This study derives novel exact traveling wave solutions for the nonlinear (1+1)-dimensional Chafee-Infante equation by synthesizing the generalized first integral method (GFIM) with Laurent polynomial expansions. As a fundamental reaction-diffusion model, the Chafee-Infante equation governs pattern formation in diverse systems—from biological to chemical and physical contexts—yet its strong nonlinearity poses persistent challenges to classical integration techniques such as the inverse scattering transform or Hirota's method. We transform the equation into an autonomous polynomial system and employ the division theorem to systematically identify its first integrals, thereby circumventing the need for auxiliary equations or ansatz-based heuristics. By introducing Laurent polynomial ansatzes of varying complexity—ranging from first-degree to higher-order expansions—we yield compact rational-exponential solutions that are both exact and computationally tractable. The validity of these solutions is confirmed through symbolic computation in Mathematica, while a detailed graphical analysis elucidates their behavior—from bounded, dissipative profiles to singular structures—across different parameter regimes, including the critical thresholds $C=0$ and $C=2$ where blow-up phenomena emerge. This work underscores the efficacy of merging Laurent series with algebraic methods, offering a powerful and generalized tool for extracting exact solutions from a broader class of intractable nonlinear partial differential equations (PDEs) arising in mathematical physics and applied mathematics.