Are all polynomial expressions an algebraic expression.

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I mean to ask that, real numbers can be written in the form of constant polynomials like, 4x^0, 2x^0 etc. If all polynomial expressions are algebraic expressions then 4x^0 = 4x^0 should be a polynomial equation implying that it is an algebraic equation too. But it does not contain any variable. After simplification it is 4 = 4. 4 is a real number so this equation can be called an arithmetic equation also. again 4 -- 4 = 0 is a zero polynomial equation because 0=0 is an equation consisting two polynomial expressions 0 and 0 separated by an equivalence. This is really confusing please help and also state the difference between algebraic equation and polynomial equation.
whether all polynomial equations are algebraic equation or not.

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1 Answer

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Algebraic equation and polynomial equation are synonyms. They are both asking for the roots of a polynomial.

But algebraic expression has a wider sense than polynomial expression (rarely used) in that it allows divisions and rational exponents. (If you allow just division, you get a rational expression.)

By decreasing generality, an algebraic expression is also a rational expression, which is also a polynomial expression, which is also an arithmetic expression.

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