the product of 23 and a number
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A polynomial is always going to be an algebraic expression, but an algebraic expression doesn't always have to be a polynomial. In another polynomial is a subset of algebraic expression.
A rational algebraic expression is the ratio of two algebraic expressions. That is, one algebraic expression divided by another. It is important that the domain is defined in such a way the the rational expression does not involve division by 0.
If an algebraic expression is equivalent to another algebraic expression then it is an equation.
To find the value of an algebraic expression is to evaluate it.