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

Verified step by step guidance

1. Recall that a polynomial function is a function that can be written in the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where each exponent is a non-negative integer and the coefficients \(a_i\) are real numbers.
2. Look at the given function: \(f(x) = \frac{x^2 + 7}{3}\). Notice that the numerator is a polynomial expression \(x^2 + 7\) and the denominator is a constant (3).
3. Since dividing a polynomial by a nonzero constant still results in a polynomial function, rewrite the function as \(f(x) = \frac{1}{3} x^2 + \frac{7}{3}\) to see it clearly in polynomial form.
4. Identify the degree of the polynomial by looking at the highest power of \(x\) in the expression. Here, the highest power is 2 from the term \(\frac{1}{3} x^2\).
5. Conclude that \(f(x)\) is a polynomial function of degree 2.

Key Concepts

Polynomial Functions

A polynomial function is a function that can be expressed as a sum of terms consisting of variables raised to non-negative integer powers, multiplied by coefficients. It has the general form f(x) = a_n x^n + ... + a_1 x + a_0, where n is a whole number. Recognizing polynomial functions involves checking for variables with whole number exponents and no variables in denominators or under roots.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial with a non-zero coefficient. It indicates the polynomial's order and affects its graph's shape and behavior. For example, in f(x) = 4x^3 + 2x^2, the degree is 3 because the highest exponent is 3.

Simplifying Functions to Identify Polynomials

To determine if a function is a polynomial, simplify the expression fully. For example, dividing a polynomial by a constant (like 3) does not change its polynomial nature. However, variables in denominators or fractional exponents disqualify it from being a polynomial. Simplification helps clarify the function's form.