By Paul Dohrman
Updated Aug 30, 2022
Written by industry expert Paul Dohrman, who has over 15 years of experience applying polynomial mathematics to finance, engineering, and scientific research.
Polynomials—equations built from sums of terms, each term a constant multiplied by one or more variables raised to a power—are the backbone of many scientific and economic models. Even simple relations like F=ma fall under the polynomial umbrella.
In finance, present‑value calculations rely on polynomial expressions that transform future cash flows into their current worth. Loan amortization schedules, company valuations, tax brackets, and interest‑rate projections can all be represented as polynomials, enabling precise modeling of complex financial scenarios.
Electronic circuit analysis frequently employs polynomial relationships. Ohm’s law, V = IR, expresses the voltage drop across a resistor as a linear polynomial of current. Power dissipation follows P = IV = I²R, another polynomial that captures the quadratic dependence on current. Kirchhoff’s junction and loop rules, which govern current conservation and voltage summation, are also expressed as polynomial equations.
Polynomials are the workhorse of regression and interpolation. In regression, data points are approximated with a polynomial—often a straight line y = mx + b—or, for multiple predictors, with a multivariate polynomial in the form of multiple linear regression. Interpolation uses short polynomials that pass exactly through all data points. Common interpolation tools include Lagrange polynomials, cubic splines, and Bezier splines.
Chemical equilibria and gas laws are naturally expressed with polynomials. The ideal gas law, PV = nRT, is a simple polynomial relating pressure, volume, temperature, and amount of substance. Equilibrium concentrations often satisfy polynomial equations; for example, the relation KC = AB links the concentrations of OH⁻ (A), H₃O⁺ (B), and H₂O (C) through the equilibrium constant K.
Physics and engineering fundamentally revolve around proportionality, many of which are captured by polynomial relationships. Classical mechanics uses F = ma, energy–mass equivalence E = mc², and Newton’s law of gravitation F = Gm₁m₂/r²—all of which are polynomial expressions describing how quantities scale with one another.
From everyday calculations to advanced scientific modeling, polynomials provide the mathematical language that turns abstract concepts into actionable insights.
