Determine which functions are polynomial functions. For those that are, identify the degree. $g\left(x\right)=7x^5-\pi x^3+\frac{1}{5}x$

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 each exponent is a non-negative integer and the coefficients \(a_i\) are real numbers.
2. Examine the given function: \(g(x) = 7x^5 - \pi x^3 + \frac{1}{5} x\).
3. Check each term to ensure the exponents on \(x\) are whole numbers (non-negative integers): the exponents are 5, 3, and 1, which are all non-negative integers.
4. Confirm that the coefficients (7, \(-\pi\), and \(\frac{1}{5}\)) are real numbers, which they are.
5. Since all terms meet the criteria, \(g(x)\) is a polynomial function. The degree of the polynomial is the highest exponent of \(x\), which is 5.

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication by constants. Each term has the form ax^n, where n is a whole number and a is a constant. Functions with variables in denominators, negative or fractional exponents, or variables inside other functions are not polynomials.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression with a nonzero coefficient. It indicates the polynomial's overall behavior and complexity. For example, in g(x) = 7x^5 − πx^3 + (1/5)x, the degree is 5, since the highest exponent of x is 5.

Identifying Terms in Polynomial Functions

Each term in a polynomial must have a variable raised to a whole number exponent and a constant coefficient. Constants alone are also considered terms with degree zero. When analyzing a function, check each term to ensure it fits this pattern to confirm the function is polynomial.