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• A polynomial method is presented for the investigation of the guided wave behaviors in FGPP plates. • The impact of uniaxial graded pre-stress is important on propagating guided modes. • Non-continuous furcation phenomenon of guided wave in FGPP material perfectly bounded to a negative magnetic permeability. The present work was undertaken to predict the guided dispersion curves in pre-stressed functionally graded piezoelectric–piezomagnetic materials (FGPPM) with negative magnetic permeability. The gradient is considered to be according the depth, not only on the material properties but also on the prestress. The linear, exponential, and Gaussian graded stress profiles are considered with different FGPPM gradient shapes. Legendre polynomial is a well known method for the roots solution. This method employs the idea that the unknown functions may be expanded into a polynomial approximation using Rodrigues' formula. It displays quick convergence and yields an accurate phase velocity based on the FGPP plate's layer number. Numerical studies are given only for open circuit surface, where the electric displacement and the magnetic induction are assumed to be zero rather than continuous. The propagating and evanescent guided modes are shown to alter depending on the type of graded stress profiles and the plate gradient shape. In addition, the fundamental Lamb mode A 0 is the most sensitive to the graded pre-stress and gradient shape of the material. In particular, results have shown that the resonance frequency for FGPP plates corresponds to the frequency where the group velocity of Lamb waves vanishes (Zero Group Velocity, "ZGV"). In the present work, the graded pre-stress observation could provide a reference for safe and better designs of smart acoustic devices made of FGPPM. [ABSTRACT FROM AUTHOR]