In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. g(x)=6x^7+πx^5+2/3 x

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression that consists of variables raised to non-negative integer powers and coefficients that are real numbers. The general form of a polynomial function is f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where n is a non-negative integer and a_n is not zero. Understanding the structure of polynomial functions is essential for identifying them and determining their properties.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It provides important information about the polynomial's behavior, such as the number of roots and the end behavior of the graph. For example, in the polynomial g(x) = 6x^7 + πx^5 + (2/3)x, the degree is 7, which indicates that the graph will have up to 7 roots and will behave in a certain way as x approaches positive or negative infinity.

Identifying Polynomial Functions

To determine if a function is a polynomial, one must check that all terms in the function are of the form ax^n, where a is a real number and n is a non-negative integer. Functions that include variables in the denominator, negative exponents, or fractional powers do not qualify as polynomial functions. In the given function g(x), all terms meet the criteria, confirming it as a polynomial function.