If someone has algebraic sense it means that they are adept at doing algebra. Algebra is the unifying thread of all of mathematics.
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Yes. That is the definition of an algebraic expression.
Algebraic expressions are terms that do not include an equality sign
They are two terms of an algebraic expression.
A polynomial is always going to be an algebraic expression, but an algebraic expression doesn't always have to be a polynomial. An algebraic expression is an expression with a variable in it, and a polynomial is an expression with multiple terms with variables in it.
You solve algebraic expressions by getting the variable by itself.