What is the difference between Algebraic Expressions and Polynomials?

"Polynomial" is a precisely defined term. A polynomial is constructed from constants and variables by adding and multiplying. One could add "subtracting", but $x-y$ is $x+(-1)y$, so adding and multiplying are enough.

"Algebraic expression" is not a precisely defined term. Algebraic expressions include many things that are not polynomials, including rational funtions, which come from dividing polynomials, and things like $\sqrt{x}$.

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Terminology

Polynomials

Abstract Algebra

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