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

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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.

Examine the given function: $g(x) = 7x^5 - \pi x^3 + \frac{1}{5} x$.

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.

Confirm that the coefficients (7, $-\pi$, and $\frac{1}{5}$) are real numbers, which they are.

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.

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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.

Determine which functions are polynomial functions. For those that are, identify the degree. g(x)=7x5−πx3+15xg(x)=7x^5-\(\pi\) x^3+\(\frac\)15x