Question

which of the options is a polynomial?
$f(x)=x^2 - 3x + \frac{2}{x^3} - 1$
$f(x)=13x^3 - 4x^5 + x^2 + 12x^{-1} + 6$
$f(x)=\frac{x + 2}{x - 3}$
$f(x)=3x^2 - 2x^4 - 2 + x$

Explanation:

To determine which function is a polynomial, we recall the definition of a polynomial: a polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non - negative integer exponents of variables.

Step 1: Analyze the green - colored function

The function is \(f(x)=x^{2}-3x + \frac{2}{x^{3}}-1\). We can rewrite \(\frac{2}{x^{3}}\) as \(2x^{-3}\). Since the exponent of \(x\) here is \(- 3\), which is a negative integer, this function is not a polynomial.

Step 2: Analyze the purple - colored function

The function is \(f(x)=13x^{3}-4x^{5}+x^{2}+12x^{-1}+6\). The term \(12x^{-1}\) can be written as \(\frac{12}{x}\), and the exponent of \(x\) is \(-1\) (a negative integer). So, this function is not a polynomial.

Step 3: Analyze the red - colored function

The function is \(f(x)=\frac{x + 2}{x-3}\). This is a rational function (a ratio of two polynomials), not a polynomial itself because it is in the form of a fraction of two polynomials.

Step 4: Analyze the teal - colored function

The function is \(f(x)=3x^{2}-2x^{4}-2 + x\). We can rewrite it in standard form as \(f(x)=-2x^{4}+3x^{2}+x - 2\). All the exponents of \(x\) (\(4\), \(2\), and \(1\)) are non - negative integers, and the operations involved are addition, subtraction, and multiplication of variables and coefficients. So, this function is a polynomial.

Answer:

The teal - colored function \(f(x)=3x^{2}-2x^{4}-2 + x\) (or \(f(x)=-2x^{4}+3x^{2}+x - 2\)) is a polynomial.