In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=(x^2+7)/x^3

Verified step by step guidance

1. Step 1: Recall the definition of a polynomial function. A polynomial function is an expression of the form \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where the coefficients \( a_i \) are real numbers and \( n \) is a non-negative integer.
2. Step 2: Examine the given function \( f(x) = \frac{x^2 + 7}{x^3} \). Notice that it is presented as a fraction.
3. Step 3: Simplify the expression by dividing each term in the numerator by the term in the denominator: \( f(x) = \frac{x^2}{x^3} + \frac{7}{x^3} = x^{-1} + 7x^{-3} \).
4. Step 4: Observe the exponents in the simplified expression. A polynomial function cannot have negative exponents.
5. Step 5: Conclude that \( f(x) = x^{-1} + 7x^{-3} \) is not a polynomial function because it contains negative exponents.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression that consists of variables raised to non-negative integer powers and coefficients. The general form is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n, a_(n-1), ..., a_0 are constants, and n is a non-negative integer. Polynomial functions do not include variables in the denominator or under radical signs.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial f(x) = 3x^4 + 2x^2 + 1, the degree is 4. The degree provides important information about the behavior of the polynomial function, including the number of roots and the end behavior of the graph.

Rational Functions

A rational function is a ratio of two polynomial functions, expressed as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. In the given function f(x) = (x^2 + 7)/x^3, the presence of x in the denominator indicates that it is a rational function, not a polynomial function. Understanding the distinction between polynomial and rational functions is crucial for correctly identifying their properties.