In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=(x^2+7)/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 \( a_n, a_{n-1}, \ldots, a_0 \) are constants and \( n \) is a non-negative integer.
2. Step 2: Examine the given function \( f(x) = \frac{x^2 + 7}{3} \).
3. Step 3: Simplify the function by dividing each term in the numerator by 3: \( f(x) = \frac{1}{3}x^2 + \frac{7}{3} \).
4. Step 4: Check if the simplified expression fits the form of a polynomial function. The expression \( \frac{1}{3}x^2 + \frac{7}{3} \) is a polynomial because it is a sum of terms with non-negative integer exponents.
5. Step 5: Identify the degree of the polynomial. The degree is the highest power of \( x \) in the expression, which is 2 in this case.

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.