To determine which of the given expressions is not a polynomial, let’s examine each one carefully:
1. Expression: [tex]\( 12x \)[/tex]
- This expression consists of a single term, [tex]\( 12x \)[/tex], where the variable [tex]\( x \)[/tex] is raised to the power of 1.
- Polynomials are expressions that consist of terms where the variables have non-negative integer exponents.
- Since [tex]\( 12x = 12x^1 \)[/tex], it fits the definition of a polynomial.
2. Expression: [tex]\( x^2 - 2x \)[/tex]
- This expression consists of two terms: [tex]\( x^2 \)[/tex] and [tex]\( -2x \)[/tex].
- The term [tex]\( x^2 \)[/tex] has the variable [tex]\( x \)[/tex] raised to the power of 2, and [tex]\( -2x \)[/tex] has the variable [tex]\( x \)[/tex] raised to the power of 1.
- Both terms have non-negative integer exponents.
- Therefore, [tex]\( x^2 - 2x \)[/tex] is a polynomial.
3. Expression: [tex]\( \frac{1}{3}x^3 \)[/tex]
- This expression consists of a single term, [tex]\( \frac{1}{3}x^3 \)[/tex], where the variable [tex]\( x \)[/tex] is raised to the power of 3.
- The coefficient [tex]\( \frac{1}{3} \)[/tex] is a constant, and the exponent of [tex]\( x \)[/tex] is a non-negative integer.
- Thus, [tex]\( \frac{1}{3}x^3 \)[/tex] is a polynomial.
4. Expression: [tex]\( \frac{x^3 + 1}{x} \)[/tex]
- This expression is given as a fraction, where the numerator is [tex]\( x^3 + 1 \)[/tex] and the denominator is [tex]\( x \)[/tex].
- To simplify this, we can rewrite the expression as [tex]\( \frac{x^3}{x} + \frac{1}{x} = x^2 + x^{-1} \)[/tex].
- The resulting expression [tex]\( x^2 + x^{-1} \)[/tex] contains a term [tex]\( x^{-1} \)[/tex], where [tex]\( -1 \)[/tex] is a negative exponent.
- Polynomials cannot have negative exponents; all exponents must be non-negative integers.
Given this examination, the expression that does not qualify as a polynomial due to the negative exponent is:
[tex]\[ \frac{x^3 + 1}{x} \][/tex]
Thus, the expression that is not a polynomial is the fourth one: [tex]\( \frac{x^3 + 1}{x} \)[/tex].
