What is the difference between Algebraic Expressions and Polynomials?

$\begingroup$

Both algebraic expressions and polynomials are a combination of terms grouped together. What is the difference between them?

$\endgroup$

2 Answers

$\begingroup$

"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 $\sqrt{x}$.

$\endgroup$ 5 $\begingroup$

Some difference is:

Algebraic Expression: may not be a continuous function on $R = (-\infty,\infty)$, but Polynomial is.Example: $\dfrac{x}{x+1}$, and $x^2-1$. The former is not defined at $x=-1$, while the latter is continous throughout $R$.

$\endgroup$