To determine which of the given expressions is not a polynomial, let's review the definition of a polynomial. A polynomial is an expression composed of variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Importantly, a polynomial cannot have variables in the denominator.
Let's analyze each given expression:
1. [tex]\( 12x \)[/tex]:
- This is a linear polynomial because it is of the form [tex]\( ax \)[/tex] where [tex]\( a \)[/tex] is a constant.
- There are no variables in the denominator.
- Polynomial.
2. [tex]\( x^2 - 2x \)[/tex]:
- This is a quadratic polynomial because it is of the form [tex]\( ax^2 + bx + c \)[/tex] where [tex]\( a, b \)[/tex], and [tex]\( c \)[/tex] are constants.
- All terms have non-negative integer exponents and no variables in the denominator.
- Polynomial.
3. [tex]\( \frac{1}{3} x^3 \)[/tex]:
- This is a cubic polynomial because it is of the form [tex]\( ax^3 + bx^2 + cx + d \)[/tex] where [tex]\( a, b, c \)[/tex], and [tex]\( d \)[/tex] are constants.
- The constant [tex]\( \frac{1}{3} \)[/tex] does not affect the polynomial nature since it is a coefficient.
- Polynomial.
4. [tex]\( \frac{x^3 + 1}{x} \)[/tex]:
- To examine this, let's simplify:
[tex]\[
\frac{x^3 + 1}{x} = \frac{x^3}{x} + \frac{1}{x} = x^2 + \frac{1}{x}
\][/tex]
- Notice that [tex]\( \frac{1}{x} \)[/tex] has a variable [tex]\( x \)[/tex] in the denominator.
- Polynomials do not have variables in the denominator.
- Not a Polynomial.
Therefore, out of the given expressions, the one that is not a polynomial is:
[tex]\[
\boxed{4}
\][/tex]
