This paper is devoted to several new results concerning (standard) octonion polynomials. The first is the determination of the roots of all right scalar multiples of octonion polynomials. The roots of left multiples are also discussed, especially over fields of characteristic not 2. We then turn to study the dynamics of monic quadratic real octonion polynomials, classifying the fixed points into attracting, repelling and ambivalent, and concluding with a discussion on the behavior of pseudo-periodic points.

• 11R52 - Quaternion and other division algebras: arithmetic, zeta functions
• 17A35 - Nonassociative division algebras
• 17A45 - Quadratic algebras (but not quadratic Jordan algebras)
• 17A75 - Composition algebras
• 17D05 - Alternative rings
• 37C25 - Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
• 37P35 - Arithmetic properties of periodic points