In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. g(x)=7x^5−πx^3+1/5 x

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1. Identify the general form of a polynomial function, which is a sum of terms of the form \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where \(a_i\) are constants and \(n\) is a non-negative integer.
2. Examine each term in the function \(g(x) = 7x^5 - \pi x^3 + \frac{1}{5} x\) to determine if it fits the polynomial form.
3. Check the first term \$7x^5\(: It is of the form \)a_n x^n\( with \)a_n = 7\( and \)n = 5$, which is a valid polynomial term.
4. Check the second term \(-\pi x^3\): It is of the form \(a_n x^n\) with \(a_n = -\pi\) and \(n = 3\), which is a valid polynomial term.
5. Check the third term \(\frac{1}{5} x\): It is of the form \(a_n x^n\) with \(a_n = \frac{1}{5}\) and \(n = 1\), which is a valid polynomial term. Therefore, \(g(x)\) is a polynomial function with the highest degree term being \$7x^5$, so the degree of the polynomial is 5.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression that involves variables raised to whole number powers, combined using addition, subtraction, and multiplication. The general form is f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_0 are constants and n is a non-negative integer. Understanding this definition is crucial for identifying polynomial functions.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It indicates the polynomial's behavior and the number of roots it can have. For example, in the polynomial g(x) = 7x^5 − πx^3 + 1/5 x, the degree is 5, as the term with the highest exponent is 7x^5.

Identifying Non-Polynomial Terms

To determine if a function is a polynomial, one must identify any non-polynomial terms. These include variables raised to negative or fractional powers, or terms involving roots or logarithms. In the given function g(x), the term 1/5 x is acceptable as it can be rewritten as (1/5)x^1, but any term that does not fit the polynomial criteria disqualifies the entire function.