The resultant Res(f, g) of two univariate polynomials f and g is the determinant of their Sylvester matrix — a structured matrix built from the coefficients of f and g. The fundamental theorem: Res(f, g) = 0 if and only if f and g share a common root (over the algebraic closure). The resultant equals a_m^deg(g) · b_n^deg(f) · Π_{f(α)=0, g(β)=0} (α - β), a product over all pairs of roots. By eliminating one variable from a polynomial system {f(x,y)=0, g(x,y)=0}, we get Res_x(f,g)(y) = 0 — a univariate polynomial whose roots are exactly the y-coordinates of intersection points. This makes resultants the classical tool for polynomial system solving and the foundation of elimination theory.