Polynomial Roots

A polynomial's roots are the x-values where it equals zero — the places its graph crosses or touches the x-axis. Finding them is the central project of algebra: every quadratic formula, every factoring trick, is a root-finding method. For higher-degree polynomials, two powerful theorems narrow the hunt from 'infinity' to 'a short list of candidates.' And the way a root behaves — does the graph cross cleanly, or just kiss and bounce? — reveals a hidden number called multiplicity.