In order to enlarge the class of equations provided by traditional polynomials over a binary algebra A to a more useful class of equations, we introduce polynomials in noncommuting or nonassociating indeterminates. We discuss algebraic properties of these formal polynomial algebras and their accompanying polynomial function algebras. We present certain basis results for polynomial algebras, which are used to address the question of zero divisors in a polynomial algebra. We give an analog of the remainder theorem and the factor theorem for polynomials. Particular emphasis is placed on showing the difference between polynomials and polynomial functions. We also provide a brief discussion of polynomial composition and formal derivatives.