥5.