Abstract

This work extends the distance–product idea, traditionally assuming all real roots, to real-coefficient polynomials with one real root and one complex–conjugate pair. For cubics, introducing the tangent line to the function at the real root shows a systematic link between the contact point and slope and the real and imaginary parts of the complex roots. The same principle generalizes to quartics and higher degrees via contact polynomials tailored to the degree, preserving a common structure. The framework simplifies definite integrals in the presence of nonreal roots by reducing them to linear combinations of standard polynomial terms.