Final answer:
The expression 3x³ - xˣ is not considered a polynomial because the second term has a variable as an exponent, which violates the definition of a polynomial that requires exponents to be whole numbers.
Step-by-step explanation:
When determining if an expression is a polynomial, we need to ensure that the exponents of the variable are whole numbers (non-negative integers). The expression 3x³ is indeed a part of a polynomial, as the exponent on x is 3, a whole number. However, the second term, xˣ, poses a problem since the exponent itself is x. This makes it a variable exponent, which is not allowed in polynomials as it can potentially be a non-whole number, depending on the value of x.
Furthermore, in order to qualify as a polynomial, an expression must have a finite number of terms. The term with a variable exponent could result in an infinite series once expanded and therefore does not meet the criteria of a polynomial. Hence, the expression 3x³ - xˣ is not a polynomial.
As examples provided in the reference suggest, when working with exponents, specific rules of exponentiation apply, such as the property xPx9 = x(p+q) or cubing of exponentials where you cube the digit term in the usual way and multiply the exponent of the exponential term by 3. Nonetheless, these examples discuss operations within polynomials, unlike the given expression which doesn't fit the definition of a polynomial.