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.
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No. A matrix polynomial is an algebraic expression in which the variable is a matrix. A polynomial matrix is a matrix in which each element is a polynomial.
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.
are the terms of the expression