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

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression that involves a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where n is a non-negative integer and a_n is not zero. Polynomial functions are characterized by their smooth curves and can be represented graphically without breaks or holes.

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) = 4x^3 + 2x^2 - x + 5, the degree is 3 because the highest exponent of x is 3. The degree provides important information about the polynomial's behavior, including the number of roots and the end behavior of its graph.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomial functions. It takes the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. In the given function f(x) = (x^2 + 7)/3, the numerator is a polynomial, but the denominator is a constant, which means the overall function is not a polynomial but rather a rational function. Understanding the distinction between polynomial and rational functions is crucial for correctly identifying their properties.