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

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression that involves variables raised to non-negative integer 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 are constants and n is a non-negative integer. Functions that include variables in the denominator or have negative exponents are not considered polynomial functions.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial 4x^5 + 3x^2 - 2, the degree is 5. The degree provides important information about the behavior of the polynomial function, including the number of roots and the end behavior of the graph.

Identifying Non-Polynomial Terms

To determine if a function is a polynomial, it is essential to identify any non-polynomial terms. For instance, terms that involve division by a variable (like 1/x) or negative exponents (like x^-1) disqualify the function from being a polynomial. Recognizing these terms is crucial for accurately classifying functions and determining their properties.