Question

progress: the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your which expression is not a polynomial? \\( x^2 - 2x \\) \\( 12x \\) \\( \frac{1}{3}x^3 \\) \\( \frac{x^3 + 1}{x} \\)

Explanation:

Step1: Recall Polynomial Definition

A polynomial is an expression of the form \(a_nx^n + a_{n - 1}x^{n - 1}+\dots+a_1x + a_0\), where \(n\) is a non - negative integer and \(a_i\) are constants. Also, polynomials do not have variables in the denominator (after simplification, if we start with a rational expression).

Step2: Analyze Each Option

• Option 1: \(x^{2}-2x\) is in the form of a polynomial with \(n = 2\) (for the \(x^{2}\) term) and \(n=1\) (for the \(- 2x\) term), and the coefficients are real numbers. So it is a polynomial.
• Option 2: \(12x\) is a polynomial with \(n = 1\) and coefficient \(12\). So it is a polynomial.
• Option 3: \(\frac{1}{3}x^{3}\) is a polynomial with \(n = 3\) and coefficient \(\frac{1}{3}\). So it is a polynomial.
• Option 4: Simplify \(\frac{x^{3}+1}{x}\). Using the rule of exponents, \(\frac{x^{3}}{x}+\frac{1}{x}=x^{2}+\frac{1}{x}=x^{2}+x^{-1}\). In a polynomial, the exponents of the variable must be non - negative integers. Here, the term \(x^{-1}\) has a negative exponent (\(-1\)), so \(\frac{x^{3}+1}{x}\) is not a polynomial.

Answer:

\(\boldsymbol{\frac{x^{3}+1}{x}}\) (the last option)