Determine which functions are polynomial functions. For those that are, identify the degree. $f\left(x\right)=(x^2+7)/x^3$

Verified step by step guidance

1. Recall that a polynomial function is a function of the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where the exponents \(n, n-1, \ldots, 1, 0\) are whole numbers (non-negative integers) and the coefficients \(a_i\) are real numbers.
2. Look at the given function: \(f(x) = \frac{x^2 + 7}{x^3}\). To analyze it, rewrite it by dividing each term in the numerator by \(x^3\) separately.
3. Rewrite the function as \(f(x) = \frac{x^2}{x^3} + \frac{7}{x^3} = x^{2-3} + 7x^{-3} = x^{-1} + 7x^{-3}\).
4. Notice that the exponents in the rewritten function are \(-1\) and \(-3\), which are negative integers. Since polynomial functions require non-negative integer exponents, this function is not a polynomial.
5. Therefore, \(f(x) = \frac{x^2 + 7}{x^3}\) is not a polynomial function, and so it does not have a degree.

Key Concepts

Definition of Polynomial Functions

A polynomial function is an expression consisting of variables raised to non-negative integer exponents, combined using addition, subtraction, and multiplication. It cannot include variables in denominators, negative exponents, or fractional powers. Understanding this helps determine if a given function qualifies as a polynomial.

Simplifying Rational Expressions

Simplifying expressions like (x^2 + 7) / x^3 involves dividing each term in the numerator by the denominator separately. This process can reveal if the function contains negative exponents, which disqualify it from being a polynomial. Simplification is key to correctly classifying the function.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable with a nonzero coefficient. After confirming a function is polynomial, identifying the degree involves finding the term with the largest exponent. This helps in understanding the function's behavior and graph.