Verified Answer :

  • A polynomial is an expression formed by a finite sum of terms \(ax^n\), where \(a\) is a constant and \(n \geq 0\) is an integer.
  • Evaluating the provided options, \(12\), \(\frac{1}{3}x^4\), and \(5x^2 - 2x\) all satisfy this criteria.
  • The expression \(\frac{3x^3 + 4}{x}\) simplifies to \(3x^2 + 4x^{-1}\), which contains a negative exponent, violating the polynomial definition.
  • The non-polynomial expression is \(\boxed{\frac{3x^3 + 4}{x}}\).

Explanation

  1. Defining Polynomials

To determine which expression is not a polynomial, we must first define what a polynomial is. A polynomial is an algebraic expression consisting of variables and coefficients that involve only the operations of addition, subtraction, multiplication, and non-negative integer exponents of the variables. Essentially, any polynomial can be written as a finite sum of terms in the form \(a \cdot x^n\), where \(a\) is a constant and \(n\) is a non-negative integer (i.e., \(n \in \{0, 1, 2, 3, \dots\}\)).

  1. Evaluating Each Expression

We examine each option individually to see if it fits the definition of a polynomial:

  1. \(12\): This can be written as \(12x^0\). Since \(0\) is a non-negative integer, this is a constant polynomial of degree \(0\).
  2. \(\frac{1}{3} x^4\): This is a term with a coefficient \(\frac{1}{3}\) and a variable \(x\) raised to the power of \(4\). Since \(4\) is a non-negative integer, this is a valid monomial (a polynomial with one term).
  3. \(5x^2 - 2x\): This is a sum/difference of two terms, \(5x^2\) and \(2x^1\). Both exponents (\(2\) and \(1\)) are non-negative integers, making this a valid polynomial.
  4. \(\frac{3x^3 + 4}{x}\): This expression involves division by the variable \(x\). We can simplify this by distributing the division: \(\frac{3x^3}{x} + \frac{4}{x} = 3x^2 + 4x^{-1}\). The second term, \(4x^{-1}\), has an exponent of \(-1\). Because \(-1\) is not a non-negative integer, this expression does not meet the criteria of a polynomial.
  1. Identifying the Non-Polynomial Expression

Based on our evaluation, the expression \(\frac{3x^3 + 4}{x}\) cannot be expressed as a sum of terms with non-negative integer exponents because it contains a term with a negative exponent (\(x^{-1}\)). Therefore, it is the only expression among the choices that is not a polynomial. The final answer is \(\boxed{\frac{3x^3 + 4}{x}}\).

Examples

In real-world applications like computer science or financial modeling, polynomials are often used to estimate complex curves or simulate growth patterns. For instance, if you are modeling the cost of production where fixed costs are constants and variable costs scale with the number of units produced (like \(C(x) = 5x^2 + 100\)), you are using polynomials. However, if the expression requires division by the input (like in the case of average cost per unit, \(\frac{C(x)}{x}\)), the resulting function might not be a polynomial, which changes the mathematical tools we use to analyze it, such as calculus for optimization.

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