Polynomial Functions

Complex Analysis · Axiom Academy

The simplest complex functions — defined everywhere, and the one place every polynomial finally factors completely. 1. A Polynomial Maps the Plane to Itself A complex polynomial takes a number and returns another complex number p(z) . So p is a mapping of the plane : feed it a path of inputs and it draws a path of outputs. Below, the blue point circles the input z -plane while the purple point traces where p(z) = z^2 + 1 sends it — the doubling of the angle (degree 2) shows up as the output looping around twice . where and the degree n is a non-negative integer 2. Why Polynomials Are So Well-Behaved Because a polynomial is built only from addition and multiplication, nothing can ever go wrong — there is no division to blow up, no branch cut to dodge. The animation grows the input radius |z| outward and the bars track |p(z)| for p(z) = z^3 - 1 : the height runs away to infinity, which is the precise meaning of "behavior at infinity." p(z) exists for every — no singularities, ever. Small changes in z make small changes in p(z) at every point. Complex-differentiable on all of — a so-called entire function. The big payoff — the Fundamental Theorem of Algebra Every non-constant polynomial of degree n has exactly n roots in , counted with multiplicity, and therefore factors completely into linear pieces. Real polynomials may have "no roots" (like z^2 + 1 ); over the roots are always there. We make this concrete next. 3. The Roots, and the Factorization They Give

This is the written version of the interactive lesson above. See the full Complex Analysis course.